papers/2407.05232v1.json

{
    "title": "PAPM: A Physics-aware Proxy Model for Process Systems",
    "abstract": "In the context of proxy modeling for process systems, traditional data-driven deep learning approaches frequently encounter significant challenges, such as substantial training costs induced by large amounts of data, and limited generalization capabilities.\nAs a promising alternative, physics-aware models incorporate partial physics knowledge to ameliorate these challenges.\nAlthough demonstrating efficacy, they fall short in terms of exploration depth and universality.\nTo address these shortcomings, we introduce a physics-aware proxy model (PAPM) that fully incorporates partial prior physics of process systems, which includes multiple input conditions and the general form of conservation relations, resulting in better out-of-sample generalization.\nAdditionally, PAPM contains a holistic temporal-spatial stepping module for flexible adaptation across various process systems.\nThrough systematic comparisons with state-of-the-art pure data-driven and physics-aware models across five two-dimensional benchmarks in nine generalization tasks, PAPM notably achieves an average performance improvement of 6.7%, while requiring fewer FLOPs, and just 1% of the parameters compared to the prior leading method.\nCode is available at https://github.com/pengwei07/PAPM.",
    "sections": [
        {
            "section_id": "1",
            "parent_section_id": null,
            "section_name": "Introduction",
            "text": "From molecular dynamics to turbulent flows, process systems are essential in numerous scientific and engineering domains (Cameron & Hangos, 2001  ###reference_b4###). Computational modeling and simulation are crucial for understanding their complex temporal-spatial dynamics, enabling precise predictions and informed decisions across various fields.\nHowever, these valuable insights are provided by traditional numerical simulations, which are often computationally intensive, especially in scenarios necessitating frequent model queries like reverse engineering forward simulation (Dijkstra & Luijten, 2021  ###reference_b7###) and optimization design (Gramacy, 2020  ###reference_b12###).\nRecent advancements in data-driven methods have paved the way to tackle computational challenges more effectively (Lu et al., 2019  ###reference_b29###; Li et al., 2020  ###reference_b24###; Kochkov et al., 2021  ###reference_b21###; Stachenfeld et al., 2021  ###reference_b42###; Hao et al., 2023b  ###reference_b16###).\nAs shown in Fig. 1  ###reference_### (left), these methods input multiple conditions of process systems to output time-dependent solutions, serving as the proxy model for process systems.\nThrough adopting a supervised learning-from-data paradigm, remarkable advancements in calculation speeds have been achieved—several orders of magnitude faster than traditional numerical simulations. This has led to significant savings in computational costs.\nWhile data-driven methods are powerful, they face two primary limitations:\n1) a dependence on extensive labeled datasets, which contrasts sharply with the high computational costs of numerical simulations,\nand 2) a presumption of train-test uniformity that leads to poor generalization, especially in out-of-sample scenarios like time extrapolation.\nThis poor generalization arises from an overemphasis on the inductive biases of network architectures based on labels, rather than a strict adherence to fundamental physical laws (Li et al., 2021  ###reference_b25###; Brandstetter et al., 2022  ###reference_b3###).\nTo ameliorate high training costs and limited generalizability, a more promising strategy, known as physics-informed deep learning (PIDL), involves prior knowledge, such as fundamental physical laws, into neural networks (NNs). This integration enhances the sample efficiency and generalizability of NNs, proving particularly vital in scenarios with limited labeled data (Li et al., 2021  ###reference_b25###; Hao et al., 2022  ###reference_b14###; Meng et al., 2022  ###reference_b32###; Cuomo et al., 2022  ###reference_b6###).\nOne of the typical methods considers complete physical laws as the loss function of NNs to construct proxy models, such as PINNs (Raissi et al., 2019  ###reference_b35###) for specific conditions, PI-DeepONet (Wang et al., 2021  ###reference_b46###), and PINO (Li et al., 2021  ###reference_b25###) for multiple sets of conditions.\nHowever, the real-world applicability of these methods is limited by an incomplete understanding of the underlying physics of specific process systems, making it challenging to derive complete physics laws.\n###figure_1### Another common approach, termed as “physics-aware” models, offers a promising solution for this scenario by incorporating partial prior knowledge alongside a small amount of labeled data. Considering the relationship between equations and their numerical schemes, the physics-aware methods convert partial prior physics laws into the corresponding numerical schemes, and embeddings them into the network structure (Long et al., 2018  ###reference_b27###; Seo et al., 2020  ###reference_b40###; Huang et al., 2023b  ###reference_b18###; Akhare et al., 2023  ###reference_b1###; Rao et al., 2023  ###reference_b36###; Huang et al., 2023a  ###reference_b17###; Kochkov et al., 2023  ###reference_b22###; Pestourie et al., 2023  ###reference_b33###; Liu et al., 2024  ###reference_b26###).\nWith these partial prior physics laws, physics-aware models only need a small number of labels to obtain excellent out-of-sample generalization performance.\nHowever, these methods focus primarily on spatial derivatives, and often neglect integral aspects such as conservation laws and constitutive relations.\nConsequently, they do not fully leverage prior physics knowledge, leading to unreliable solutions.\nBesides, these methods are generally tailored for a specific process system with limited universality.\nRecognizing that process system modeling often requires the incorporation of conservation relations grounded in diffusion, convection, and source flows, this work aims to integrate this general physics law into the network architecture. By reinforcing the inductive biases in this manner, we can achieve better out-of-sample generalization.\nAdditionally, as different process systems correspond to specific conservation or constitutive equations based on inherent system characteristics (Cameron & Hangos, 2001  ###reference_b4###; Takamoto et al., 2022  ###reference_b44###; Hao et al., 2023a  ###reference_b15###), it is beneficial to identify both similarities and differences among process systems. Such an approach can offer a general temporal-spatial stepping module to combine various process systems flexibly.\nAs illustrated in Fig. 1  ###reference_###, we propose a physics-aware proxy model (PAPM) for process systems, which incorporates multiple conditions to output solutions at arbitrary time points.\nPAPM fully utilizes partial prior knowledge, including multiple conditions, and the general form of conservation relations, alongside a small amount of label data, to model the dynamics of systems through the proposed temporal-spatial stepping module .\nNotably, PAPM leverages the direction of data flow based on this general form, a distinction often overlooked by other physics-aware methods.\nFurthermore, PAPM focuses on out-of-sample scenarios, such as time extrapolation, aligning with the capabilities of alternative methods.\nThe core contributions of this work are:\nThe proposal of PAPM, a versatile physics-aware architecture design that fully incorporates partial prior knowledge such as multiple conditions, and the general form of conservation relations. This design proves to be superior in both training efficiency and out-of-sample generalizability.\nThe introduction of a holistic temporal-spatial stepping module (TSSM) for flexible adaptation across various process systems. It aligns with the distinct equation characteristics of different process systems by employing stepping schemes via temporal and spatial operations, whether in physical or spectral space.\nA systematic evaluation of state-of-the-art pure data-driven models alongside physics-aware models, spanning five two-dimensional non-trivial benchmarks. Notably, PAPM achieved an average absolute performance boost of 6.7% with fewer FLOPs and only 1%-10% of the parameters compared to alternative methods."
        },
        {
            "section_id": "2",
            "parent_section_id": null,
            "section_name": "Related Work",
            "text": "Pure Data-driven Method.\nThere are various neural network designs, where CNNs (Yu et al., 2017  ###reference_b47###; Bhatnagar et al., 2019  ###reference_b2###; Stachenfeld et al., 2021  ###reference_b42###) and GNNs (Sanchez-Gonzalez et al., 2020  ###reference_b39###; Li & Farimani, 2022  ###reference_b23###) target spatial dynamics within mesh grids, while RNN (Kochkov et al., 2021  ###reference_b21###) and LSTM (Shi et al., 2015  ###reference_b41###; Zhang et al., 2020  ###reference_b48###) focus on temporal progression.\nAnother line is the neural operator, excelling in mapping between temporal-spatial functional spaces, demonstrating success across various PDEs. Fourier neural operator (FNO) (Li et al., 2020  ###reference_b24###) learns the operator by harnessing the spectral domain alongside the Fast Fourier Transform. DeepONet (Lu et al., 2019  ###reference_b29###) approximates various nonlinear operators by leveraging branch and trunk networks for input functions and query points. Building upon this, MIONet (Jin et al., 2022  ###reference_b19###) addresses the challenges of multiple input functions within the DeepONet framework. Moreover, U-FNets (Gupta & Brandstetter, 2022  ###reference_b13###) and convolutional neural operators (CNO) (Raonić et al., 2023  ###reference_b37###) are modified U-Net (Ronneberger et al., 2015  ###reference_b38###) variants, where the former replace U-Net s layers by FNO s Fourier blocks, and the latter replace them by predefined convolutional block.\nPhysics-aware Method.\nContrary to the method of integrating complete physics knowledge into its loss function, the physics-aware method only leverages labels while explicitly incorporating either entire or partial mechanistic knowledge into the network architecture.\nInspired by the finite volume method, FINN (Karlbauer et al., 2022  ###reference_b20###) innovatively employs flux and state kernels for modeling components of advection-diffusion equations in each volume, which is one class of process systems, and has an explicit form. Since FINN is conducted for each volume, FINN is computationally inefficient in the whole spatial region. PiNDiff (Akhare et al., 2023  ###reference_b1###), PeRCNN (Rao et al., 2023  ###reference_b36###), and PPNN (Liu et al., 2024  ###reference_b26###) are inspired by the finite difference method.\nPiNDiff (Akhare et al., 2023  ###reference_b1###) and NeuralGCM (Kochkov et al., 2023  ###reference_b22###) integrate partial physics knowledge into the NN block for forecasting the systems  future states, ensuring mathematical integrity via differentiable programming.\nPeRCNN (Rao et al., 2023  ###reference_b36###) employs convolutional operations to approximate unknown nonlinear terms for reaction-diffusion systems while incorporating known terms through difference schemes.\nPPNN (Liu et al., 2024  ###reference_b26###) bakes prior-knowledge terms from low-resolution data, estimates unknown parts with the trainable network, and uses the Euler time-stepping difference scheme to form a regression model for updating states.\nLearned correction methods.\nAs highlighted in (Rackauckas et al., 2020  ###reference_b34###; Um et al., 2020  ###reference_b45###; Dresdner et al., 2022  ###reference_b8###; Sun et al., 2023  ###reference_b43###; McGreivy & Hakim, 2023  ###reference_b31###), the core idea of learned correction methods is to approximate the known part with specific fixed modules (such as numerical methods) and approximate the unknown part with a neural network, which often yields superior results compared to fully learned approaches. However, the current learned correction methods typically rely on known equations and are optimized for specific conditions, which is somewhat different from our model s broader objective of generalizing across various conditions and conservation relations. The second difference between our work and the mentioned literature lies in the precision of the known parts. Specifically, PAPM only knows that the different terms in the equation adhere to the general form of conservation relations, without exact knowledge of each term s specific composition. In contrast, the literature deals with cases where the known parts are precise."
        },
        {
            "section_id": "3",
            "parent_section_id": null,
            "section_name": "Preliminaries",
            "text": "This section presents the foundational description of process systems, known as the process model. Additionally, further clarification is provided on the specific problem in this work.\nProcess Model.\nPivotal in engineering disciplines, process models represent and predict the dynamics of diverse process systems. This model s mathematical foundation relies on two essential sets of equations: conservation equations, governing the dynamic behavior of fundamental quantities, and constitutive equations, which describe the interactions among different variables. Further details are provided in Appendix B  ###reference_###.\nEq. 1  ###reference_### and Eq. 2  ###reference_### represent the universal conservation and constitutive equations, respectively.\nwhere  is the physical quantity, denoting the system s state.\nEq. 1  ###reference_### comprises four essential elements: the diffusion flows , convection flows , the internal source , and the external source . In Eq. 2  ###reference_###,  denotes the velocity of the physical quantity being transmitted,  is the diffusion coefficient,  denotes the coefficients, and  is the input of the external source term. Here, the corresponding linear or nonlinear mapping is the , , and .\nThe structure of PAPM is depicted in Fig. 2  ###reference_### at time  .\nOur goal is to use partial prior knowledge (the general form of Eq. 1  ###reference_###) and a small amount of label data to establish a proxy model, which takes these various input conditions and outputs system time-dependent solutions (), as shown in Fig 1  ###reference_###. Notably, the general form of Eq. 1  ###reference_### is only known, which is also the data flow, while the specific item is unknown, such as the mappings of , , and , aligning with most real-world scenarios (Karlbauer et al., 2022  ###reference_b20###; Huang et al., 2023a  ###reference_b17###; Liu et al., 2024  ###reference_b26###).\n###figure_2### Problem Formulation.\nUnder different initial and boundary conditions, external sources, and coefficients, the following -step trajectory should be predicted.\nMoreover, due to the high cost of generating labeled data, we focused on out-of-sample scenarios, e.g. time extrapolation, where the training dataset only contains the subsequent -step trajectory, and .\nFormally, the dataset , where .\nHere  contains a set of inputs, that is, initial condition , boundary conditions, such as Robin conditions, external sources , and coefficients .\n is the following trajectory, and the mapping  is our goal to learn.\nEach  is a vector, which consists of  physical quantities, such as velocity, vorticity, pressure, and temperature.\nWe discretize each quantity  on the grid . In a nutshell, for modeling this operator , we use a parameterized neural network  with parameters , inputs , and outputs , where .\nOur goal is to minimize the  relative error loss between the prediction  and real data  in the training dataset as,\nwhere  is the training time-step size, and  is the size of the training dataset.  is the mean  relative error loss at time  of index .  is a set of the network parameters and  is the parameter space."
        },
        {
            "section_id": "4",
            "parent_section_id": null,
            "section_name": "Methodology",
            "text": "This section presents PAPM s architecture specifically tailored to conservation and constitutive relations. Then, a holistic temporal-spatial stepping module (TSSM) is proposed, adapting to the unique equation characteristics of various process systems.\nThe Appendix D.1  ###reference_### provides the pseudo-code for the entire training process, offering a comprehensive understanding of our approach."
        },
        {
            "section_id": "4.1",
            "parent_section_id": "4",
            "section_name": "PAPM Overview",
            "text": "Aligning with the general form of Eq. 1  ###reference_### and Eq. 2  ###reference_###, there are four elements corresponding to Diffusive Flows (DF), Convective Flows (CF), Internal Source Term (IST), and External Source Term (EST) in PAPM s structure diagram, as illustrated in Fig. 2  ###reference_###.\nThe versatile general structure of PAPM could enable it to work effectively across different process systems.\nThe input contains a set of inputs, that is, coefficient , initial state , external source input , and boundary conditions, which are multiple conditions of process systems.\nThe sequence of embedding this prior knowledge unfolds as follows:\n###table_1### 1) Embedding BCs. Using the given boundary conditions, the physical quantity  is updated, yielding . A padding strategy is employed to integrate four different boundary conditions in four different directions into PAPM. Further details are provided in Appendix C  ###reference_###.\n2) Diffusive Flows (DF). Using  and coefficients , we represent the directionless diffusive flow. The diffusion flow and its gradient are obtained as  and  via a symmetric gradient operator, respectively.\n3) Convective Flows (CF). The pattern  is derived from . Once  is determined, its sign indicates the direction of the flows, enabling computation of  and  through a directional gradient operator.\n4) Internal Source Term (IST) & External Source Term (EST). Generally, IST and EST present a complex interplay between physical quantities  and external inputs . Often, this part in real systems doesn t have a clear physics-based relation, prompting the use of NNs to capture this intricate relationship.\n5) ODE solver. From DF, CF, IST, and EST, the dynamic  are derived.\nBy doing so, the Eq. 1  ###reference_### can be reduced to an ordinary differential equation (ODE), and the ODE solver is used to approximate the evolving state as .\nFig. 2  ###reference_### above illustrates the data flow at time . Then, during the training or inference phase, PAPM performs autoregressive predictions as , where , with  during training and  during inference.\nIn short, PAPM takes different conditions, including initial conditions, boundary conditions, external sources, and coefficients, and interactively propagates the dynamics of process systems forward using five distinct components. The purpose of such a structured design is to reinforce the inductive biases concerning strict physical laws."
        },
        {
            "section_id": "4.2",
            "parent_section_id": "4",
            "section_name": "Temporal-Spatial Stepping Module (TSSM)",
            "text": "Due to the diversity of process systems, we develop a holistic Temporal-Spatial Stepping Module (TSSM) to align with the unique characteristics of different process systems, which forms the specific network structure for each component in PAPM. As shown in Tab. 1  ###reference_###, TSSM is categorized into three types based on structures of process systems, where each type decomposes temporal and spatial components, i.e., structure-preserved localized operator, spectral operator, and hybrid operator. Notably, all three approaches can employ a common temporal operation through ODE solvers.\n###figure_3### Temporal Operation. After obtaining the dynamic state derivative, , the subsequent state  can be computed through numerical integration over different time spans. Due to the numerical instability associated with first-order explicit methods like the Euler method (Gottlieb et al., 2001  ###reference_b11###; Fatunla, 2014  ###reference_b9###), we adopt the neural ordinary differential equation approach (Neural ODE (Chen et al., 2018  ###reference_b5###)), which employs the Runge–Kutta time-stepping strategy to enhance stability. The computed state  is then recursively fed back into the network as the input for the subsequent time step, continuing this process until the final time step is reached.\nStructure-preserved Localized Operator.\nFor systems with explicit structures, such as the Burgers and RD equations, typified by expressions like , convolutional kernels in the physical space are employed to capture system dynamics.\nWe opt for either fixed or trainable kernels, illustrated in Fig. 3  ###reference_### (Left), depending on our understanding of the system.\nSpecifically, the fixed one is based on the predefined convolution kernel derived from difference schemes, and further details are provided in Appendix D.2.1  ###reference_.SSS1###.\nMoreover, the trainable version tailors its design to essential features of convection (upper or lower triangular) and diffusion (symmetric).\nOnce set, the localized operator is depicted in Fig. 3  ###reference_### (Mid). We could represent nonlinear terms in DF and CF using predefined convolution kernels alongside partially known physics (the general form of Eq. 1  ###reference_### and Eq. 2  ###reference_###).\nAny unknown specific terms, such as source, are then addressed through a shallow ResNet block.\nStructure-preserved Spectral Operator.\nFor systems with implicit structures, such as the Navier-Stokes Equation in vorticity form, represented like , we adopt a sequential process, as shown in Fig. 3  ###reference_###(Right).\nRecognizing the implicit linkage between velocity and vorticity,  is initially processed to extract the flow function, subsequently leading to the velocity derivation.\nThe spectral space dimensions ( and ) and spectral quantity (denoted as ), are obtained by leveraging the FFT.\nAssociating  and  with , differential operators like  are represented via E-Conv (e.g., element-wise product), and then mapped back to the physical space using IFFT. Doing so can represent the nonlinear terms in DF and CF via simple computations such as addition and multiplication in the spectral domain.\nMoreover, the spectral convolution (S-Conv) fo FNO (Li et al., 2020  ###reference_b24###) is introduced to learn unknown components. This process can be further detailed in Appendix D.2.2  ###reference_.SSS2###.\nStructure-preserved Hybrid Operator. For systems with a hybrid structure, such as the Navier-Stokes Equation in general form (e.g., ), given the implicit interrelation between pressure  and velocity , a combination of the method above is employed. Explicit constituents, such as  and , are addressed through the localized operator. Meanwhile, implicit relations are resolved similarly by the spectral operator. For unknown components, either of the two operators can be engaged. We generally favor the localized operator as it allows direct operations without requiring transitions between different spaces."
        },
        {
            "section_id": "5",
            "parent_section_id": null,
            "section_name": "Experiments",
            "text": ""
        },
        {
            "section_id": "5.1",
            "parent_section_id": "5",
            "section_name": "Experimental setup and evaluation protocol",
            "text": "Datasets. We employ five datasets spanning diverse domains, such as fluid dynamics and heat conduction, detailed in Appendix E  ###reference_###. These datasets are categorized based on the Temporal-Spatial Stepping Module used in PAPM to highlight distinct equation characteristics in various process systems.\nLocalized Category:\nBurgers2d (Huang et al., 2023a  ###reference_b17###) is a 2D benchmark PDE with periodic boundary conditions, various initial conditions, and viscosity, while the source remains unknown.\nRD2d (Takamoto et al., 2022  ###reference_b44###) addresses a 2D FitzHugh-Nagumo reaction-diffusion equation with no-flow Neumann boundary condition, diverse initial conditions, and unknown source terms.\nSpectral Category:\nNS2d (Li et al., 2020  ###reference_b24###) explores incompressible fluid dynamics in vorticity form with varied initial conditions and unknown sources.\nHybrid Category:\nLid2d focuses on incompressible lid-driven cavity flow, characterized by differing viscosity and BCs. NSM2d deals with incompressible fluid dynamics within an unknown magnetic field source, featuring time-varying BCs, various initial conditions, and viscosity.\nBaselines. We compared our approach with eight SOTA baselines for a comprehensive evaluation. ConvLSTM (Shi et al., 2015  ###reference_b41###) is a classical time series modeling technique that captures dynamics via CNN and LSTM.\nDil-ResNet (Stachenfeld et al., 2021  ###reference_b42###) adopts the encoder-process-decoder process with dilated-ConvResNet for dynamic data through an autoregressive stepping manner.\ntime-FNO2D (Li et al., 2020  ###reference_b24###) and MIONet (Jin et al., 2022  ###reference_b19###) are two typical neural operators in learning dynamics.\nU-FNet (Gupta & Brandstetter, 2022  ###reference_b13###) and CNO (Raonić et al., 2023  ###reference_b37###) are modified U-Net (Ronneberger et al., 2015  ###reference_b38###) variants.\nPeRCNN (Rao et al., 2023  ###reference_b36###) incorporates specific physical structures into a neural network, ideal for sparse data scenarios. PPNN (Liu et al., 2024  ###reference_b26###) is a novel autoregressive framework preserving known PDEs using multi-resolution convolutional blocks.\nMetrics.\nAccording to Eq. 3  ###reference_###, we adopt the mean  relative error, abbreviated as , as our evaluation metric for validation and testing datasets.  is formulated as follows:\nEvaluation Protocol.\nBased on the experimental setting of time extrapolation, we further conducted experiments in the following two parts: coefficient interpolation (referred to as C Int.) and coefficient extrapolation (referred to as C Ext.).\nMore information about the evaluation protocol, the hyper-parameters of baselines, and our methods can be further detailed in Appendix F  ###reference_###."
        },
        {
            "section_id": "5.2",
            "parent_section_id": "5",
            "section_name": "Main Results",
            "text": "Performance Comparisons. Tab. 3  ###reference_### and Tab. 3  ###reference_### present the primary experimental outcomes, the number of trainable parameters (), and computational cost (FLOPs) for each baseline across datasets.\nHere, Bold and Underline indicate the best and second best performance, respectively. Notably, lower results mean better performance because the metric is the mean  relative error. Our observations from the results are as follows:\nFirstly, PAPM exhibits the most balanced trade-off between performance, parameter count, and computational cost among all methods evaluated, from explicit structures (Burgers2d, RD2d) to implicit (NS2d) and more complex hybrid structures (Lid2d, NSM2d).\nRemarkably, even though PAPM requires significantly fewer FLOPs and only 1% of the parameters employed by the prior leading method, PPNN, it still outperforms it by a large margin. In a nutshell, our model enhances the performance by an average of 6.7% over nine tasks, affirming PAPM as a versatile and efficient framework suitable for diverse process systems.\nSecondly, PAPM s structured treatment of system inputs and states leads to a remarkable 9.6% performance boost in three coefficient-extrapolation tasks. This highlights its superior generalization capability in out-of-sample scenarios. Unlike models like PPNN, which directly use system-specific inputs, PAPM integrates coefficient data more intricately within conservation and constitutive relations, boosting its adaptability to varying coefficients.\nThirdly, data-driven methods are less effective than physics-aware methods like PPNN and our PAPM in time extrapolation tasks, where incorporating prior physical knowledge through structured network design enhances a model s generalization ability. Notably, PeRCNN uses  convolution to approximate nonlinear terms, but experimental results suggest limited performance. Further details are available in Appendix F.2  ###reference_###.\n###figure_4### ###figure_5### Visualization. Fig. 4  ###reference_### showcases the stepwise relative error of PAPM during the extrapolation process in the test dataset, using Burgers2d s C Int. as a representative example.\nCompared to the two best-performing baselines, our model (depicted by the red line) exhibits superior performance throughout the extrapolation process, with the least error accumulation. Turning our attention to the more challenging NSM2d dataset, Fig. 5  ###reference_### presents the results across five extrapolation time slices. While FNO demonstrates commendable accuracy within the training domain (), its performance falters significantly outside of it (). On the other hand, physics-aware methods (PPNN), and PAPM in particular, consistently capture the evolving patterns with a greater degree of robustness. Notably, our method emerges as a leader in terms of precision. Additional visual results can be found in Appendix G.1  ###reference_###.\n###figure_6###"
        },
        {
            "section_id": "5.3",
            "parent_section_id": "5",
            "section_name": "Efficiency",
            "text": "Training and Inference Cost: Dataset generation for our work is notably resource-intensive, with inference times ranging from  to  seconds for public datasets, and up to  seconds for those datasets we generated using COMSOL Multiphysics.\nIn stark contrast, both baselines and PAPM register inference times between  to  seconds (detailed in Appendix G.2  ###reference_###), achieving an improvement of 3 to 5 orders of magnitude. Notably, PAPM s time cost rivals or even surpasses baselines across different datasets. PAPM s efficiency remains competitive with other data-driven methods.\nData Efficiency. Owing to PAPM s structured design, data utilization is significantly enhanced. To evaluate data efficiency, we conducted tests using RD2d as a representative example, with Dil-ResNet and PPNN symbolizing pure data-driven and physics-aware methods. The results, displayed in Fig. 6  ###reference_###, depict PAPM s efficiency concerning data volume and label data step size in training.\n(1) Amount of Data: With a fixed 20% reserved for the test set, the remaining 80% of the total data is allocated to the training set. We systematically varied the training data volume, ranging from initially utilizing only 5% of the training set and progressively increasing it to the entire 100%.\nPAPM s relative error distinctly outperforms other baselines, especially with limited data (5%). As depicted in Fig. 6  ###reference_### (Left), PAPM s error consistently surpasses other methods, stabilizing below 2% as the training data volume increases.\n(2) Time Step Size: We varied the time step size from a tenth to half of the total duration, increasing it in increments of tenths. As shown in Fig. 6  ###reference_### (Right), PAPM demonstrates the capability for long-range time extrapolation with fewer dynamic steps. It consistently outperforms other methods, achieving superior results even with shorter training time step sizes."
        },
        {
            "section_id": "5.4",
            "parent_section_id": "5",
            "section_name": "Ablation Studies",
            "text": "Different blocks impacts.\nTab. 4  ###reference_### displays our selection of the Burger2d dataset for ablation studies, chosen for its representation of diffusion, convection, and source terms. We defined several configurations to assess the impact of individual components. The  relative error on the boundary (BC ) is introduced to highlight the importance of physics embedding further. no_DF excludes diffusion, whereas no_CF omits convection. In no_Phy, we retain only a structure with a residual connection, thereby eliminating both diffusion and convection. no_BCs setup removes the explicit embedding of boundary conditions, no_NODE replaces the Neural ODE with the Euler stepping scheme, and no_All adopts a purely data-driven approach. Additional ablation results can be found in Appendix G.3  ###reference_###.\nKey findings include the crucial roles of diffusion and convection in representing system dynamics, as evidenced in the no_DF, no_CF, and no_Phy configurations. Specifically, the no_DF configuration demonstrated the importance of integrating the viscosity coefficient with the diffusion term, with its absence leading to significant errors. The necessity of adhering to physical laws in boundary conditions was highlighted in the no_BCs, notably reducing errors on the boundary (BC ). Lastly, the no_NODE results indicate that different temporal stepping schemes significantly impact the outcomes, underscoring the effectiveness of neural ODEs in continuous-time modeling.\nDifferent blocks validations.\nTaking the Burgers2d and RD2d datasets as examples to demonstrate the fact that the convection/diffusion/source terms could actually learn those parts in the equations. We use high-fidelity FDM and FVM to compute the corresponding terms to obtain the detailed term for convection/diffusion/source terms.\nThe  relative error between ground truth and numerical results are  and  in all time steps for Burgers2d and RD2d, respectively. Thus, we can use the results obtained by the numerical methods as reference values to verify this.\nAs shown in Tab. 5  ###reference_###, the different terms of PAPM can be used to learn the equation s convection/diffusion/source parts. Additional visualization results can be found in Appendix G.1  ###reference_### (Fig. 9  ###reference_### and Fig. 10  ###reference_###)."
        },
        {
            "section_id": "6",
            "parent_section_id": null,
            "section_name": "Conclusion",
            "text": "To address the challenges of physics-aware models falling short regarding exploration depth and universality, we have proposed PAPM.\nIt fully incorporates partial prior physics of process systems, which includes multiple input conditions and the general form of conservation relations, resulting in better training efficiency and out-of-sample generalization.\nAdditionally, PAPM contains a holistic temporal-spatial stepping module for flexible adaptation across various process systems.\nThe efficacy of PAPM s structured design and holistic module was extensively validated across five datasets within distinct out-of-sample tasks.\nNotably, PAPM achieved an average performance boost of 6.7% with fewer FLOPs and only 1% of the parameters employed by the prior leading method.\nThrough such analysis, PAPM exhibits the most balanced trade-off between accuracy and computational efficiency among all evaluated methods, alongside impressive out-of-sample generalization capabilities."
        },
        {
            "section_id": "7",
            "parent_section_id": null,
            "section_name": "Limitation and Future Work",
            "text": "We aim to extend our model to more complex, industrially relevant systems, moving beyond 2D spatio-temporal dynamics to scenarios like 3D-industry-standard aerodynamics, plasma discharge, and multi-physics couplings (e.g., fluid-structure and thermal fluid-structure interactions).\nDespite our model s proven balance in accuracy and efficiency, we aim to challenge it further in these intricate environments. Additionally, we plan to adapt PAPM for irregular grid scenarios, typical in the industry, by integrating graph neural networks. This will enhance PAPM s versatility, allowing it to handle diverse data structures and complex dynamic processes such as convection, diffusion, and source interactions."
        }
    ],
    "appendix": [
        {
            "section_id": "Appendix 1",
            "parent_section_id": null,
            "section_name": "Appendix A Table of notations",
            "text": "A table of notations is given in Tab.6  ###reference_###."
        },
        {
            "section_id": "Appendix 2",
            "parent_section_id": null,
            "section_name": "Appendix B Process models",
            "text": "Pivotal in engineering disciplines, process models serve to represent and predict the dynamics of diverse process systems, from entire plants to single equipment pieces. These models primarily rely on the interplay between conservation and constitutive equations, ensuring an accurate depiction of system dynamics. Conservation equations dictate the model s dynamics using partial differential equations that govern primary physical quantities like mass, energy, and momentum. On the other hand, constitutive equations relate potentials to extensive variables via algebraic equations, such as flows, temperatures, pressures, concentrations, and enthalpies, enriching the model s comprehensiveness. Additionally, accounting for initial and boundary conditions ensures the model s reliability, making these four components interdependently integral to the model s solid mathematical framework.\nConservation equations. The general form in differential representation is:\nwhere , , and  represents the state of the model, which is the object of our modeling,  and , .  represents convective flows, and  represents diffusive or molecular flows.  describes the convective flow pattern into or out of the system volume.  represents the diffusion coefficient. , the internal source term, for example, the chemical reaction for component mass conservation, where species appear or are consumed due to reactions within the space of interest. Other internal source terms arise from energy dissipation, conversion, compressibility, or density changes. , the external source term, including gravitational, electrical, and magnetic fields as well as pressure fields.\nConstitutive equations. For the internal source term, it usually depends on the state of the model and can be expressed as . The external source term is usually related to the external effects imposed and can be expressed as , where  is a vector of parameters imposed externally, which may include voltages, pressures, etc. For convective flows, the velocity  may be determined by the state of the model, which can be expressed as .\nInitial conditions (IC). Every process or system evolves over time, but we need a reference or a starting point to predict or understand this evolution. The initial conditions provide this starting point. For example, in the context of a reactor, initial conditions might describe the concentration of various reactants at . Mathematically, IC can be represented as .\nBoundary conditions (BC). Initial conditions set the foundation at , while boundary conditions inform how a system evolves and interacts with its environment, for instance, by specifying heat flux at a heat exchanger s boundary or flow rate at a reactor s inlet. These boundary conditions can be categorized as Dirichlet, prescribing specific values like temperature on the boundary; Neumann, defining derivatives or fluxes such as the heat flux; and Robin, which combines aspects of both Dirichlet and Neumann, encompassing parameters like both heat transfer rates and surface temperatures. Regardless of the type, they re mathematically expressed as , where ."
        },
        {
            "section_id": "Appendix 3",
            "parent_section_id": null,
            "section_name": "Appendix C Embedding Boundary Conditions",
            "text": "This part covers the method of embedding four different boundary conditions (Dirichlet, Neumann, Robin, and Periodic) into neural networks via convolution padding. Let s consider a rectangular region in a  space, , which can be discretized into an  grid, , . Each grid point can be represented as , where  and . Hence, we can transform the continuous space into a discrete grid of points.\nBoundary Conditions on the -axis. The direction vector is , which means the boundary conditions are the same for each  value.\nDirichlet: If the boundary condition is given as , the discrete form would be , and we can use a padding method in the convolution kernel .\nNeumann: If the boundary condition is given as , the discrete form would be  and we can use a padding method in the convolution kernel .\nRobin: If the boundary condition is given as , the discrete form would be . We can use a padding method in the convolution kernel .\nPeriodic: If the boundary condition is given as , where  denotes the left boundary and  the right boundary, the discrete form would be . We can use a padding method in the convolution kernel .\nBoundary Conditions on the -axis. The direction vector is . The basic handling method is similar to the -direction case but with the grid spacing replaced with , and the boundary conditions applied to the upper and lower boundaries, i.e.,  and . The corresponding -direction expressions can be derived by replacing  with  in the -direction expressions and swapping  with .\nArbitrary Direction Boundary Conditions in the Rectangular Area. The direction vector . Both  and  directions need to be considered, resulting in the following expressions for each of the four boundary conditions:\nDirichlet:Given the condition , its discrete form remains . The corresponding padding method in the convolution kernel is .\nNeumann: For the boundary condition , the discrete form can be represented as . The corresponding padding method in the convolution kernel can be written as  and .\nRobin: Given the condition , the discrete form becomes . The corresponding padding method in the convolution kernel is  and .\nPeriodic: For the condition , the discrete form is  and . The corresponding padding method in the convolution kernel is  and .\nDirectionless Boundary Conditions in the Rectangular Area. The following strategies are employed for handling the Neumann and Robin boundary conditions:\nFor , the discrete form is  and . We can use a padding method in the convolution kernel where  and .\nFor , the discrete form is  and . We can use a padding method in the convolution kernel where  and ."
        },
        {
            "section_id": "Appendix 4",
            "parent_section_id": null,
            "section_name": "Appendix D Supplemental TSSM",
            "text": ""
        },
        {
            "section_id": "Appendix 5",
            "parent_section_id": null,
            "section_name": "Appendix E Datasets",
            "text": "As shown in Tab.9  ###reference_###, we employ five datasets spanning diverse domains, such as fluid dynamics and heat conduction. Based on the TSSM scheme employed by PAPM, we categorize the aforementioned five datasets into three types: Burgers2d and RD2d fall under the localized category, NS2d is classified as spectral, while Lid2d and NSM2d are designated as hybrid. The generations of Lid2d and NSM2d are detailed via COMSOL multiphysics in E.2  ###reference_###. We are particularly keen to make Lid2d and NSM2d publicly available, anticipating various research endeavors on these datasets by the community."
        },
        {
            "section_id": "Appendix 6",
            "parent_section_id": null,
            "section_name": "Appendix F Details for experimental setup and models’ hyper-parameters",
            "text": ""
        },
        {
            "section_id": "Appendix 7",
            "parent_section_id": null,
            "section_name": "Appendix G Additional experimental results",
            "text": "In this section, some additional experimental results are shown, which are data visualization (G.1  ###reference_###), training and inference time cost-specific details (G.2  ###reference_###), and detailed ablation studies (G.3  ###reference_###)."
        }
    ],
    "tables": {
        "1": {
            "table_html": "<figure class=\"ltx_table\" id=\"S4.T1\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 1: </span>Temporal-Spatial Stepping Module (TSSM).</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_align_middle\" id=\"S4.T1.3\">\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S4.T1.3.4.1\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" colspan=\"2\" id=\"S4.T1.3.4.1.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S4.T1.3.4.1.1.1\" style=\"font-size:90%;\">Category</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"S4.T1.3.4.1.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\">\n<table class=\"ltx_tabular ltx_align_middle\" id=\"S4.T1.3.4.1.2.1\">\n<tr class=\"ltx_tr\" id=\"S4.T1.3.4.1.2.1.1\">\n<td class=\"ltx_td ltx_nopad_r ltx_align_center\" id=\"S4.T1.3.4.1.2.1.1.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S4.T1.3.4.1.2.1.1.1.1\" style=\"font-size:90%;\">Localized</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.3.4.1.2.1.2\">\n<td class=\"ltx_td ltx_nopad_r ltx_align_center\" id=\"S4.T1.3.4.1.2.1.2.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\">\n<span class=\"ltx_text\" id=\"S4.T1.3.4.1.2.1.2.1.1\" style=\"font-size:90%;\">Fig.</span><a class=\"ltx_ref\" href=\"https://arxiv.org/html/2407.05232v1#S4.F3\" style=\"font-size:90%;\" title=\"Figure 3 ‣ 4.2 Temporal-Spatial Stepping Module (TSSM) ‣ 4 Methodology ‣ PAPM: A Physics-aware Proxy Model for Process Systems\"><span class=\"ltx_text ltx_ref_tag\">3</span></a><span class=\"ltx_text\" id=\"S4.T1.3.4.1.2.1.2.1.2\" style=\"font-size:90%;\"> (Left and Mid), Alg. </span><a class=\"ltx_ref\" href=\"https://arxiv.org/html/2407.05232v1#alg1\" style=\"font-size:90%;\" title=\"Algorithm 1 ‣ D.1 Algorithm Display ‣ Appendix D Supplemental TSSM ‣ PAPM: A Physics-aware Proxy Model for Process Systems\"><span class=\"ltx_text ltx_ref_tag\">1</span></a>\n</td>\n</tr>\n</table>\n</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"S4.T1.3.4.1.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\">\n<table class=\"ltx_tabular ltx_align_middle\" id=\"S4.T1.3.4.1.3.1\">\n<tr class=\"ltx_tr\" id=\"S4.T1.3.4.1.3.1.1\">\n<td class=\"ltx_td ltx_nopad_r ltx_align_center\" id=\"S4.T1.3.4.1.3.1.1.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S4.T1.3.4.1.3.1.1.1.1\" style=\"font-size:90%;\">Spectral</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.3.4.1.3.1.2\">\n<td class=\"ltx_td ltx_nopad_r ltx_align_center\" id=\"S4.T1.3.4.1.3.1.2.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\">\n<span class=\"ltx_text\" id=\"S4.T1.3.4.1.3.1.2.1.1\" style=\"font-size:90%;\">Fig.</span><a class=\"ltx_ref\" href=\"https://arxiv.org/html/2407.05232v1#S4.F3\" style=\"font-size:90%;\" title=\"Figure 3 ‣ 4.2 Temporal-Spatial Stepping Module (TSSM) ‣ 4 Methodology ‣ PAPM: A Physics-aware Proxy Model for Process Systems\"><span class=\"ltx_text ltx_ref_tag\">3</span></a><span class=\"ltx_text\" id=\"S4.T1.3.4.1.3.1.2.1.2\" style=\"font-size:90%;\"> (Right), Alg. </span><a class=\"ltx_ref\" href=\"https://arxiv.org/html/2407.05232v1#alg2\" style=\"font-size:90%;\" title=\"Algorithm 2 ‣ D.1 Algorithm Display ‣ Appendix D Supplemental TSSM ‣ PAPM: A Physics-aware Proxy Model for Process Systems\"><span class=\"ltx_text ltx_ref_tag\">2</span></a>\n</td>\n</tr>\n</table>\n</td>\n<td class=\"ltx_td ltx_align_center ltx_border_tt\" id=\"S4.T1.3.4.1.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S4.T1.3.4.1.4.1\" style=\"font-size:90%;\">Hybrid</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.3.5.2\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" colspan=\"2\" id=\"S4.T1.3.5.2.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.5.2.1.1\" style=\"font-size:90%;\">Characteristic</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.3.5.2.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.5.2.2.1\" style=\"font-size:90%;\">Explicit</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.3.5.2.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.5.2.3.1\" style=\"font-size:90%;\">Implicit</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S4.T1.3.5.2.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.5.2.4.1\" style=\"font-size:90%;\">Explicit+Implicit</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.3.3\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" colspan=\"2\" id=\"S4.T1.3.3.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.3.4.1\" style=\"font-size:90%;\">Example</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.1.1.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.2.2.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S4.T1.3.3.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.3.6.3\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.3.6.3.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.6.3.1.1\" style=\"font-size:90%;\">Temporal</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.3.6.3.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.6.3.2.1\" style=\"font-size:90%;\">ODE solver</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" colspan=\"3\" id=\"S4.T1.3.6.3.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.6.3.3.1\" style=\"font-size:90%;\">Neural ODE</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.3.7.4\">\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S4.T1.3.7.4.1\" rowspan=\"3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.7.4.1.1\" style=\"font-size:90%;\">Spatial</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.3.7.4.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.7.4.2.1\" style=\"font-size:90%;\">DF</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.3.7.4.3\" rowspan=\"2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.7.4.3.1\" style=\"font-size:90%;\">\n<span class=\"ltx_tabular ltx_align_middle\" id=\"S4.T1.3.7.4.3.1.1\">\n<span class=\"ltx_tr\" id=\"S4.T1.3.7.4.3.1.1.1\">\n<span class=\"ltx_td ltx_nopad_r ltx_align_center\" id=\"S4.T1.3.7.4.3.1.1.1.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\">Pre-defined convolution</span></span>\n</span></span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S4.T1.3.7.4.4\" rowspan=\"2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.7.4.4.1\" style=\"font-size:90%;\">E-Conv</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S4.T1.3.7.4.5\" rowspan=\"2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.7.4.5.1\" style=\"font-size:90%;\">\n<span class=\"ltx_tabular ltx_align_middle\" id=\"S4.T1.3.7.4.5.1.1\">\n<span class=\"ltx_tr\" id=\"S4.T1.3.7.4.5.1.1.1\">\n<span class=\"ltx_td ltx_nopad_r ltx_align_center\" id=\"S4.T1.3.7.4.5.1.1.1.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\">Pre-defined convolution</span></span>\n</span></span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.3.8.5\">\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S4.T1.3.8.5.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.8.5.1.1\" style=\"font-size:90%;\">CF</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S4.T1.3.9.6\">\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S4.T1.3.9.6.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.9.6.1.1\" style=\"font-size:90%;\">IST/EST</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S4.T1.3.9.6.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.9.6.2.1\" style=\"font-size:90%;\">ResNet block</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S4.T1.3.9.6.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.9.6.3.1\" style=\"font-size:90%;\">S-Conv block</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"S4.T1.3.9.6.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S4.T1.3.9.6.4.1\" style=\"font-size:90%;\">ResNet/S-Conv block</span></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 1: Temporal-Spatial Stepping Module (TSSM)."
        },
        "2": {
            "table_html": "<figure class=\"ltx_table\" id=\"S5.T3\">\n<div class=\"ltx_flex_figure ltx_flex_table\">\n<div class=\"ltx_flex_cell ltx_flex_size_2\">\n<figure class=\"ltx_figure ltx_figure_panel ltx_minipage ltx_align_center ltx_align_top\" id=\"S5.T3.8\" style=\"width:260.2pt;\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_figure\">Table 2: </span> (Eq. 4) across different datasets in time extrapolation task.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"S5.T3.8.8\">\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.7.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_tt\" id=\"S5.T3.8.8.7.1.1\" rowspan=\"2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.7.1.1.1\" style=\"font-size:70%;\">Config</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" colspan=\"2\" id=\"S5.T3.8.8.7.1.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.7.1.2.1\" style=\"font-size:70%;\">Burgers2d</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"S5.T3.8.8.7.1.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.7.1.3.1\" style=\"font-size:70%;\">RD2d</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" colspan=\"3\" id=\"S5.T3.8.8.7.1.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.7.1.4.1\" style=\"font-size:70%;\">NS2d</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"S5.T3.8.8.7.1.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.7.1.5.1\" style=\"font-size:70%;\">Lid2d</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_tt\" colspan=\"2\" id=\"S5.T3.8.8.7.1.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.7.1.6.1\" style=\"font-size:70%;\">NSM2d</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.6\">\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.6.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.6.7.1\" style=\"font-size:70%;\">C Int.</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.6.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.6.8.1\" style=\"font-size:70%;\">C Ext.</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.6.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.6.9.1\" style=\"font-size:70%;\">C Int.</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.4.4.2.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.4.4.2.2.2\" style=\"font-size:70%;\">=-3</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.6.6.4.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.6.6.4.4.2\" style=\"font-size:70%;\">=-4</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.6.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.6.6.2\" style=\"font-size:70%;\">=-5</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.6.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.6.10.1\" style=\"font-size:70%;\">C Ext.</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.6.11\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.6.11.1\" style=\"font-size:70%;\">C Int.</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.8.8.6.12\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.6.12.1\" style=\"font-size:70%;\">C Ext.</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.8.2\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.1.1\" style=\"font-size:70%;\">ConvLSTM</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.2.1\" style=\"font-size:70%;\">0.314</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.3.1\" style=\"font-size:70%;\">0.551</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.4.1\" style=\"font-size:70%;\">0.815</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.5.1\" style=\"font-size:70%;\">0.781</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.6.1\" style=\"font-size:70%;\">0.877</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.7.1\" style=\"font-size:70%;\">0.788</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.8.1\" style=\"font-size:70%;\">1.323</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.8.2.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.9.1\" style=\"font-size:70%;\">0.910</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S5.T3.8.8.8.2.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.8.2.10.1\" style=\"font-size:70%;\">1.102</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.9.3\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.8.8.9.3.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.1.1\" style=\"font-size:70%;\">Dil-ResNet</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.9.3.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.2.1\" style=\"font-size:70%;\">0.071</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.9.3.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.3.1\" style=\"font-size:70%;\">0.136</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.9.3.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.9.3.4.1\" style=\"font-size:70%;\">0.021</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.9.3.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.5.1\" style=\"font-size:70%;\">0.152</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.9.3.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.6.1\" style=\"font-size:70%;\">0.511</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.9.3.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.7.1\" style=\"font-size:70%;\">0.199</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.9.3.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.8.1\" style=\"font-size:70%;\">0.261</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.9.3.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.9.1\" style=\"font-size:70%;\">0.288</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.8.8.9.3.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.9.3.10.1\" style=\"font-size:70%;\">0.314</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.10.4\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.8.8.10.4.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.10.4.1.1\" style=\"font-size:70%;\">time-FNO2D</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.10.4.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.10.4.2.1\" style=\"font-size:70%;\">0.173</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.10.4.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.10.4.3.1\" style=\"font-size:70%;\">0.233</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.10.4.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.10.4.4.1\" style=\"font-size:70%;\">0.333</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.10.4.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.10.4.5.1\" style=\"font-size:70%;\">0.118</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.10.4.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.10.4.6.1\" style=\"font-size:70%;\">0.100</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.10.4.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.10.4.7.1\" style=\"font-size:70%;\">0.033</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.10.4.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.10.4.8.1\" style=\"font-size:70%;\">0.265</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.10.4.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.10.4.9.1\" style=\"font-size:70%;\">0.341</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.8.8.10.4.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.10.4.10.1\" style=\"font-size:70%;\">0.443</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.11.5\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.8.8.11.5.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.1.1\" style=\"font-size:70%;\">MIONet</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.11.5.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.2.1\" style=\"font-size:70%;\">0.181</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.11.5.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.3.1\" style=\"font-size:70%;\">0.212</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.11.5.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.4.1\" style=\"font-size:70%;\">0.247</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.11.5.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.5.1\" style=\"font-size:70%;\">0.139</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.11.5.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.6.1\" style=\"font-size:70%;\">0.114</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.11.5.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.7.1\" style=\"font-size:70%;\">0.051</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.11.5.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.8.1\" style=\"font-size:70%;\">0.221</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.11.5.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.9.1\" style=\"font-size:70%;\">0.268</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.8.8.11.5.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.11.5.10.1\" style=\"font-size:70%;\">0.440</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.12.6\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.8.8.12.6.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.1.1\" style=\"font-size:70%;\">U-FNet</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.12.6.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.2.1\" style=\"font-size:70%;\">0.109</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.12.6.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.3.1\" style=\"font-size:70%;\">0.433</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.12.6.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.4.1\" style=\"font-size:70%;\">0.239</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.12.6.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.5.1\" style=\"font-size:70%;\">0.191</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.12.6.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.6.1\" style=\"font-size:70%;\">0.190</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.12.6.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.7.1\" style=\"font-size:70%;\">0.256</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.12.6.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.8.1\" style=\"font-size:70%;\">0.192</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.12.6.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.9.1\" style=\"font-size:70%;\">0.257</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.8.8.12.6.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.12.6.10.1\" style=\"font-size:70%;\">0.457</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.13.7\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.8.8.13.7.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.13.7.1.1\" style=\"font-size:70%;\">CNO</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.13.7.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.13.7.2.1\" style=\"font-size:70%;\">0.112</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.13.7.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.13.7.3.1\" style=\"font-size:70%;\">0.126</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.13.7.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.13.7.4.1\" style=\"font-size:70%;\">0.258</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.13.7.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.13.7.5.1\" style=\"font-size:70%;\">0.125</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.13.7.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.13.7.6.1\" style=\"font-size:70%;\">0.148</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.13.7.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.13.7.7.1\" style=\"font-size:70%;\">0.030</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.13.7.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.13.7.8.1\" style=\"font-size:70%;\">0.218</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.13.7.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.13.7.9.1\" style=\"font-size:70%;\">0.197</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.8.8.13.7.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.13.7.10.1\" style=\"font-size:70%;\">0.355</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.14.8\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.8.8.14.8.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.1.1\" style=\"font-size:70%;\">PeRCNN</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.14.8.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.2.1\" style=\"font-size:70%;\">0.212</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.14.8.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.3.1\" style=\"font-size:70%;\">0.282</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.14.8.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.4.1\" style=\"font-size:70%;\">0.773</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.14.8.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.5.1\" style=\"font-size:70%;\">0.571</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.14.8.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.6.1\" style=\"font-size:70%;\">0.591</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.14.8.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.7.1\" style=\"font-size:70%;\">0.275</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.14.8.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.8.1\" style=\"font-size:70%;\">0.534</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.14.8.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.9.1\" style=\"font-size:70%;\">0.493</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.8.8.14.8.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.14.8.10.1\" style=\"font-size:70%;\">0.493</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.15.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.8.8.15.9.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.15.9.1.1\" style=\"font-size:70%;\">PPNN</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.15.9.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.15.9.2.1\" style=\"font-size:70%;\">0.047</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.15.9.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.15.9.3.1\" style=\"font-size:70%;\">0.132</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.15.9.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.15.9.4.1\" style=\"font-size:70%;\">0.030</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.15.9.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.15.9.5.1\" style=\"font-size:70%;\">0.365</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.15.9.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.15.9.6.1\" style=\"font-size:70%;\">0.357</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.15.9.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.15.9.7.1\" style=\"font-size:70%;\">0.046</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.15.9.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.15.9.8.1\" style=\"font-size:70%;\">0.163</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.8.8.15.9.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.15.9.9.1\" style=\"font-size:70%;\">0.206</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.8.8.15.9.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.8.8.15.9.10.1\" style=\"font-size:70%;\">0.264</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.8.8.16.10\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.1.1\" style=\"font-size:70%;\">PAPM (Our)</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.2.1\" style=\"font-size:70%;\">0.039</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.3.1\" style=\"font-size:70%;\">0.101</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.4.1\" style=\"font-size:70%;\">0.018</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.5.1\" style=\"font-size:70%;\">0.110</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.6\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.6.1\" style=\"font-size:70%;\">0.097</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.7\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.8.8.16.10.7.1\" style=\"font-size:70%;\">0.034</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.8\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.8.1\" style=\"font-size:70%;\">0.160</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.8.8.16.10.9\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.9.1\" style=\"font-size:70%;\">0.189</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"S5.T3.8.8.16.10.10\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.8.8.16.10.10.1\" style=\"font-size:70%;\">0.245</span></td>\n</tr>\n</tbody>\n</table>\n</figure>\n</div>\n<div class=\"ltx_flex_cell ltx_flex_size_2\">\n<figure class=\"ltx_figure ltx_figure_panel ltx_minipage ltx_align_center ltx_align_top\" id=\"S5.T3.9\" style=\"width:169.1pt;\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_figure\">Table 3: </span>Comparison of parameters and FLOPs.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"S5.T3.9.1\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt\" id=\"S5.T3.9.1.1.2\" rowspan=\"2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.9.1.1.2.1\" style=\"font-size:70%;\">Config</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" colspan=\"3\" id=\"S5.T3.9.1.1.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt\" id=\"S5.T3.9.1.1.3\" rowspan=\"2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.1.3.1\" style=\"font-size:70%;\">FLOPs/M</span></th>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.2.1\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r\" id=\"S5.T3.9.1.2.1.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.2.1.1.1\" style=\"font-size:70%;\">Localized/M</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r\" id=\"S5.T3.9.1.2.1.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.2.1.2.1\" style=\"font-size:70%;\">Spectra/M</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r\" id=\"S5.T3.9.1.2.1.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.2.1.3.1\" style=\"font-size:70%;\">Hybrid/M</span></th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.3.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t\" id=\"S5.T3.9.1.3.1.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.3.1.1.1\" style=\"font-size:70%;\">ConvLSTM</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.9.1.3.1.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.3.1.2.1\" style=\"font-size:70%;\">0.175</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.9.1.3.1.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.3.1.3.1\" style=\"font-size:70%;\">0.139</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T3.9.1.3.1.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.3.1.4.1\" style=\"font-size:70%;\">0.211</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S5.T3.9.1.3.1.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.3.1.5.1\" style=\"font-size:70%;\">32.75</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.4.2\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.9.1.4.2.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.4.2.1.1\" style=\"font-size:70%;\">Dil-ResNet</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.4.2.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.4.2.2.1\" style=\"font-size:70%;\">0.150</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.4.2.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.4.2.3.1\" style=\"font-size:70%;\">0.148</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.4.2.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.4.2.4.1\" style=\"font-size:70%;\">0.152</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.9.1.4.2.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.4.2.5.1\" style=\"font-size:70%;\">62.40</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.5.3\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.9.1.5.3.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.5.3.1.1\" style=\"font-size:70%;\">time-FNO2D</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.5.3.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.5.3.2.1\" style=\"font-size:70%;\">0.464</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.5.3.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.5.3.3.1\" style=\"font-size:70%;\">0.463</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.5.3.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.5.3.4.1\" style=\"font-size:70%;\">0.464</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.9.1.5.3.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.5.3.5.1\" style=\"font-size:70%;\">6.88</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.6.4\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.9.1.6.4.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.6.4.1.1\" style=\"font-size:70%;\">MIONet</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.6.4.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.6.4.2.1\" style=\"font-size:70%;\">0.261</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.6.4.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.6.4.3.1\" style=\"font-size:70%;\">0.261</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.6.4.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.6.4.4.1\" style=\"font-size:70%;\">0.261</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.9.1.6.4.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.6.4.5.1\" style=\"font-size:70%;\">10.01</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.7.5\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.9.1.7.5.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.7.5.1.1\" style=\"font-size:70%;\">U-FNet</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.7.5.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.7.5.2.1\" style=\"font-size:70%;\">9.853</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.7.5.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.7.5.3.1\" style=\"font-size:70%;\">9.851</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.7.5.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.7.5.4.1\" style=\"font-size:70%;\">9.854</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.9.1.7.5.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.7.5.5.1\" style=\"font-size:70%;\">559.89</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.8.6\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.9.1.8.6.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.8.6.1.1\" style=\"font-size:70%;\">CNO</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.8.6.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.8.6.2.1\" style=\"font-size:70%;\">2.606</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.8.6.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.8.6.3.1\" style=\"font-size:70%;\">2.600</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.8.6.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.8.6.4.1\" style=\"font-size:70%;\">2.612</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.9.1.8.6.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.8.6.5.1\" style=\"font-size:70%;\">835.37</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.9.7\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.9.1.9.7.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.9.7.1.1\" style=\"font-size:70%;\">PeRCNN</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.9.7.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.9.1.9.7.2.1\" style=\"font-size:70%;\">0.001</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.9.7.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.9.1.9.7.3.1\" style=\"font-size:70%;\">0.001</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.9.7.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.9.1.9.7.4.1\" style=\"font-size:70%;\">0.001</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.9.1.9.7.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.9.1.9.7.5.1\" style=\"font-size:70%;\">3.44</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.10.8\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T3.9.1.10.8.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.10.8.1.1\" style=\"font-size:70%;\">PPNN</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.10.8.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.10.8.2.1\" style=\"font-size:70%;\">1.201</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.10.8.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.10.8.3.1\" style=\"font-size:70%;\">1.190</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T3.9.1.10.8.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.10.8.4.1\" style=\"font-size:70%;\">1.213</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T3.9.1.10.8.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text\" id=\"S5.T3.9.1.10.8.5.1\" style=\"font-size:70%;\">348.56</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T3.9.1.11.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.9.1.11.9.1\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.9.1.11.9.1.1\" style=\"font-size:70%;\">PAPM</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.9.1.11.9.2\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.9.1.11.9.2.1\" style=\"font-size:70%;\">0.014</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.9.1.11.9.3\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.9.1.11.9.3.1\" style=\"font-size:70%;\">0.034</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T3.9.1.11.9.4\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"S5.T3.9.1.11.9.4.1\" style=\"font-size:70%;\">0.035</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"S5.T3.9.1.11.9.5\" style=\"padding-left:3.0pt;padding-right:3.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T3.9.1.11.9.5.1\" style=\"font-size:70%;\">1.23</span></td>\n</tr>\n</tbody>\n</table>\n</figure>\n</div>\n</div>\n</figure>",
            "capture": "Table 2:  (Eq. 4) across different datasets in time extrapolation task."
        },
        "3": {
            "table_html": "<figure class=\"ltx_table\" id=\"S5.T4\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 4: </span>Ablation comparison of  and  on the boundary (BC ).</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"S5.T4.10\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"S5.T4.10.5.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt\" id=\"S5.T4.10.5.1.1\" rowspan=\"2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T4.10.5.1.1.1\" style=\"font-size:90%;\">Config</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" colspan=\"2\" id=\"S5.T4.10.5.1.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T4.10.5.1.2.1\" style=\"font-size:90%;\">C Int.</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt\" colspan=\"2\" id=\"S5.T4.10.5.1.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T4.10.5.1.3.1\" style=\"font-size:90%;\">C Ext.</span></th>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T4.10.4\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"S5.T4.7.1.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r\" id=\"S5.T4.8.2.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\">\n<span class=\"ltx_text\" id=\"S5.T4.8.2.2.1\" style=\"font-size:90%;\">BC </span>\n</th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"S5.T4.9.3.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"S5.T4.10.4.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\">\n<span class=\"ltx_text\" id=\"S5.T4.10.4.4.1\" style=\"font-size:90%;\">BC </span>\n</th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S5.T4.10.6.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t\" id=\"S5.T4.10.6.1.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.6.1.1.1\" style=\"font-size:90%;\">no_DF</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S5.T4.10.6.1.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.6.1.2.1\" style=\"font-size:90%;\">0.067</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T4.10.6.1.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.6.1.3.1\" style=\"font-size:90%;\">0.051</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S5.T4.10.6.1.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.6.1.4.1\" style=\"font-size:90%;\">0.207</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S5.T4.10.6.1.5\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.6.1.5.1\" style=\"font-size:90%;\">0.067</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T4.10.7.2\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T4.10.7.2.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.7.2.1.1\" style=\"font-size:90%;\">no_CF</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.7.2.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.7.2.2.1\" style=\"font-size:90%;\">0.062</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T4.10.7.2.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.7.2.3.1\" style=\"font-size:90%;\">0.043</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.7.2.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.7.2.4.1\" style=\"font-size:90%;\">0.131</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.7.2.5\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.7.2.5.1\" style=\"font-size:90%;\">0.054</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T4.10.8.3\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T4.10.8.3.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.8.3.1.1\" style=\"font-size:90%;\">no_Phy</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.8.3.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.8.3.2.1\" style=\"font-size:90%;\">0.149</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T4.10.8.3.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.8.3.3.1\" style=\"font-size:90%;\">0.051</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.8.3.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.8.3.4.1\" style=\"font-size:90%;\">0.210</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.8.3.5\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.8.3.5.1\" style=\"font-size:90%;\">0.144</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T4.10.9.4\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T4.10.9.4.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.9.4.1.1\" style=\"font-size:90%;\">no_BCs</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.9.4.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.9.4.2.1\" style=\"font-size:90%;\">0.068</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T4.10.9.4.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.9.4.3.1\" style=\"font-size:90%;\">0.097</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.9.4.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.9.4.4.1\" style=\"font-size:90%;\">0.136</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.9.4.5\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.9.4.5.1\" style=\"font-size:90%;\">0.193</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T4.10.10.5\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T4.10.10.5.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.10.5.1.1\" style=\"font-size:90%;\">no_NODE</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.10.5.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.10.5.2.1\" style=\"font-size:90%;\">0.053</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T4.10.10.5.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.10.5.3.1\" style=\"font-size:90%;\">0.041</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.10.5.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.10.5.4.1\" style=\"font-size:90%;\">0.150</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.10.5.5\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.10.5.5.1\" style=\"font-size:90%;\">0.045</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T4.10.11.6\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"S5.T4.10.11.6.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.11.6.1.1\" style=\"font-size:90%;\">no_All</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.11.6.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.11.6.2.1\" style=\"font-size:90%;\">0.162</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"S5.T4.10.11.6.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.11.6.3.1\" style=\"font-size:90%;\">0.195</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.11.6.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.11.6.4.1\" style=\"font-size:90%;\">0.216</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"S5.T4.10.11.6.5\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text\" id=\"S5.T4.10.11.6.5.1\" style=\"font-size:90%;\">0.250</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T4.10.12.7\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T4.10.12.7.1\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T4.10.12.7.1.1\" style=\"font-size:90%;\">PAPM</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"S5.T4.10.12.7.2\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T4.10.12.7.2.1\" style=\"font-size:90%;\">0.039</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"S5.T4.10.12.7.3\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T4.10.12.7.3.1\" style=\"font-size:90%;\">0.037</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"S5.T4.10.12.7.4\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T4.10.12.7.4.1\" style=\"font-size:90%;\">0.101</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"S5.T4.10.12.7.5\" style=\"padding-left:8.0pt;padding-right:8.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T4.10.12.7.5.1\" style=\"font-size:90%;\">0.043</span></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 4: Ablation comparison of  and  on the boundary (BC )."
        },
        "4": {
            "table_html": "<figure class=\"ltx_table\" id=\"S5.T5\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 5: </span>Comparison of  for different blocks on different datasets.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"S5.T5.6\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"S5.T5.6.4\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt\" id=\"S5.T5.6.4.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T5.6.4.5.1\" style=\"font-size:90%;\">Datasets</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" id=\"S5.T5.3.1.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" id=\"S5.T5.4.2.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"S5.T5.4.2.2.1\" style=\"font-size:90%;\">convection </span>\n</th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" id=\"S5.T5.5.3.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"S5.T5.5.3.3.1\" style=\"font-size:90%;\">diffusion </span>\n</th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt\" id=\"S5.T5.6.4.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"S5.T5.6.4.4.1\" style=\"font-size:90%;\">source </span>\n</th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"S5.T5.6.5.1\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_r ltx_border_t\" id=\"S5.T5.6.5.1.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T5.6.5.1.1.1\" style=\"font-size:90%;\">Burgers2d</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T5.6.5.1.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T5.6.5.1.2.1\" style=\"font-size:90%;\">0.039</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T5.6.5.1.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T5.6.5.1.3.1\" style=\"font-size:90%;\">0.037</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"S5.T5.6.5.1.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T5.6.5.1.4.1\" style=\"font-size:90%;\">0.041</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"S5.T5.6.5.1.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T5.6.5.1.5.1\" style=\"font-size:90%;\">0.069</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"S5.T5.6.6.2\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_row ltx_border_bb ltx_border_r\" id=\"S5.T5.6.6.2.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T5.6.6.2.1.1\" style=\"font-size:90%;\">RD2d</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r\" id=\"S5.T5.6.6.2.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"S5.T5.6.6.2.2.1\" style=\"font-size:90%;\">0.018</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r\" id=\"S5.T5.6.6.2.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T5.6.6.2.3.1\" style=\"font-size:90%;\">-</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r\" id=\"S5.T5.6.6.2.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T5.6.6.2.4.1\" style=\"font-size:90%;\">0.025</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb\" id=\"S5.T5.6.6.2.5\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"S5.T5.6.6.2.5.1\" style=\"font-size:90%;\">0.012</span></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 5: Comparison of  for different blocks on different datasets."
        },
        "5": {
            "table_html": "<figure class=\"ltx_table\" id=\"A0.T6\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 6: </span>Table of notations.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"A0.T6.42\">\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"A0.T6.42.43.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_tt\" id=\"A0.T6.42.43.1.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"A0.T6.42.43.1.1.1\" style=\"font-size:90%;\">Notation</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_tt\" id=\"A0.T6.42.43.1.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text ltx_font_bold\" id=\"A0.T6.42.43.1.2.1\" style=\"font-size:90%;\">Meaning</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.44.2\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t\" id=\"A0.T6.42.44.2.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.44.2.1.1\" style=\"font-size:90%;\">Process model</span></th>\n<td class=\"ltx_td ltx_border_t\" id=\"A0.T6.42.44.2.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.1.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t\" id=\"A0.T6.1.1.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A0.T6.1.1.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.1.1.2.1\" style=\"font-size:90%;\">the physical quantity, also as the system’s state</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.2.2\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.2.2.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.2.2.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.2.2.2.1\" style=\"font-size:90%;\">the diffusion flows in the conservation relations</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.3.3\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.3.3.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.3.3.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.3.3.2.1\" style=\"font-size:90%;\">the convection flows in the conservation relations</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.4.4\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.4.4.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.4.4.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.4.4.2.1\" style=\"font-size:90%;\">the internal source in the conservation relations</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.5.5\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.5.5.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.5.5.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.5.5.2.1\" style=\"font-size:90%;\">the external source in the conservation relations</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.6.6\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.6.6.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.6.6.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.6.6.2.1\" style=\"font-size:90%;\">the velocity of the physical quantity being transmitted</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.7.7\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.7.7.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.7.7.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.7.7.2.1\" style=\"font-size:90%;\">the coefficients</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.8.8\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.8.8.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.8.8.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.8.8.2.1\" style=\"font-size:90%;\">the coefficients, such as viscosity</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.9.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.9.9.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.9.9.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.9.9.2.1\" style=\"font-size:90%;\">a vector of external sources, such as voltages</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.10.10\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.10.10.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.10.10.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.10.10.2.1\" style=\"font-size:90%;\">the initial condition</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.45.3\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.42.45.3.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.45.3.1.1\" style=\"font-size:90%;\">ICs</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.42.45.3.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.45.3.2.1\" style=\"font-size:90%;\">Initial conditions</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.46.4\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.42.46.4.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.46.4.1.1\" style=\"font-size:90%;\">BCs</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.42.46.4.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.46.4.2.1\" style=\"font-size:90%;\">Boundary conditions</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.47.5\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t\" id=\"A0.T6.42.47.5.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.47.5.1.1\" style=\"font-size:90%;\">Problem formulation</span></th>\n<td class=\"ltx_td ltx_border_t\" id=\"A0.T6.42.47.5.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.14.14\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t\" id=\"A0.T6.11.11.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A0.T6.14.14.4\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.14.14.4.1\" style=\"font-size:90%;\">a vector, which consists of </span><span class=\"ltx_text\" id=\"A0.T6.14.14.4.2\" style=\"font-size:90%;\"> physical quantities of index </span><span class=\"ltx_text\" id=\"A0.T6.14.14.4.3\" style=\"font-size:90%;\"> at time </span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.17.17\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.15.15.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.17.17.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.17.17.3.1\" style=\"font-size:90%;\">initial condition of index </span><span class=\"ltx_text\" id=\"A0.T6.17.17.3.2\" style=\"font-size:90%;\"> at time </span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.18.18\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.18.18.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.18.18.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.18.18.2.1\" style=\"font-size:90%;\">a physical quantity, such as vorticity</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.19.19\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.19.19.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.19.19.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.19.19.2.1\" style=\"font-size:90%;\">the grid (also as discretized spatial coordinates)</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.20.20\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.20.20.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.20.20.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.20.20.2.1\" style=\"font-size:90%;\">the input conditions space</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.22.22\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.21.21.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.22.22.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.22.22.2.1\" style=\"font-size:90%;\">a set input conditions of index </span><span class=\"ltx_text\" id=\"A0.T6.22.22.2.2\" style=\"font-size:90%;\">, containing initial condition, and other conditions</span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.23.23\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.23.23.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.23.23.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.23.23.2.1\" style=\"font-size:90%;\">the solutions space</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.25.25\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.24.24.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.25.25.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.25.25.2.1\" style=\"font-size:90%;\">a set output of index of index </span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.27.27\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.26.26.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.27.27.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.27.27.2.1\" style=\"font-size:90%;\">the dataset, where </span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.28.28\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.28.28.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.28.28.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.28.28.2.1\" style=\"font-size:90%;\">the mapping of our goal to learn</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.30.30\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.29.29.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.30.30.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.30.30.2.1\" style=\"font-size:90%;\">a parameterized neural network with parameters </span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.31.31\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.31.31.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.31.31.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.31.31.2.1\" style=\"font-size:90%;\">the parameter space</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.32.32\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.32.32.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.32.32.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.32.32.2.1\" style=\"font-size:90%;\">the time-step size for inference</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.34.34\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.33.33.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.34.34.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.34.34.2.1\" style=\"font-size:90%;\">the time-step size of the training dataset, where </span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.48.6\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t\" id=\"A0.T6.42.48.6.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.48.6.1.1\" style=\"font-size:90%;\">Methodology</span></th>\n<td class=\"ltx_td ltx_border_t\" id=\"A0.T6.42.48.6.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.35.35\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t\" id=\"A0.T6.35.35.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A0.T6.35.35.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.35.35.2.1\" style=\"font-size:90%;\">the updated physical quantity by using the given boundary conditions</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.49.7\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.42.49.7.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.49.7.1.1\" style=\"font-size:90%;\">DF</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.42.49.7.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.49.7.2.1\" style=\"font-size:90%;\">Diffusive Flows</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.50.8\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.42.50.8.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.50.8.1.1\" style=\"font-size:90%;\">CF</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.42.50.8.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.50.8.2.1\" style=\"font-size:90%;\">Convective Flows</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.51.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.42.51.9.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.51.9.1.1\" style=\"font-size:90%;\">IST</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.42.51.9.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.51.9.2.1\" style=\"font-size:90%;\">Internal Source Term</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.52.10\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.42.52.10.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.52.10.1.1\" style=\"font-size:90%;\">EST</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.42.52.10.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.52.10.2.1\" style=\"font-size:90%;\">External Source Term</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.53.11\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.42.53.11.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.53.11.1.1\" style=\"font-size:90%;\">TSSM</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.42.53.11.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.53.11.2.1\" style=\"font-size:90%;\">Temporal-Spatial Stepping Module</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.37.37\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.37.37.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.37.37.2.1\" style=\"font-size:90%;\">, </span>\n</th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.37.37.3\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.37.37.3.1\" style=\"font-size:90%;\">the spectral space dimensions</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.54.12\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t\" id=\"A0.T6.42.54.12.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.54.12.1.1\" style=\"font-size:90%;\">Experiments</span></th>\n<td class=\"ltx_td ltx_border_t\" id=\"A0.T6.42.54.12.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.55.13\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_t\" id=\"A0.T6.42.55.13.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.55.13.1.1\" style=\"font-size:90%;\">C Int.</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A0.T6.42.55.13.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.55.13.2.1\" style=\"font-size:90%;\">the task of coefficient interpolation</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.56.14\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.42.56.14.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.56.14.1.1\" style=\"font-size:90%;\">C Ext.</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.42.56.14.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.56.14.2.1\" style=\"font-size:90%;\">the task of coefficient extrapolation</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.39.39\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.38.38.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.39.39.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.39.39.2.1\" style=\"font-size:90%;\">the mean </span><span class=\"ltx_text\" id=\"A0.T6.39.39.2.2\" style=\"font-size:90%;\"> relative error</span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.41.41\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row\" id=\"A0.T6.40.40.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.40.40.1.1\" style=\"font-size:90%;\">BC </span>\n</th>\n<td class=\"ltx_td ltx_align_center\" id=\"A0.T6.41.41.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\">\n<span class=\"ltx_text\" id=\"A0.T6.41.41.2.1\" style=\"font-size:90%;\">the mean </span><span class=\"ltx_text\" id=\"A0.T6.41.41.2.2\" style=\"font-size:90%;\"> relative error on the boundary</span>\n</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A0.T6.42.42\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb\" id=\"A0.T6.42.42.1\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"></th>\n<td class=\"ltx_td ltx_align_center ltx_border_bb\" id=\"A0.T6.42.42.2\" style=\"padding-left:4.0pt;padding-right:4.0pt;\"><span class=\"ltx_text\" id=\"A0.T6.42.42.2.1\" style=\"font-size:90%;\">the number of trainable parameters</span></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 6: Table of notations."
        },
        "6": {
            "table_html": "<figure class=\"ltx_table\" id=\"A4.T7\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 7: </span>The choice of different paths.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_align_middle\" id=\"A4.T7.3\">\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"A4.T7.3.4.1\">\n<td class=\"ltx_td ltx_border_r ltx_border_tt\" id=\"A4.T7.3.4.1.1\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A4.T7.3.4.1.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A4.T7.3.4.1.2.1\">Localized</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A4.T7.3.4.1.3\"><span class=\"ltx_text ltx_font_bold\" id=\"A4.T7.3.4.1.3.1\">Spectral</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_tt\" id=\"A4.T7.3.4.1.4\"><span class=\"ltx_text ltx_font_bold\" id=\"A4.T7.3.4.1.4.1\">Hybrid</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T7.3.5.2\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.3.5.2.1\">Characteristic</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.3.5.2.2\">Explicit</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.3.5.2.3\">Implicit</td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A4.T7.3.5.2.4\">Explicit+Implicit</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T7.3.3\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.3.3.4\">Example</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.1.1.1\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.2.2.2\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A4.T7.3.3.3\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T7.3.6.3\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.3.6.3.1\">Diffusive Flows/Convective Flows</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.3.6.3.2\">Pre-defined convolution</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T7.3.6.3.3\">E-Conv</td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A4.T7.3.6.3.4\">Pre-defined convolution</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T7.3.7.4\">\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A4.T7.3.7.4.1\">Internal Source Term/External Source Term</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A4.T7.3.7.4.2\">ResNet block</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A4.T7.3.7.4.3\">S-Conv block</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A4.T7.3.7.4.4\">ResNet/S-Conv block</td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 7: The choice of different paths."
        },
        "7": {
            "table_html": "<figure class=\"ltx_table\" id=\"A4.T8\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 8: </span>Impact of Mismatched Path Selection on Performance.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_align_middle\" id=\"A4.T8.1\">\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"A4.T8.1.2.1\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A4.T8.1.2.1.1\"><span class=\"ltx_text ltx_font_bold\" id=\"A4.T8.1.2.1.1.1\">Datasets</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A4.T8.1.2.1.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A4.T8.1.2.1.2.1\">Category</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A4.T8.1.2.1.3\">Localized</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A4.T8.1.2.1.4\">Spectral</td>\n<td class=\"ltx_td ltx_align_center ltx_border_tt\" id=\"A4.T8.1.2.1.5\">Hybrid</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T8.1.3.2\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T8.1.3.2.1\">Burgers2d</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T8.1.3.2.2\">Localized</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T8.1.3.2.3\">0.039</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T8.1.3.2.4\">0.043</td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A4.T8.1.3.2.5\"><span class=\"ltx_text ltx_font_bold\" id=\"A4.T8.1.3.2.5.1\">0.037</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T8.1.1\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T8.1.1.1\">NS2d (-5)</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T8.1.1.2\">Spectral</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T8.1.1.3\">0.061</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T8.1.1.4\"><span class=\"ltx_text ltx_font_bold\" id=\"A4.T8.1.1.4.1\">0.034</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A4.T8.1.1.5\">0.048</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T8.1.4.3\">\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A4.T8.1.4.3.1\">NSM2d</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A4.T8.1.4.3.2\">Hybrid</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A4.T8.1.4.3.3\">0.205</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A4.T8.1.4.3.4\">0.196</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A4.T8.1.4.3.5\"><span class=\"ltx_text ltx_font_bold\" id=\"A4.T8.1.4.3.5.1\">0.189</span></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 8: Impact of Mismatched Path Selection on Performance."
        },
        "8": {
            "table_html": "<figure class=\"ltx_table\" id=\"A4.T9\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 9: </span>The difference of five datasets.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"A4.T9.1\">\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"A4.T9.1.1.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_tt\" id=\"A4.T9.1.1.1.1\" rowspan=\"2\"><span class=\"ltx_text\" id=\"A4.T9.1.1.1.1.1\">Dataset</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A4.T9.1.1.1.2\" rowspan=\"2\"><span class=\"ltx_text\" id=\"A4.T9.1.1.1.2.1\">Category</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_tt\" colspan=\"4\" id=\"A4.T9.1.1.1.3\">Various Conditions</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T9.1.2.2\">\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A4.T9.1.2.2.1\">Initial conditions</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A4.T9.1.2.2.2\">Boundary conditions</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A4.T9.1.2.2.3\">Coefficients</td>\n<td class=\"ltx_td ltx_align_center\" id=\"A4.T9.1.2.2.4\">External sources</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T9.1.3.3\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t\" id=\"A4.T9.1.3.3.1\">Burgers2d</th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.3.3.2\">Localized</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.3.3.3\">✓</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.3.3.4\">Periodic</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.3.3.5\">✓</td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A4.T9.1.3.3.6\">unknown</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T9.1.4.4\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"A4.T9.1.4.4.1\">RD2d</th>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A4.T9.1.4.4.2\">Localized</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A4.T9.1.4.4.3\">✓</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A4.T9.1.4.4.4\">No-flow Neumann</td>\n<td class=\"ltx_td ltx_border_r\" id=\"A4.T9.1.4.4.5\"></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A4.T9.1.4.4.6\">unknown</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T9.1.5.5\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t\" id=\"A4.T9.1.5.5.1\">NS2d</th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.5.5.2\">Spectral</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.5.5.3\">✓</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.5.5.4\">Periodic</td>\n<td class=\"ltx_td ltx_border_r ltx_border_t\" id=\"A4.T9.1.5.5.5\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A4.T9.1.5.5.6\">unknown</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T9.1.6.6\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t\" id=\"A4.T9.1.6.6.1\">Lid2d</th>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.6.6.2\">Hybrid</td>\n<td class=\"ltx_td ltx_border_r ltx_border_t\" id=\"A4.T9.1.6.6.3\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.6.6.4\">Dirichlet, Neumann</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A4.T9.1.6.6.5\">✓</td>\n<td class=\"ltx_td ltx_border_t\" id=\"A4.T9.1.6.6.6\"></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A4.T9.1.7.7\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_r\" id=\"A4.T9.1.7.7.1\">NSM2d</th>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r\" id=\"A4.T9.1.7.7.2\">Hybrid</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r\" id=\"A4.T9.1.7.7.3\">✓</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r\" id=\"A4.T9.1.7.7.4\">Dirichlet, Neumann</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r\" id=\"A4.T9.1.7.7.5\">✓</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb\" id=\"A4.T9.1.7.7.6\">unknown</td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 9: The difference of five datasets."
        },
        "9": {
            "table_html": "<figure class=\"ltx_table\" id=\"A7.T10\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 10: </span>Training and inference time cost (epoch/second) of different baselines.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"A7.T10.1\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"A7.T10.1.1.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_column ltx_th_row ltx_border_r ltx_border_tt\" id=\"A7.T10.1.1.1.1\" rowspan=\"2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.1.1.1.1\" style=\"font-size:90%;\">Config</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" colspan=\"2\" id=\"A7.T10.1.1.1.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.1.1.2.1\" style=\"font-size:90%;\">Burgers2d</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" colspan=\"2\" id=\"A7.T10.1.1.1.3\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.1.1.3.1\" style=\"font-size:90%;\">RD2d</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" colspan=\"2\" id=\"A7.T10.1.1.1.4\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.1.1.4.1\" style=\"font-size:90%;\">NS2d</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" colspan=\"2\" id=\"A7.T10.1.1.1.5\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.1.1.5.1\" style=\"font-size:90%;\">Lid2d</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt\" colspan=\"2\" id=\"A7.T10.1.1.1.6\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.1.1.6.1\" style=\"font-size:90%;\">NSM2d</span></th>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.2.2\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"A7.T10.1.2.2.1\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.1.1\" style=\"font-size:90%;\">Train</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r\" id=\"A7.T10.1.2.2.2\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.2.1\" style=\"font-size:90%;\">Infer</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"A7.T10.1.2.2.3\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.3.1\" style=\"font-size:90%;\">Train</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r\" id=\"A7.T10.1.2.2.4\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.4.1\" style=\"font-size:90%;\">Infer</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"A7.T10.1.2.2.5\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.5.1\" style=\"font-size:90%;\">Train</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r\" id=\"A7.T10.1.2.2.6\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.6.1\" style=\"font-size:90%;\">Infer</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"A7.T10.1.2.2.7\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.7.1\" style=\"font-size:90%;\">Train</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r\" id=\"A7.T10.1.2.2.8\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.8.1\" style=\"font-size:90%;\">Infer</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"A7.T10.1.2.2.9\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.9.1\" style=\"font-size:90%;\">Train</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column\" id=\"A7.T10.1.2.2.10\"><span class=\"ltx_text\" id=\"A7.T10.1.2.2.10.1\" style=\"font-size:90%;\">Infer</span></th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"A7.T10.1.3.1\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r ltx_border_t\" id=\"A7.T10.1.3.1.1\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.1.1\" style=\"font-size:90%;\">ConvLSTM</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T10.1.3.1.2\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.2.1\" style=\"font-size:90%;\">5.41</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T10.1.3.1.3\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.3.1\" style=\"font-size:90%;\">1.12</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T10.1.3.1.4\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.4.1\" style=\"font-size:90%;\">21.41</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T10.1.3.1.5\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.5.1\" style=\"font-size:90%;\">3.94</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T10.1.3.1.6\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.6.1\" style=\"font-size:90%;\">7.05</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T10.1.3.1.7\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.7.1\" style=\"font-size:90%;\">0.86</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T10.1.3.1.8\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.3.1.8.1\" style=\"font-size:90%;\">8.68</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T10.1.3.1.9\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.9.1\" style=\"font-size:90%;\">3.22</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T10.1.3.1.10\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.10.1\" style=\"font-size:90%;\">4.39</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T10.1.3.1.11\"><span class=\"ltx_text\" id=\"A7.T10.1.3.1.11.1\" style=\"font-size:90%;\">0.92</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.4.2\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"A7.T10.1.4.2.1\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.1.1\" style=\"font-size:90%;\">Dil-ResNet</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.4.2.2\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.2.1\" style=\"font-size:90%;\">6.64</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.4.2.3\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.3.1\" style=\"font-size:90%;\">1.73</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.4.2.4\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.4.1\" style=\"font-size:90%;\">27.06</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.4.2.5\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.5.1\" style=\"font-size:90%;\">4.06</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.4.2.6\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.6.1\" style=\"font-size:90%;\">9.96</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.4.2.7\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.7.1\" style=\"font-size:90%;\">1.19</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.4.2.8\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.8.1\" style=\"font-size:90%;\">10.90</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.4.2.9\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.9.1\" style=\"font-size:90%;\">3.66</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.4.2.10\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.10.1\" style=\"font-size:90%;\">6.34</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.4.2.11\"><span class=\"ltx_text\" id=\"A7.T10.1.4.2.11.1\" style=\"font-size:90%;\">1.03</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.5.3\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"A7.T10.1.5.3.1\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.1.1\" style=\"font-size:90%;\">time-FNO2D</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.5.3.2\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.2.1\" style=\"font-size:90%;\">4.87</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.5.3.3\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.3.1\" style=\"font-size:90%;\">1.46</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.5.3.4\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.4.1\" style=\"font-size:90%;\">8.94</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.5.3.5\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.5.1\" style=\"font-size:90%;\">1.95</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.5.3.6\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.6.1\" style=\"font-size:90%;\">5.16</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.5.3.7\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.7.1\" style=\"font-size:90%;\">0.79</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.5.3.8\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.8.1\" style=\"font-size:90%;\">10.41</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.5.3.9\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.9.1\" style=\"font-size:90%;\">2.11</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.5.3.10\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.5.3.10.1\" style=\"font-size:90%;\">3.35</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.5.3.11\"><span class=\"ltx_text\" id=\"A7.T10.1.5.3.11.1\" style=\"font-size:90%;\">0.69</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.6.4\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"A7.T10.1.6.4.1\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.1.1\" style=\"font-size:90%;\">MIONet</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.6.4.2\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.2.1\" style=\"font-size:90%;\">5.69</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.6.4.3\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.3.1\" style=\"font-size:90%;\">1.58</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.6.4.4\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.4.1\" style=\"font-size:90%;\">8.69</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.6.4.5\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.5.1\" style=\"font-size:90%;\">2.03</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.6.4.6\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.6.1\" style=\"font-size:90%;\">5.05</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.6.4.7\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.7.1\" style=\"font-size:90%;\">0.89</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.6.4.8\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.8.1\" style=\"font-size:90%;\">10.54</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.6.4.9\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.9.1\" style=\"font-size:90%;\">3.02</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.6.4.10\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.6.4.10.1\" style=\"font-size:90%;\">4.03</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.6.4.11\"><span class=\"ltx_text\" id=\"A7.T10.1.6.4.11.1\" style=\"font-size:90%;\">0.76</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.7.5\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"A7.T10.1.7.5.1\"><span class=\"ltx_text\" id=\"A7.T10.1.7.5.1.1\" style=\"font-size:90%;\">U-FNet</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.7.5.2\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.7.5.2.1\" style=\"font-size:90%;\">3.64</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.7.5.3\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.7.5.3.1\" style=\"font-size:90%;\">0.52</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.7.5.4\"><span class=\"ltx_text\" id=\"A7.T10.1.7.5.4.1\" style=\"font-size:90%;\">14.56</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.7.5.5\"><span class=\"ltx_text\" id=\"A7.T10.1.7.5.5.1\" style=\"font-size:90%;\">1.96</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.7.5.6\"><span class=\"ltx_text\" id=\"A7.T10.1.7.5.6.1\" style=\"font-size:90%;\">6.67</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.7.5.7\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.7.5.7.1\" style=\"font-size:90%;\">0.51</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.7.5.8\"><span class=\"ltx_text\" id=\"A7.T10.1.7.5.8.1\" style=\"font-size:90%;\">10.42</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.7.5.9\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.7.5.9.1\" style=\"font-size:90%;\">1.14</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.7.5.10\"><span class=\"ltx_text\" id=\"A7.T10.1.7.5.10.1\" style=\"font-size:90%;\">6.96</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.7.5.11\"><span class=\"ltx_text\" id=\"A7.T10.1.7.5.11.1\" style=\"font-size:90%;\">0.82</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.8.6\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"A7.T10.1.8.6.1\"><span class=\"ltx_text\" id=\"A7.T10.1.8.6.1.1\" style=\"font-size:90%;\">CNO</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.8.6.2\"><span class=\"ltx_text\" id=\"A7.T10.1.8.6.2.1\" style=\"font-size:90%;\">4.02</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.8.6.3\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.8.6.3.1\" style=\"font-size:90%;\">0.60</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.8.6.4\"><span class=\"ltx_text\" id=\"A7.T10.1.8.6.4.1\" style=\"font-size:90%;\">15.72</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.8.6.5\"><span class=\"ltx_text\" id=\"A7.T10.1.8.6.5.1\" style=\"font-size:90%;\">2.28</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.8.6.6\"><span class=\"ltx_text\" id=\"A7.T10.1.8.6.6.1\" style=\"font-size:90%;\">4.92</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.8.6.7\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.8.6.7.1\" style=\"font-size:90%;\">0.44</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.8.6.8\"><span class=\"ltx_text\" id=\"A7.T10.1.8.6.8.1\" style=\"font-size:90%;\">11.08</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.8.6.9\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.8.6.9.1\" style=\"font-size:90%;\">1.12</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.8.6.10\"><span class=\"ltx_text\" id=\"A7.T10.1.8.6.10.1\" style=\"font-size:90%;\">5.90</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.8.6.11\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.8.6.11.1\" style=\"font-size:90%;\">0.68</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.9.7\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"A7.T10.1.9.7.1\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.1.1\" style=\"font-size:90%;\">PeRCNN</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.9.7.2\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.2.1\" style=\"font-size:90%;\">5.02</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.9.7.3\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.3.1\" style=\"font-size:90%;\">1.72</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.9.7.4\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.9.7.4.1\" style=\"font-size:90%;\">5.73</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.9.7.5\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.9.7.5.1\" style=\"font-size:90%;\">1.47</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.9.7.6\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.6.1\" style=\"font-size:90%;\">6.53</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.9.7.7\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.7.1\" style=\"font-size:90%;\">0.84</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.9.7.8\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.8.1\" style=\"font-size:90%;\">17.44</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.9.7.9\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.9.1\" style=\"font-size:90%;\">4.08</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.9.7.10\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.10.1\" style=\"font-size:90%;\">4.24</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.9.7.11\"><span class=\"ltx_text\" id=\"A7.T10.1.9.7.11.1\" style=\"font-size:90%;\">0.82</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.10.8\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_r\" id=\"A7.T10.1.10.8.1\"><span class=\"ltx_text\" id=\"A7.T10.1.10.8.1.1\" style=\"font-size:90%;\">PPNN</span></th>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.10.8.2\"><span class=\"ltx_text\" id=\"A7.T10.1.10.8.2.1\" style=\"font-size:90%;\">5.07</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.10.8.3\"><span class=\"ltx_text\" id=\"A7.T10.1.10.8.3.1\" style=\"font-size:90%;\">0.96</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.10.8.4\"><span class=\"ltx_text\" id=\"A7.T10.1.10.8.4.1\" style=\"font-size:90%;\">8.88</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.10.8.5\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.10.8.5.1\" style=\"font-size:90%;\">1.19</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.10.8.6\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.10.8.6.1\" style=\"font-size:90%;\">4.87</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.10.8.7\"><span class=\"ltx_text\" id=\"A7.T10.1.10.8.7.1\" style=\"font-size:90%;\">0.91</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.10.8.8\"><span class=\"ltx_text\" id=\"A7.T10.1.10.8.8.1\" style=\"font-size:90%;\">15.58</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r\" id=\"A7.T10.1.10.8.9\"><span class=\"ltx_text\" id=\"A7.T10.1.10.8.9.1\" style=\"font-size:90%;\">3.44</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.10.8.10\"><span class=\"ltx_text\" id=\"A7.T10.1.10.8.10.1\" style=\"font-size:90%;\">8.08</span></td>\n<td class=\"ltx_td ltx_align_center\" id=\"A7.T10.1.10.8.11\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.10.8.11.1\" style=\"font-size:90%;\">0.64</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T10.1.11.9\">\n<th class=\"ltx_td ltx_align_left ltx_th ltx_th_row ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T10.1.11.9.1\"><span class=\"ltx_text\" id=\"A7.T10.1.11.9.1.1\" style=\"font-size:90%;\">PAPM</span></th>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A7.T10.1.11.9.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.11.9.2.1\" style=\"font-size:90%;\">3.44</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T10.1.11.9.3\"><span class=\"ltx_text\" id=\"A7.T10.1.11.9.3.1\" style=\"font-size:90%;\">0.93</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A7.T10.1.11.9.4\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.11.9.4.1\" style=\"font-size:90%;\">8.62</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T10.1.11.9.5\"><span class=\"ltx_text\" id=\"A7.T10.1.11.9.5.1\" style=\"font-size:90%;\">2.07</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A7.T10.1.11.9.6\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T10.1.11.9.6.1\" style=\"font-size:90%;\">3.70</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T10.1.11.9.7\"><span class=\"ltx_text\" id=\"A7.T10.1.11.9.7.1\" style=\"font-size:90%;\">1.27</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A7.T10.1.11.9.8\"><span class=\"ltx_text ltx_framed ltx_framed_underline\" id=\"A7.T10.1.11.9.8.1\" style=\"font-size:90%;\">8.91</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T10.1.11.9.9\"><span class=\"ltx_text\" id=\"A7.T10.1.11.9.9.1\" style=\"font-size:90%;\">2.94</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A7.T10.1.11.9.10\"><span class=\"ltx_text\" id=\"A7.T10.1.11.9.10.1\" style=\"font-size:90%;\">5.13</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A7.T10.1.11.9.11\"><span class=\"ltx_text\" id=\"A7.T10.1.11.9.11.1\" style=\"font-size:90%;\">0.88</span></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 10: Training and inference time cost (epoch/second) of different baselines."
        },
        "10": {
            "table_html": "<figure class=\"ltx_table\" id=\"A7.T11\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 11: </span>Performance comparison for different configurations.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_guessed_headers ltx_align_middle\" id=\"A7.T11.1\">\n<thead class=\"ltx_thead\">\n<tr class=\"ltx_tr\" id=\"A7.T11.1.1.1\">\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" id=\"A7.T11.1.1.1.1\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T11.1.1.1.1.1\">config</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_r ltx_border_tt\" id=\"A7.T11.1.1.1.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T11.1.1.1.2.1\">Burgers2d</span></th>\n<th class=\"ltx_td ltx_align_center ltx_th ltx_th_column ltx_border_tt\" id=\"A7.T11.1.1.1.3\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T11.1.1.1.3.1\">RD2d</span></th>\n</tr>\n</thead>\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"A7.T11.1.2.1\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T11.1.2.1.1\">fixed</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T11.1.2.1.2\">0.082</td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T11.1.2.1.3\">0.049</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T11.1.3.2\">\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T11.1.3.2.1\">trainable</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T11.1.3.2.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T11.1.3.2.2.1\">0.039</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A7.T11.1.3.2.3\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T11.1.3.2.3.1\">0.018</span></td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 11: Performance comparison for different configurations."
        },
        "11": {
            "table_html": "<figure class=\"ltx_table\" id=\"A7.T12\">\n<figcaption class=\"ltx_caption ltx_centering\"><span class=\"ltx_tag ltx_tag_table\">Table 12: </span>Performance metrics across different datasets with various modifications.</figcaption>\n<table class=\"ltx_tabular ltx_centering ltx_align_middle\" id=\"A7.T12.1\">\n<tbody class=\"ltx_tbody\">\n<tr class=\"ltx_tr\" id=\"A7.T12.1.1\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A7.T12.1.1.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.1.2.1\">Datasets</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A7.T12.1.1.1\"></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A7.T12.1.1.3\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.1.3.1\">no_DF</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A7.T12.1.1.4\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.1.4.1\">no_CF</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A7.T12.1.1.5\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.1.5.1\">no_Phy</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A7.T12.1.1.6\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.1.6.1\">no_IST</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_tt\" id=\"A7.T12.1.1.7\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.1.7.1\">no_EST</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_tt\" id=\"A7.T12.1.1.8\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.1.8.1\">no_BCs</span></td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T12.1.2.1\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.2.1.1\">Burgers2d</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.2.1.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.2.1.2.1\">0.039</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.2.1.3\">0.067</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.2.1.4\">0.062</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.2.1.5\">0.149</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.2.1.6\">0.174</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.2.1.7\">-</td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T12.1.2.1.8\">0.068</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T12.1.3.2\">\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.3.2.1\">RD2d</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.3.2.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.3.2.2.1\">0.018</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.3.2.3\">0.102</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.3.2.4\">-</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.3.2.5\">0.102</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.3.2.6\">0.281</td>\n<td class=\"ltx_td ltx_align_center ltx_border_r ltx_border_t\" id=\"A7.T12.1.3.2.7\">-</td>\n<td class=\"ltx_td ltx_align_center ltx_border_t\" id=\"A7.T12.1.3.2.8\">0.083</td>\n</tr>\n<tr class=\"ltx_tr\" id=\"A7.T12.1.4.3\">\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T12.1.4.3.1\">NSM2d</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T12.1.4.3.2\"><span class=\"ltx_text ltx_font_bold\" id=\"A7.T12.1.4.3.2.1\">0.189</span></td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T12.1.4.3.3\">0.273</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T12.1.4.3.4\">0.212</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T12.1.4.3.5\">0.299</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T12.1.4.3.6\">0.392</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_r ltx_border_t\" id=\"A7.T12.1.4.3.7\">0.311</td>\n<td class=\"ltx_td ltx_align_center ltx_border_bb ltx_border_t\" id=\"A7.T12.1.4.3.8\">0.201</td>\n</tr>\n</tbody>\n</table>\n</figure>",
            "capture": "Table 12: Performance metrics across different datasets with various modifications."
        }
    },
    "image_paths": {
        "1": {
            "figure_path": "2407.05232v1_figure_1.png",
            "caption": "Figure 1: Overview of the PAPM s pipeline. The model takes the multiple conditions of process systems for time extrapolation and outputs solutions at an arbitrary time point. The core is the temporal-spatial stepping module (TSSM) (𝑼t=i→𝑼t=i+1)→superscript𝑼𝑡𝑖superscript𝑼𝑡𝑖1(\\bm{U}^{t=i}\\rightarrow\\bm{U}^{t=i+1})( bold_italic_U start_POSTSUPERSCRIPT italic_t = italic_i end_POSTSUPERSCRIPT → bold_italic_U start_POSTSUPERSCRIPT italic_t = italic_i + 1 end_POSTSUPERSCRIPT ). Spatially, a structure-preserved operation aligns with the specific equation characteristics of different process systems. Temporally, it utilizes a continuous-time modeling framework through an ODE solver.",
            "url": "http://arxiv.org/html/2407.05232v1/x1.png"
        },
        "2": {
            "figure_path": "2407.05232v1_figure_2.png",
            "caption": "Figure 2: A detailed structure of PAPM at time t𝑡titalic_t.\nHere, v𝑣vitalic_v, hOsubscriptℎ𝑂h_{O}italic_h start_POSTSUBSCRIPT italic_O end_POSTSUBSCRIPT, and hFsubscriptℎ𝐹h_{F}italic_h start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT are the corresponding unknown mapping, and the neural networks are needed for learning. We propose a temporal-spatial stepping module (TSSM) for DF, CF, IST, and EST in section 4.2, which aligns with the distinct equation characteristics of different process systems.",
            "url": "http://arxiv.org/html/2407.05232v1/x2.png"
        },
        "3": {
            "figure_path": "2407.05232v1_figure_3.png",
            "caption": "Figure 3: Left: Pre-defined convolutional kernels, where fixed and trainable correspond to the matrices at the top and bottom, respectively. The bottom kernels approximate the unidirectional convection (upwind scheme) and directionless diffusion (central scheme). Symbols ∗bold-∗\\bm{\\ast}bold_∗ and ⋆bold-⋆\\bm{\\star}bold_⋆ indicate trainable parameters corresponding to the upper triangular and symmetric matrices.\nMid: Structure-preserved localized operator.\nRight: Structure-preserved spatial operator.",
            "url": "http://arxiv.org/html/2407.05232v1/x3.png"
        },
        "4": {
            "figure_path": "2407.05232v1_figure_4.png",
            "caption": "Figure 4: ϵitalic-ϵ\\epsilonitalic_ϵ (Eq. 4) of predicted each time step on Burgers2d, where in is the same as the training, out is the time extrapolation.",
            "url": "http://arxiv.org/html/2407.05232v1/x4.png"
        },
        "5": {
            "figure_path": "2407.05232v1_figure_5.png",
            "caption": "Figure 5: Predicted flow velocity (|𝒖|2subscript𝒖2|\\bm{u}|_{2}| bold_italic_u | start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) snapshots by FNO, PPNN, and PAPM (Ours) vs. Ground Truth (GT) on NSM2d dataset in T Ext. task.",
            "url": "http://arxiv.org/html/2407.05232v1/x5.png"
        },
        "6": {
            "figure_path": "2407.05232v1_figure_6.png",
            "caption": "Figure 6: ϵitalic-ϵ\\epsilonitalic_ϵ (Eq. 4) comparison by the leading method, PPNN, Dil-Resnet, PAPM (Ours) on the RD2d dataset. Left: varying amount of training data. Right: varying train step size.",
            "url": "http://arxiv.org/html/2407.05232v1/x6.png"
        },
        "7": {
            "figure_path": "2407.05232v1_figure_7.png",
            "caption": "Figure 7: Predicted flow velocity (‖𝒖‖2subscriptnorm𝒖2\\|\\bm{u}\\|_{2}∥ bold_italic_u ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) snapshots by PPNN, and PAPM (Ours) vs. Ground Truth (GT) on Burgers2d dataset in T Ext. task.",
            "url": "http://arxiv.org/html/2407.05232v1/x7.png"
        },
        "8": {
            "figure_path": "2407.05232v1_figure_8.png",
            "caption": "Figure 8: Predicted flow velocity (‖𝒖‖2subscriptnorm𝒖2\\|\\bm{u}\\|_{2}∥ bold_italic_u ∥ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT) snapshots by PPNN, and PAPM (Ours) vs. Ground Truth (GT) on RD2d dataset in T Ext. task.",
            "url": "http://arxiv.org/html/2407.05232v1/x8.png"
        },
        "9": {
            "figure_path": "2407.05232v1_figure_9.png",
            "caption": "Figure 9: Different terms on the Burgers2d dataset in T Ext. task.",
            "url": "http://arxiv.org/html/2407.05232v1/x9.png"
        },
        "10": {
            "figure_path": "2407.05232v1_figure_10.png",
            "caption": "Figure 10: Different terms on the RD2d dataset in T Ext. task.",
            "url": "http://arxiv.org/html/2407.05232v1/x10.png"
        }
    },
    "validation": true,
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                "author": "Stachenfeld, K., Fielding, D. B., Kochkov, D., Cranmer, M., Pfaff, T., Godwin,\nJ., Cui, C., Ho, S., Battaglia, P., and Sanchez-Gonzalez, A.",
                "venue": "arXiv preprint arXiv:2112.15275, 2021.",
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        {
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                "title": "A neural pde solver with temporal stencil modeling.",
                "author": "Sun, Z., Yang, Y., and Yoo, S.",
                "venue": "In International Conference on Machine Learning, pp. 33135–33155. PMLR, 2023.",
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                "title": "Pdebench: An extensive benchmark for scientific machine learning.",
                "author": "Takamoto, M., Praditia, T., Leiteritz, R., MacKinlay, D., Alesiani, F.,\nPflüger, D., and Niepert, M.",
                "venue": "Advances in Neural Information Processing Systems,\n35:1596–1611, 2022.",
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        {
            "45": {
                "title": "Solver-in-the-loop: Learning from differentiable physics to interact\nwith iterative pde-solvers.",
                "author": "Um, K., Brand, R., Fei, Y. R., Holl, P., and Thuerey, N.",
                "venue": "Advances in Neural Information Processing Systems,\n33:6111–6122, 2020.",
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        },
        {
            "46": {
                "title": "Learning the solution operator of parametric partial differential\nequations with physics-informed deeponets.",
                "author": "Wang, S., Wang, H., and Perdikaris, P.",
                "venue": "Science advances, 7(40):eabi8605, 2021.",
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        }
    ],
    "url": "http://arxiv.org/html/2407.05232v1",
    "Chart": [
        "4",
        "6"
    ],
    "Diagram": [
        "1",
        "2"
    ],
    "extract_figure_info": {
        "1": {
            "1": [
                "From molecular dynamics to turbulent flows, process systems are essential in numerous scientific and engineering domains (Cameron & Hangos, 2001 ).",
                "Computational modeling and simulation are crucial for understanding their complex temporal-spatial dynamics, enabling precise predictions and informed decisions across various fields.",
                "However, these valuable insights are provided by traditional numerical simulations, which are often computationally intensive, especially in scenarios necessitating frequent model queries like reverse engineering forward simulation (Dijkstra & Luijten, 2021 ) and optimization design (Gramacy, 2020 ).",
                "Recent advancements in data-driven methods have paved the way to tackle computational challenges more effectively (Lu et al., 2019 ; Li et al., 2020 ; Kochkov et al., 2021 ; Stachenfeld et al., 2021 ; Hao et al., 2023b ).",
                "As shown in Figure 1 (left), these methods input multiple conditions of process systems to output time-dependent solutions, serving as the proxy model for process systems.",
                "Through adopting a supervised learning-from-data paradigm, remarkable advancements in calculation speeds have been achieved—several orders of magnitude faster than traditional numerical simulations.",
                "This has led to significant savings in computational costs.",
                "While data-driven methods are powerful, they face two primary limitations: 1) a dependence on extensive labeled datasets, which contrasts sharply with the high computational costs of numerical simulations, and 2) a presumption of train-test uniformity that leads to poor generalization, especially in out-of-sample scenarios like time extrapolation.",
                "This poor generalization arises from an overemphasis on the inductive biases of network architectures based on labels, rather than a strict adherence to fundamental physical laws (Li et al., 2021 ; Brandstetter et al., 2022 ).",
                "To ameliorate high training costs and limited generalizability, a more promising strategy, known as physics-informed deep learning (PIDL), involves prior knowledge, such as fundamental physical laws, into neural networks (NNs).",
                "This integration enhances the sample efficiency and generalizability of NNs, proving particularly vital in scenarios with limited labeled data (Li et al., 2021 ; Hao et al., 2022 ; Meng et al., 2022 ; Cuomo et al., 2022 ).",
                "One of the typical methods considers complete physical laws as the loss function of NNs to construct proxy models, such as PINNs (Raissi et al., 2019 ) for specific conditions, PI-DeepONet (Wang et al., 2021 ), and PINO (Li et al., 2021 ) for multiple sets of conditions.",
                "However, the real-world applicability of these methods is limited by an incomplete understanding of the underlying physics of specific process systems, making it challenging to derive complete physics laws.",
                "Another common approach, termed as “physics-aware” models, offers a promising solution for this scenario by incorporating partial prior knowledge alongside a small amount of labeled data.",
                "Considering the relationship between equations and their numerical schemes, the physics-aware methods convert partial prior physics laws into the corresponding numerical schemes, and embeddings them into the network structure (Long et al., 2018 ; Seo et al., 2020 ; Huang et al., 2023b ; Akhare et al., 2023 ; Rao et al., 2023 ; Huang et al., 2023a ; Kochkov et al., 2023 ; Pestourie et al., 2023 ; Liu et al., 2024 ).",
                "With these partial prior physics laws, physics-aware models only need a small number of labels to obtain excellent out-of-sample generalization performance.",
                "However, these methods focus primarily on spatial derivatives, and often neglect integral aspects such as conservation laws and constitutive relations.",
                "Consequently, they do not fully leverage prior physics knowledge, leading to unreliable solutions.",
                "Besides, these methods are generally tailored for a specific process system with limited universality.",
                "Recognizing that process system modeling often requires the incorporation of conservation relations grounded in diffusion, convection, and source flows, this work aims to integrate this general physics law into the network architecture.",
                "By reinforcing the inductive biases in this manner, we can achieve better out-of-sample generalization.",
                "Additionally, as different process systems correspond to specific conservation or constitutive equations based on inherent system characteristics (Cameron & Hangos, 2001 ; Takamoto et al., 2022 ; Hao et al., 2023a ), it is beneficial to identify both similarities and differences among process systems.",
                "Such an approach can offer a general temporal-spatial stepping module to combine various process systems flexibly.",
                "As illustrated in Figure 1 , we propose a physics-aware proxy model (PAPM) for process systems, which incorporates multiple conditions to output solutions at arbitrary time points.",
                "PAPM fully utilizes partial prior knowledge, including multiple conditions, and the general form of conservation relations, alongside a small amount of label data, to model the dynamics of systems through the proposed temporal-spatial stepping module .",
                "Notably, PAPM leverages the direction of data flow based on this general form, a distinction often overlooked by other physics-aware methods.",
                "Furthermore, PAPM focuses on out-of-sample scenarios, such as time extrapolation, aligning with the capabilities of alternative methods.",
                "The core contributions of this work are: The proposal of PAPM, a versatile physics-aware architecture design that fully incorporates partial prior knowledge such as multiple conditions, and the general form of conservation relations.",
                "This design proves to be superior in both training efficiency and out-of-sample generalizability.",
                "The introduction of a holistic temporal-spatial stepping module (TSSM) for flexible adaptation across various process systems.",
                "It aligns with the distinct equation characteristics of different process systems by employing stepping schemes via temporal and spatial operations, whether in physical or spectral space.",
                "A systematic evaluation of state-of-the-art pure data-driven models alongside physics-aware models, spanning five two-dimensional non-trivial benchmarks.",
                "Notably, PAPM achieved an average absolute performance boost of 6.7% with fewer FLOPs and only 1%-10% of the parameters compared to alternative methods."
            ]
        },
        "2": {
            "3": [
                "This section presents the foundational description of process systems, known as the process model.",
                "Additionally, further clarification is provided on the specific problem in this work.",
                "Process Model.",
                "Pivotal in engineering disciplines, process models represent and predict the dynamics of diverse process systems.",
                "This model s mathematical foundation relies on two essential sets of equations: conservation equations, governing the dynamic behavior of fundamental quantities, and constitutive equations, which describe the interactions among different variables.",
                "Further details are provided in Appendix B .",
                "1 and Eq.",
                "2 represent the universal conservation and constitutive equations, respectively.",
                "where is the physical quantity, denoting the system s state.",
                "1 comprises four essential elements: the diffusion flows , convection flows , the internal source , and the external source .",
                "In Eq.",
                "2 , denotes the velocity of the physical quantity being transmitted, is the diffusion coefficient, denotes the coefficients, and is the input of the external source term.",
                "Here, the corresponding linear or nonlinear mapping is the , , and .",
                "The structure of PAPM is depicted in Figure 2 at time .",
                "Our goal is to use partial prior knowledge (the general form of Eq.",
                "1 ) and a small amount of label data to establish a proxy model, which takes these various input conditions and outputs system time-dependent solutions (), as shown in Fig 1 .",
                "Notably, the general form of Eq.",
                "1 is only known, which is also the data flow, while the specific item is unknown, such as the mappings of , , and , aligning with most real-world scenarios (Karlbauer et al., 2022 ; Huang et al., 2023a ; Liu et al., 2024 ).",
                "Problem Formulation.",
                "Under different initial and boundary conditions, external sources, and coefficients, the following -step trajectory should be predicted.",
                "Moreover, due to the high cost of generating labeled data, we focused on out-of-sample scenarios, e.g.",
                "time extrapolation, where the training dataset only contains the subsequent -step trajectory, and .",
                "Formally, the dataset , where .",
                "Here contains a set of inputs, that is, initial condition , boundary conditions, such as Robin conditions, external sources , and coefficients .",
                "is the following trajectory, and the mapping is our goal to learn.",
                "Each is a vector, which consists of physical quantities, such as velocity, vorticity, pressure, and temperature.",
                "We discretize each quantity on the grid .",
                "In a nutshell, for modeling this operator , we use a parameterized neural network with parameters , inputs , and outputs , where .",
                "Our goal is to minimize the relative error loss between the prediction and real data in the training dataset as, where is the training time-step size, and is the size of the training dataset.",
                "is the mean relative error loss at time of index .",
                "is a set of the network parameters and is the parameter space."
            ],
            "4.1": [
                "Aligning with the general form of Eq.",
                "1 and Eq.",
                "2 , there are four elements corresponding to Diffusive Flows (DF), Convective Flows (CF), Internal Source Term (IST), and External Source Term (EST) in PAPM s structure diagram, as illustrated in Figure 2 .",
                "The versatile general structure of PAPM could enable it to work effectively across different process systems.",
                "The input contains a set of inputs, that is, coefficient , initial state , external source input , and boundary conditions, which are multiple conditions of process systems.",
                "The sequence of embedding this prior knowledge unfolds as follows: 1) Embedding BCs.",
                "Using the given boundary conditions, the physical quantity is updated, yielding .",
                "A padding strategy is employed to integrate four different boundary conditions in four different directions into PAPM.",
                "Further details are provided in Appendix C .",
                "2) Diffusive Flows (DF).",
                "Using and coefficients , we represent the directionless diffusive flow.",
                "The diffusion flow and its gradient are obtained as and via a symmetric gradient operator, respectively.",
                "3) Convective Flows (CF).",
                "The pattern is derived from .",
                "Once is determined, its sign indicates the direction of the flows, enabling computation of and through a directional gradient operator.",
                "4) Internal Source Term (IST) & External Source Term (EST).",
                "Generally, IST and EST present a complex interplay between physical quantities and external inputs .",
                "Often, this part in real systems doesn t have a clear physics-based relation, prompting the use of NNs to capture this intricate relationship.",
                "5) ODE solver.",
                "From DF, CF, IST, and EST, the dynamic are derived.",
                "By doing so, the Eq.",
                "1 can be reduced to an ordinary differential equation (ODE), and the ODE solver is used to approximate the evolving state as .",
                "Figure 2 above illustrates the data flow at time .",
                "Then, during the training or inference phase, PAPM performs autoregressive predictions as , where , with during training and during inference.",
                "In short, PAPM takes different conditions, including initial conditions, boundary conditions, external sources, and coefficients, and interactively propagates the dynamics of process systems forward using five distinct components.",
                "The purpose of such a structured design is to reinforce the inductive biases concerning strict physical laws."
            ]
        },
        "3": {
            "4.2": [
                "Due to the diversity of process systems, we develop a holistic Temporal-Spatial Stepping Module (TSSM) to align with the unique characteristics of different process systems, which forms the specific network structure for each component in PAPM.",
                "As shown in Tab.",
                "1 , TSSM is categorized into three types based on structures of process systems, where each type decomposes temporal and spatial components, i.e., structure-preserved localized operator, spectral operator, and hybrid operator.",
                "Notably, all three approaches can employ a common temporal operation through ODE solvers.",
                "Temporal Operation.",
                "After obtaining the dynamic state derivative, , the subsequent state can be computed through numerical integration over different time spans.",
                "Due to the numerical instability associated with first-order explicit methods like the Euler method (Gottlieb et al., 2001 ; Fatunla, 2014 ), we adopt the neural ordinary differential equation approach (Neural ODE (Chen et al., 2018 )), which employs the Runge–Kutta time-stepping strategy to enhance stability.",
                "The computed state is then recursively fed back into the network as the input for the subsequent time step, continuing this process until the final time step is reached.",
                "Structure-preserved Localized Operator.",
                "For systems with explicit structures, such as the Burgers and RD equations, typified by expressions like , convolutional kernels in the physical space are employed to capture system dynamics.",
                "We opt for either fixed or trainable kernels, illustrated in Figure 3 (Left), depending on our understanding of the system.",
                "Specifically, the fixed one is based on the predefined convolution kernel derived from difference schemes, and further details are provided in Appendix D.2.1 .",
                "Moreover, the trainable version tailors its design to essential features of convection (upper or lower triangular) and diffusion (symmetric).",
                "Once set, the localized operator is depicted in Figure 3 (Mid).",
                "We could represent nonlinear terms in DF and CF using predefined convolution kernels alongside partially known physics (the general form of Eq.",
                "1 and Eq.",
                "Any unknown specific terms, such as source, are then addressed through a shallow ResNet block.",
                "Structure-preserved Spectral Operator.",
                "For systems with implicit structures, such as the Navier-Stokes Equation in vorticity form, represented like , we adopt a sequential process, as shown in Figure 3 (Right).",
                "Recognizing the implicit linkage between velocity and vorticity, is initially processed to extract the flow function, subsequently leading to the velocity derivation.",
                "The spectral space dimensions ( and ) and spectral quantity (denoted as ), are obtained by leveraging the FFT.",
                "Associating and with , differential operators like are represented via E-Conv (e.g., element-wise product), and then mapped back to the physical space using IFFT.",
                "Doing so can represent the nonlinear terms in DF and CF via simple computations such as addition and multiplication in the spectral domain.",
                "Moreover, the spectral convolution (S-Conv) fo FNO (Li et al., 2020 ) is introduced to learn unknown components.",
                "This process can be further detailed in Appendix D.2.2 .",
                "Structure-preserved Hybrid Operator.",
                "For systems with a hybrid structure, such as the Navier-Stokes Equation in general form (e.g., ), given the implicit interrelation between pressure and velocity , a combination of the method above is employed.",
                "Explicit constituents, such as and , are addressed through the localized operator.",
                "Meanwhile, implicit relations are resolved similarly by the spectral operator.",
                "For unknown components, either of the two operators can be engaged.",
                "We generally favor the localized operator as it allows direct operations without requiring transitions between different spaces."
            ]
        },
        "4": {
            "5.2": [
                "Performance Comparisons.",
                "3 and Tab.",
                "3 present the primary experimental outcomes, the number of trainable parameters (), and computational cost (FLOPs) for each baseline across datasets.",
                "Here, Bold and Underline indicate the best and second best performance, respectively.",
                "Notably, lower results mean better performance because the metric is the mean relative error.",
                "Our observations from the results are as follows: Firstly, PAPM exhibits the most balanced trade-off between performance, parameter count, and computational cost among all methods evaluated, from explicit structures (Burgers2d, RD2d) to implicit (NS2d) and more complex hybrid structures (Lid2d, NSM2d).",
                "Remarkably, even though PAPM requires significantly fewer FLOPs and only 1% of the parameters employed by the prior leading method, PPNN, it still outperforms it by a large margin.",
                "In a nutshell, our model enhances the performance by an average of 6.7% over nine tasks, affirming PAPM as a versatile and efficient framework suitable for diverse process systems.",
                "Secondly, PAPM s structured treatment of system inputs and states leads to a remarkable 9.6% performance boost in three coefficient-extrapolation tasks.",
                "This highlights its superior generalization capability in out-of-sample scenarios.",
                "Unlike models like PPNN, which directly use system-specific inputs, PAPM integrates coefficient data more intricately within conservation and constitutive relations, boosting its adaptability to varying coefficients.",
                "Thirdly, data-driven methods are less effective than physics-aware methods like PPNN and our PAPM in time extrapolation tasks, where incorporating prior physical knowledge through structured network design enhances a model s generalization ability.",
                "Notably, PeRCNN uses convolution to approximate nonlinear terms, but experimental results suggest limited performance.",
                "Further details are available in Appendix F.2 .",
                "Visualization.",
                "Figure 4 showcases the stepwise relative error of PAPM during the extrapolation process in the test dataset, using Burgers2d s C Int.",
                "as a representative example.",
                "Compared to the two best-performing baselines, our model (depicted by the red line) exhibits superior performance throughout the extrapolation process, with the least error accumulation.",
                "Turning our attention to the more challenging NSM2d dataset, Figure 5 presents the results across five extrapolation time slices.",
                "While FNO demonstrates commendable accuracy within the training domain (), its performance falters significantly outside of it ().",
                "On the other hand, physics-aware methods (PPNN), and PAPM in particular, consistently capture the evolving patterns with a greater degree of robustness.",
                "Notably, our method emerges as a leader in terms of precision.",
                "Additional visual results can be found in Appendix G.1 ."
            ]
        },
        "5": {
            "5.2": [
                "Performance Comparisons.",
                "3 and Tab.",
                "3 present the primary experimental outcomes, the number of trainable parameters (), and computational cost (FLOPs) for each baseline across datasets.",
                "Here, Bold and Underline indicate the best and second best performance, respectively.",
                "Notably, lower results mean better performance because the metric is the mean relative error.",
                "Our observations from the results are as follows: Firstly, PAPM exhibits the most balanced trade-off between performance, parameter count, and computational cost among all methods evaluated, from explicit structures (Burgers2d, RD2d) to implicit (NS2d) and more complex hybrid structures (Lid2d, NSM2d).",
                "Remarkably, even though PAPM requires significantly fewer FLOPs and only 1% of the parameters employed by the prior leading method, PPNN, it still outperforms it by a large margin.",
                "In a nutshell, our model enhances the performance by an average of 6.7% over nine tasks, affirming PAPM as a versatile and efficient framework suitable for diverse process systems.",
                "Secondly, PAPM s structured treatment of system inputs and states leads to a remarkable 9.6% performance boost in three coefficient-extrapolation tasks.",
                "This highlights its superior generalization capability in out-of-sample scenarios.",
                "Unlike models like PPNN, which directly use system-specific inputs, PAPM integrates coefficient data more intricately within conservation and constitutive relations, boosting its adaptability to varying coefficients.",
                "Thirdly, data-driven methods are less effective than physics-aware methods like PPNN and our PAPM in time extrapolation tasks, where incorporating prior physical knowledge through structured network design enhances a model s generalization ability.",
                "Notably, PeRCNN uses convolution to approximate nonlinear terms, but experimental results suggest limited performance.",
                "Further details are available in Appendix F.2 .",
                "Visualization.",
                "Figure 4 showcases the stepwise relative error of PAPM during the extrapolation process in the test dataset, using Burgers2d s C Int.",
                "as a representative example.",
                "Compared to the two best-performing baselines, our model (depicted by the red line) exhibits superior performance throughout the extrapolation process, with the least error accumulation.",
                "Turning our attention to the more challenging NSM2d dataset, Figure 5 presents the results across five extrapolation time slices.",
                "While FNO demonstrates commendable accuracy within the training domain (), its performance falters significantly outside of it ().",
                "On the other hand, physics-aware methods (PPNN), and PAPM in particular, consistently capture the evolving patterns with a greater degree of robustness.",
                "Notably, our method emerges as a leader in terms of precision.",
                "Additional visual results can be found in Appendix G.1 ."
            ]
        },
        "6": {
            "5.3": [
                "Training and Inference Cost: Dataset generation for our work is notably resource-intensive, with inference times ranging from to seconds for public datasets, and up to seconds for those datasets we generated using COMSOL Multiphysics.",
                "In stark contrast, both baselines and PAPM register inference times between to seconds (detailed in Appendix G.2 ), achieving an improvement of 3 to 5 orders of magnitude.",
                "Notably, PAPM s time cost rivals or even surpasses baselines across different datasets.",
                "PAPM s efficiency remains competitive with other data-driven methods.",
                "Data Efficiency.",
                "Owing to PAPM s structured design, data utilization is significantly enhanced.",
                "To evaluate data efficiency, we conducted tests using RD2d as a representative example, with Dil-ResNet and PPNN symbolizing pure data-driven and physics-aware methods.",
                "The results, displayed in Figure 6 , depict PAPM s efficiency concerning data volume and label data step size in training.",
                "(1) Amount of Data: With a fixed 20% reserved for the test set, the remaining 80% of the total data is allocated to the training set.",
                "We systematically varied the training data volume, ranging from initially utilizing only 5% of the training set and progressively increasing it to the entire 100%.",
                "PAPM s relative error distinctly outperforms other baselines, especially with limited data (5%).",
                "As depicted in Figure 6 (Left), PAPM s error consistently surpasses other methods, stabilizing below 2% as the training data volume increases.",
                "(2) Time Step Size: We varied the time step size from a tenth to half of the total duration, increasing it in increments of tenths.",
                "As shown in Figure 6 (Right), PAPM demonstrates the capability for long-range time extrapolation with fewer dynamic steps.",
                "It consistently outperforms other methods, achieving superior results even with shorter training time step sizes."
            ]
        },
        "7": {},
        "8": {},
        "9": {
            "5.4": [
                "Different blocks impacts.",
                "4 displays our selection of the Burger2d dataset for ablation studies, chosen for its representation of diffusion, convection, and source terms.",
                "We defined several configurations to assess the impact of individual components.",
                "The relative error on the boundary (BC ) is introduced to highlight the importance of physics embedding further.",
                "no_DF excludes diffusion, whereas no_CF omits convection.",
                "In no_Phy, we retain only a structure with a residual connection, thereby eliminating both diffusion and convection.",
                "no_BCs setup removes the explicit embedding of boundary conditions, no_NODE replaces the Neural ODE with the Euler stepping scheme, and no_All adopts a purely data-driven approach.",
                "Additional ablation results can be found in Appendix G.3 .",
                "Key findings include the crucial roles of diffusion and convection in representing system dynamics, as evidenced in the no_DF, no_CF, and no_Phy configurations.",
                "Specifically, the no_DF configuration demonstrated the importance of integrating the viscosity coefficient with the diffusion term, with its absence leading to significant errors.",
                "The necessity of adhering to physical laws in boundary conditions was highlighted in the no_BCs, notably reducing errors on the boundary (BC ).",
                "Lastly, the no_NODE results indicate that different temporal stepping schemes significantly impact the outcomes, underscoring the effectiveness of neural ODEs in continuous-time modeling.",
                "Different blocks validations.",
                "Taking the Burgers2d and RD2d datasets as examples to demonstrate the fact that the convection/diffusion/source terms could actually learn those parts in the equations.",
                "We use high-fidelity FDM and FVM to compute the corresponding terms to obtain the detailed term for convection/diffusion/source terms.",
                "The relative error between ground truth and numerical results are and in all time steps for Burgers2d and RD2d, respectively.",
                "Thus, we can use the results obtained by the numerical methods as reference values to verify this.",
                "As shown in Tab.",
                "5 , the different terms of PAPM can be used to learn the equation s convection/diffusion/source parts.",
                "Additional visualization results can be found in Appendix G.1 (Figure 9 and Figure 10 )."
            ]
        },
        "10": {
            "5.4": [
                "Different blocks impacts.",
                "4 displays our selection of the Burger2d dataset for ablation studies, chosen for its representation of diffusion, convection, and source terms.",
                "We defined several configurations to assess the impact of individual components.",
                "The relative error on the boundary (BC ) is introduced to highlight the importance of physics embedding further.",
                "no_DF excludes diffusion, whereas no_CF omits convection.",
                "In no_Phy, we retain only a structure with a residual connection, thereby eliminating both diffusion and convection.",
                "no_BCs setup removes the explicit embedding of boundary conditions, no_NODE replaces the Neural ODE with the Euler stepping scheme, and no_All adopts a purely data-driven approach.",
                "Additional ablation results can be found in Appendix G.3 .",
                "Key findings include the crucial roles of diffusion and convection in representing system dynamics, as evidenced in the no_DF, no_CF, and no_Phy configurations.",
                "Specifically, the no_DF configuration demonstrated the importance of integrating the viscosity coefficient with the diffusion term, with its absence leading to significant errors.",
                "The necessity of adhering to physical laws in boundary conditions was highlighted in the no_BCs, notably reducing errors on the boundary (BC ).",
                "Lastly, the no_NODE results indicate that different temporal stepping schemes significantly impact the outcomes, underscoring the effectiveness of neural ODEs in continuous-time modeling.",
                "Different blocks validations.",
                "Taking the Burgers2d and RD2d datasets as examples to demonstrate the fact that the convection/diffusion/source terms could actually learn those parts in the equations.",
                "We use high-fidelity FDM and FVM to compute the corresponding terms to obtain the detailed term for convection/diffusion/source terms.",
                "The relative error between ground truth and numerical results are and in all time steps for Burgers2d and RD2d, respectively.",
                "Thus, we can use the results obtained by the numerical methods as reference values to verify this.",
                "As shown in Tab.",
                "5 , the different terms of PAPM can be used to learn the equation s convection/diffusion/source parts.",
                "Additional visualization results can be found in Appendix G.1 (Figure 9 and Figure 10 )."
            ]
        }
    },
    "extract_table_info": {
        "1": {
            "4.2": [
                "Due to the diversity of process systems, we develop a holistic Temporal-Spatial Stepping Module (TSSM) to align with the unique characteristics of different process systems, which forms the specific network structure for each component in PAPM.",
                "As shown in Tab.",
                "1 , TSSM is categorized into three types based on structures of process systems, where each type decomposes temporal and spatial components, i.e., structure-preserved localized operator, spectral operator, and hybrid operator.",
                "Notably, all three approaches can employ a common temporal operation through ODE solvers.",
                "Temporal Operation.",
                "After obtaining the dynamic state derivative, , the subsequent state can be computed through numerical integration over different time spans.",
                "Due to the numerical instability associated with first-order explicit methods like the Euler method (Gottlieb et al., 2001 ; Fatunla, 2014 ), we adopt the neural ordinary differential equation approach (Neural ODE (Chen et al., 2018 )), which employs the Runge–Kutta time-stepping strategy to enhance stability.",
                "The computed state is then recursively fed back into the network as the input for the subsequent time step, continuing this process until the final time step is reached.",
                "Structure-preserved Localized Operator.",
                "For systems with explicit structures, such as the Burgers and RD equations, typified by expressions like , convolutional kernels in the physical space are employed to capture system dynamics.",
                "We opt for either fixed or trainable kernels, illustrated in Figure 3 (Left), depending on our understanding of the system.",
                "Specifically, the fixed one is based on the predefined convolution kernel derived from difference schemes, and further details are provided in Appendix D.2.1 .",
                "Moreover, the trainable version tailors its design to essential features of convection (upper or lower triangular) and diffusion (symmetric).",
                "Once set, the localized operator is depicted in Figure 3 (Mid).",
                "We could represent nonlinear terms in DF and CF using predefined convolution kernels alongside partially known physics (the general form of Eq.",
                "1 and Eq.",
                "Any unknown specific terms, such as source, are then addressed through a shallow ResNet block.",
                "Structure-preserved Spectral Operator.",
                "For systems with implicit structures, such as the Navier-Stokes Equation in vorticity form, represented like , we adopt a sequential process, as shown in Figure 3 (Right).",
                "Recognizing the implicit linkage between velocity and vorticity, is initially processed to extract the flow function, subsequently leading to the velocity derivation.",
                "The spectral space dimensions ( and ) and spectral quantity (denoted as ), are obtained by leveraging the FFT.",
                "Associating and with , differential operators like are represented via E-Conv (e.g., element-wise product), and then mapped back to the physical space using IFFT.",
                "Doing so can represent the nonlinear terms in DF and CF via simple computations such as addition and multiplication in the spectral domain.",
                "Moreover, the spectral convolution (S-Conv) fo FNO (Li et al., 2020 ) is introduced to learn unknown components.",
                "This process can be further detailed in Appendix D.2.2 .",
                "Structure-preserved Hybrid Operator.",
                "For systems with a hybrid structure, such as the Navier-Stokes Equation in general form (e.g., ), given the implicit interrelation between pressure and velocity , a combination of the method above is employed.",
                "Explicit constituents, such as and , are addressed through the localized operator.",
                "Meanwhile, implicit relations are resolved similarly by the spectral operator.",
                "For unknown components, either of the two operators can be engaged.",
                "We generally favor the localized operator as it allows direct operations without requiring transitions between different spaces."
            ]
        },
        "2": {},
        "3": {
            "5.2": [
                "Performance Comparisons.",
                "3 and Tab.",
                "3 present the primary experimental outcomes, the number of trainable parameters (), and computational cost (FLOPs) for each baseline across datasets.",
                "Here, Bold and Underline indicate the best and second best performance, respectively.",
                "Notably, lower results mean better performance because the metric is the mean relative error.",
                "Our observations from the results are as follows: Firstly, PAPM exhibits the most balanced trade-off between performance, parameter count, and computational cost among all methods evaluated, from explicit structures (Burgers2d, RD2d) to implicit (NS2d) and more complex hybrid structures (Lid2d, NSM2d).",
                "Remarkably, even though PAPM requires significantly fewer FLOPs and only 1% of the parameters employed by the prior leading method, PPNN, it still outperforms it by a large margin.",
                "In a nutshell, our model enhances the performance by an average of 6.7% over nine tasks, affirming PAPM as a versatile and efficient framework suitable for diverse process systems.",
                "Secondly, PAPM s structured treatment of system inputs and states leads to a remarkable 9.6% performance boost in three coefficient-extrapolation tasks.",
                "This highlights its superior generalization capability in out-of-sample scenarios.",
                "Unlike models like PPNN, which directly use system-specific inputs, PAPM integrates coefficient data more intricately within conservation and constitutive relations, boosting its adaptability to varying coefficients.",
                "Thirdly, data-driven methods are less effective than physics-aware methods like PPNN and our PAPM in time extrapolation tasks, where incorporating prior physical knowledge through structured network design enhances a model s generalization ability.",
                "Notably, PeRCNN uses convolution to approximate nonlinear terms, but experimental results suggest limited performance.",
                "Further details are available in Appendix F.2 .",
                "Visualization.",
                "Figure 4 showcases the stepwise relative error of PAPM during the extrapolation process in the test dataset, using Burgers2d s C Int.",
                "as a representative example.",
                "Compared to the two best-performing baselines, our model (depicted by the red line) exhibits superior performance throughout the extrapolation process, with the least error accumulation.",
                "Turning our attention to the more challenging NSM2d dataset, Figure 5 presents the results across five extrapolation time slices.",
                "While FNO demonstrates commendable accuracy within the training domain (), its performance falters significantly outside of it ().",
                "On the other hand, physics-aware methods (PPNN), and PAPM in particular, consistently capture the evolving patterns with a greater degree of robustness.",
                "Notably, our method emerges as a leader in terms of precision.",
                "Additional visual results can be found in Appendix G.1 ."
            ]
        },
        "4": {
            "5.4": [
                "Different blocks impacts.",
                "4 displays our selection of the Burger2d dataset for ablation studies, chosen for its representation of diffusion, convection, and source terms.",
                "We defined several configurations to assess the impact of individual components.",
                "The relative error on the boundary (BC ) is introduced to highlight the importance of physics embedding further.",
                "no_DF excludes diffusion, whereas no_CF omits convection.",
                "In no_Phy, we retain only a structure with a residual connection, thereby eliminating both diffusion and convection.",
                "no_BCs setup removes the explicit embedding of boundary conditions, no_NODE replaces the Neural ODE with the Euler stepping scheme, and no_All adopts a purely data-driven approach.",
                "Additional ablation results can be found in Appendix G.3 .",
                "Key findings include the crucial roles of diffusion and convection in representing system dynamics, as evidenced in the no_DF, no_CF, and no_Phy configurations.",
                "Specifically, the no_DF configuration demonstrated the importance of integrating the viscosity coefficient with the diffusion term, with its absence leading to significant errors.",
                "The necessity of adhering to physical laws in boundary conditions was highlighted in the no_BCs, notably reducing errors on the boundary (BC ).",
                "Lastly, the no_NODE results indicate that different temporal stepping schemes significantly impact the outcomes, underscoring the effectiveness of neural ODEs in continuous-time modeling.",
                "Different blocks validations.",
                "Taking the Burgers2d and RD2d datasets as examples to demonstrate the fact that the convection/diffusion/source terms could actually learn those parts in the equations.",
                "We use high-fidelity FDM and FVM to compute the corresponding terms to obtain the detailed term for convection/diffusion/source terms.",
                "The relative error between ground truth and numerical results are and in all time steps for Burgers2d and RD2d, respectively.",
                "Thus, we can use the results obtained by the numerical methods as reference values to verify this.",
                "As shown in Tab.",
                "5 , the different terms of PAPM can be used to learn the equation s convection/diffusion/source parts.",
                "Additional visualization results can be found in Appendix G.1 (Figure 9 and Figure 10 )."
            ]
        },
        "5": {
            "5.4": [
                "Different blocks impacts.",
                "4 displays our selection of the Burger2d dataset for ablation studies, chosen for its representation of diffusion, convection, and source terms.",
                "We defined several configurations to assess the impact of individual components.",
                "The relative error on the boundary (BC ) is introduced to highlight the importance of physics embedding further.",
                "no_DF excludes diffusion, whereas no_CF omits convection.",
                "In no_Phy, we retain only a structure with a residual connection, thereby eliminating both diffusion and convection.",
                "no_BCs setup removes the explicit embedding of boundary conditions, no_NODE replaces the Neural ODE with the Euler stepping scheme, and no_All adopts a purely data-driven approach.",
                "Additional ablation results can be found in Appendix G.3 .",
                "Key findings include the crucial roles of diffusion and convection in representing system dynamics, as evidenced in the no_DF, no_CF, and no_Phy configurations.",
                "Specifically, the no_DF configuration demonstrated the importance of integrating the viscosity coefficient with the diffusion term, with its absence leading to significant errors.",
                "The necessity of adhering to physical laws in boundary conditions was highlighted in the no_BCs, notably reducing errors on the boundary (BC ).",
                "Lastly, the no_NODE results indicate that different temporal stepping schemes significantly impact the outcomes, underscoring the effectiveness of neural ODEs in continuous-time modeling.",
                "Different blocks validations.",
                "Taking the Burgers2d and RD2d datasets as examples to demonstrate the fact that the convection/diffusion/source terms could actually learn those parts in the equations.",
                "We use high-fidelity FDM and FVM to compute the corresponding terms to obtain the detailed term for convection/diffusion/source terms.",
                "The relative error between ground truth and numerical results are and in all time steps for Burgers2d and RD2d, respectively.",
                "Thus, we can use the results obtained by the numerical methods as reference values to verify this.",
                "As shown in Tab.",
                "5 , the different terms of PAPM can be used to learn the equation s convection/diffusion/source parts.",
                "Additional visualization results can be found in Appendix G.1 (Figure 9 and Figure 10 )."
            ]
        },
        "6": {},
        "7": {},
        "8": {},
        "9": {},
        "10": {},
        "11": {}
    },
    "chart_result": {
        "4": {
            "5.2": []
        },
        "6": {
            "5.3": [
                0,
                1,
                2,
                3,
                4,
                5,
                6,
                7,
                10,
                14,
                15,
                16
            ]
        }
    },
    "section_info": {
        "Introduction_and_related_work": [
            1,
            2
        ],
        "Method_Design": [
            3,
            4
        ],
        "Experiments": [
            5
        ],
        "Conclusion_and_limitation": [
            6,
            7
        ]
    }
}