MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation

Qi Wang, Yuan Mi, Wang Haoyun, Yi Zhang, Ruizhi Chengze, Hongsheng Liu, Ji-Rong Wen, Hao Sun
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:63929-63951, 2025.

Abstract

Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but struggle with weak generalizability, interpretability, and data dependency, as well as suffer in long-term prediction. To this end, we propose a PDE-embedded network with multiscale time stepping (MultiPDENet), which fuses the scheme of numerical methods and machine learning, for accelerated simulation of flows. In particular, we design a convolutional filter based on the structure of finite difference stencils with a small number of parameters to optimize, which estimates the equivalent form of spatial derivative on a coarse grid to minimize the equation’s residual. A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction. To alleviate the curse of temporal error accumulation in long-term prediction, we introduce a multiscale time integration approach, where a neural network is used to correct the prediction error at a coarse time scale. Experiments across various PDE systems, including the Navier-Stokes equations, demonstrate that MultiPDENet can accurately predict long-term spatiotemporal dynamics, even given small and incomplete training data, e.g., spatiotemporally down-sampled datasets. MultiPDENet achieves the state-of-the-art performance compared with other neural baseline models, also with clear speedup compared to classical numerical methods

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-wang25bz, title = {{M}ulti{PDEN}et: {PDE}-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation}, author = {Wang, Qi and Mi, Yuan and Haoyun, Wang and Zhang, Yi and Chengze, Ruizhi and Liu, Hongsheng and Wen, Ji-Rong and Sun, Hao}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {63929--63951}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/wang25bz/wang25bz.pdf}, url = {https://proceedings.mlr.press/v267/wang25bz.html}, abstract = {Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but struggle with weak generalizability, interpretability, and data dependency, as well as suffer in long-term prediction. To this end, we propose a PDE-embedded network with multiscale time stepping (MultiPDENet), which fuses the scheme of numerical methods and machine learning, for accelerated simulation of flows. In particular, we design a convolutional filter based on the structure of finite difference stencils with a small number of parameters to optimize, which estimates the equivalent form of spatial derivative on a coarse grid to minimize the equation’s residual. A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction. To alleviate the curse of temporal error accumulation in long-term prediction, we introduce a multiscale time integration approach, where a neural network is used to correct the prediction error at a coarse time scale. Experiments across various PDE systems, including the Navier-Stokes equations, demonstrate that MultiPDENet can accurately predict long-term spatiotemporal dynamics, even given small and incomplete training data, e.g., spatiotemporally down-sampled datasets. MultiPDENet achieves the state-of-the-art performance compared with other neural baseline models, also with clear speedup compared to classical numerical methods} }
Endnote
%0 Conference Paper %T MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation %A Qi Wang %A Yuan Mi %A Wang Haoyun %A Yi Zhang %A Ruizhi Chengze %A Hongsheng Liu %A Ji-Rong Wen %A Hao Sun %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-wang25bz %I PMLR %P 63929--63951 %U https://proceedings.mlr.press/v267/wang25bz.html %V 267 %X Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but struggle with weak generalizability, interpretability, and data dependency, as well as suffer in long-term prediction. To this end, we propose a PDE-embedded network with multiscale time stepping (MultiPDENet), which fuses the scheme of numerical methods and machine learning, for accelerated simulation of flows. In particular, we design a convolutional filter based on the structure of finite difference stencils with a small number of parameters to optimize, which estimates the equivalent form of spatial derivative on a coarse grid to minimize the equation’s residual. A Physics Block with a 4th-order Runge-Kutta integrator at the fine time scale is established that embeds the structure of PDEs to guide the prediction. To alleviate the curse of temporal error accumulation in long-term prediction, we introduce a multiscale time integration approach, where a neural network is used to correct the prediction error at a coarse time scale. Experiments across various PDE systems, including the Navier-Stokes equations, demonstrate that MultiPDENet can accurately predict long-term spatiotemporal dynamics, even given small and incomplete training data, e.g., spatiotemporally down-sampled datasets. MultiPDENet achieves the state-of-the-art performance compared with other neural baseline models, also with clear speedup compared to classical numerical methods
APA
Wang, Q., Mi, Y., Haoyun, W., Zhang, Y., Chengze, R., Liu, H., Wen, J. & Sun, H.. (2025). MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:63929-63951 Available from https://proceedings.mlr.press/v267/wang25bz.html.

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