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

• URL: http://arxiv.org/abs/2501.15987v1
• Date: Mon, 27 Jan 2025 12:15:51 GMT
• Title: MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation
• Authors: Qi Wang, Yuan Mi, Haoyun Wang, Yi Zhang, Ruizhi Chengze, Hongsheng Liu, Ji-Rong Wen, Hao Sun,
• Abstract summary: We propose a PDE-embedded network with multiscale time stepping (MultiPDENet) In particular, we design a convolutional filter based on the structure of finite difference with a small number of parameters to optimize. 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.
• Score: 48.41289705783405
• License:
• 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.

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