A Neural PDE Solver with Temporal Stencil Modeling

Zhiqing Sun, Yiming Yang, Shinjae Yoo

Keywords: neural PDE solver, Navier-Stokes equation, turbulent flow, Computational Fluid Dynamics, CFD

Abstract: Numerical simulation of non-linear partial differential equations plays a crucial role in modeling physical science and engineering phenomena, such as weather, climate, and aerodynamics. Recent Machine Learning (ML) models trained on low-resolution spatio-temporal signals have shown new promises in capturing important dynamics in high-resolution signals, under the condition that the models can effectively recover the missing details. However, this study shows that significant information is often lost in the low-resolution down-sampled features. To address such issues, we propose a new approach, namely Temporal Stencil Modeling (TSM), which combines the strengths of advanced time-series sequence modeling (with the HiPPO features) and state-of-the-art neural PDE solvers (with learnable stencil modeling). TSM aims to recover the lost information from the PDE trajectories and can be regarded as a temporal generalization of classic finite volume methods such as WENO. Our experimental results show that TSM achieves the new state-of-the-art simulation accuracy for 2-D incompressible Navier-Stokes turbulent flows: it significantly outperforms the previously reported best results by 19.9% in terms of the highly-correlated duration time, and reduces the inference latency into 80%. We also show a strong generalization ability of the proposed method to various out-of-distribution turbulent flow settings.

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TL;DR: We propose a novel Temporal Stencil Modeling (TSM) method for solving time-dependent PDEs in conservation form.

Community Implementations: [ 4 code implementations](https://www.catalyzex.com/paper/arxiv:2302.08105/code)