A Neural PDE Solver with Temporal Stencil Modeling

TL;DR

This paper introduces Temporal Stencil Modeling (TSM), a neural PDE solver that combines advanced time-series modeling with learnable stencils, achieving state-of-the-art accuracy in simulating turbulent flows and demonstrating strong generalization capabilities.

Contribution

The paper presents TSM, a novel neural PDE solver that effectively recovers lost information in low-resolution data by integrating HiPPO features and learnable stencil modeling, improving simulation accuracy.

Findings

Achieves 19.9% improvement in simulation accuracy for 2D Navier-Stokes flows.

Reduces inference latency by 80%.

Demonstrates strong generalization to out-of-distribution turbulent flows.

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. Our code is available at "https://github.com/Edward-Sun/TSM-PDE".