Physics-embedded Fourier Neural Network for Partial Differential Equations

TL;DR

This paper introduces Physics-embedded Fourier Neural Networks (PeFNN) that incorporate physical laws like momentum conservation into neural network architectures to improve interpretability and accuracy in solving complex PDEs, including real-world flood simulations.

Contribution

The paper presents a novel PeFNN model with momentum-conserving Fourier layers that enforce physical laws, enhancing interpretability and generalization in PDE solutions.

Findings

PeFNN achieves state-of-the-art accuracy on standard PDE benchmarks.

PeFNN generalizes well across different input resolutions.

PeFNN performs effectively in large-scale flood simulation applications.

Abstract

We consider solving complex spatiotemporal dynamical systems governed by partial differential equations (PDEs) using frequency domain-based discrete learning approaches, such as Fourier neural operators. Despite their widespread use for approximating nonlinear PDEs, the majority of these methods neglect fundamental physical laws and lack interpretability. We address these shortcomings by introducing Physics-embedded Fourier Neural Networks (PeFNN) with flexible and explainable error control. PeFNN is designed to enforce momentum conservation and yields interpretable nonlinear expressions by utilizing unique multi-scale momentum-conserving Fourier (MC-Fourier) layers and an element-wise product operation. The MC-Fourier layer is by design translation- and rotation-invariant in the frequency domain, serving as a plug-and-play module that adheres to the laws of momentum conservation. PeFNN establishes a new state-of-the-art in solving widely employed spatiotemporal PDEs and generalizes well across input resolutions. Further, we demonstrate its outstanding performance for challenging real-world applications such as large-scale flood simulations.