Group Equivariant Fourier Neural Operators for Partial Differential Equations

TL;DR

This paper introduces a novel Fourier neural operator architecture that encodes physical symmetries like rotations, translations, and reflections directly in the frequency domain, improving PDE solving performance and generalization.

Contribution

It extends group convolutions to the frequency domain, creating Fourier layers that are equivariant to key symmetries, which was previously under-explored.

Findings

Generalizes well across input resolutions

Performs effectively with varying symmetry levels

Code available in AIRS library

Abstract

We consider solving partial differential equations (PDEs) with Fourier neural operators (FNOs), which operate in the frequency domain. Since the laws of physics do not depend on the coordinate system used to describe them, it is desirable to encode such symmetries in the neural operator architecture for better performance and easier learning. While encoding symmetries in the physical domain using group theory has been studied extensively, how to capture symmetries in the frequency domain is under-explored. In this work, we extend group convolutions to the frequency domain and design Fourier layers that are equivariant to rotations, translations, and reflections by leveraging the equivariance property of the Fourier transform. The resulting $G$-FNO architecture generalizes well across input resolutions and performs well in settings with varying levels of symmetry. Our code is publicly available as part of the AIRS library (https://github.com/divelab/AIRS).