Group Equivariant Fourier Neural Operators for Partial Differential Equations
This work addresses the challenge of incorporating physical symmetries into neural operators for PDEs, which is incremental as it builds on existing group theory and FNO methods by adapting them to the frequency domain.
The authors tackled the problem of solving partial differential equations (PDEs) by extending Fourier neural operators (FNOs) to encode symmetries like rotations, translations, and reflections in the frequency domain, resulting in a $G$-FNO architecture that generalizes well across input resolutions and performs effectively in varying symmetry settings.
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).