Isotropic Fourier Neural Operators

TL;DR

The paper introduces Isotropic Fourier Neural Operators, which incorporate spatial symmetry considerations into Fourier layers, enhancing performance and reducing parameters in PDE learning tasks.

Contribution

It proposes a modification to Fourier Neural Operators that enforces isotropy, improving accuracy and parameter efficiency.

Findings

Improved model performance with symmetry-respecting transformations.

Parameter reduction by up to 16 times in 2D and 96 times in 3D.

Enhanced learning of PDEs with respect to physical symmetries.

Abstract

Fourier Neural Operators are deep learning models that learn mappings between function spaces and can be used to learn and solve partial differential equations (PDEs), in some cases significantly faster than traditional PDE solvers. Within the model are Fourier layers, which apply linear transformations directly to the Fourier modes, with parameters depending on the wave numbers. However, most physical systems are isotropic, with the results being independent of the coordinate system chosen, but the linear transformations do not necessarily respect these symmetries. We propose a modification to the linear transformations that ensures spatial symmetries are respected, called the Isotropic Fourier Neural Operator, which both improves model performance and reduces the number of parameters by up to a factor of 16 in 2D and 96 in 3D.