Fourier Neural Operator for Parametric Partial Differential Equations

TL;DR

This paper introduces a Fourier neural operator that efficiently learns mappings between function spaces for PDEs, enabling fast, accurate solutions and zero-shot super-resolution for turbulent flows.

Contribution

It proposes a novel neural operator parameterized in Fourier space, significantly improving efficiency and accuracy over previous methods for solving parametric PDEs.

Findings

Up to 1000x faster than traditional PDE solvers

First ML method to model turbulent flows with super-resolution

Achieves superior accuracy at fixed resolution

Abstract

The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.