Fourier Neural Operator for Parametric Partial Differential Equations

TLDR

Neural networks have traditionally mapped finite‑dimensional Euclidean spaces, but recent neural operators extend this to mappings between function spaces, allowing PDE solvers to learn the entire family of equations rather than a single instance. We introduce a Fourier neural operator that parameterizes the integral kernel directly in Fourier space, yielding an expressive and efficient architecture. The operator is evaluated on Burgers’, Darcy flow, and Navier‑Stokes equations. It is the first ML‑based method to model turbulent flows with zero‑shot super‑resolution, runs up to three orders of magnitude faster than conventional PDE solvers, and achieves higher 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.