On universal approximation and error bounds for Fourier Neural Operators

TL;DR

This paper establishes the universal approximation capabilities of Fourier neural operators (FNOs) for operators in infinite-dimensional spaces and provides explicit error bounds demonstrating their efficiency in approximating PDE-related operators.

Contribution

It proves the universality of FNOs for continuous operators and derives explicit error bounds, highlighting their efficiency in approximating PDE operators.

Findings

FNOs can approximate any continuous operator to desired accuracy.

Explicit error bounds show sub (log)-linear growth of FNO size with inverse error.

FNOs efficiently approximate operators in PDEs like Darcy and Navier-Stokes.

Abstract

Fourier neural operators (FNOs) have recently been proposed as an effective framework for learning operators that map between infinite-dimensional spaces. We prove that FNOs are universal, in the sense that they can approximate any continuous operator to desired accuracy. Moreover, we suggest a mechanism by which FNOs can approximate operators associated with PDEs efficiently. Explicit error bounds are derived to show that the size of the FNO, approximating operators associated with a Darcy type elliptic PDE and with the incompressible Navier-Stokes equations of fluid dynamics, only increases sub (log)-linearly in terms of the reciprocal of the error. Thus, FNOs are shown to efficiently approximate operators arising in a large class of PDEs.