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.
@article{arxiv.2107.07562,
title = {On universal approximation and error bounds for Fourier Neural Operators},
author = {Nikola Kovachki and Samuel Lanthaler and Siddhartha Mishra},
journal= {arXiv preprint arXiv:2107.07562},
year = {2021}
}