On universal approximation and error bounds for Fourier Neural Operators

Nikola Kovachki; Samuel Lanthaler; Siddhartha Mishra

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arXiv · 2107.07562

On universal approximation and error bounds for Fourier Neural Operators

Nikola Kovachki·Samuel Lanthaler·Siddhartha Mishra

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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.

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Nikola Kovachki, Samuel Lanthaler, Siddhartha Mishra. 2021-07-15. On universal approximation and error bounds for Fourier Neural Operators. https://arxiv.org/abs/2107.07562

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