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2021

On universal approximation and error bounds for Fourier Neural Operators

Kovachki, Nikola, Lanthaler, Samuel, Mishra, Siddhartha

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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.

On universal approximation and error bounds for Fourier Neural Operators · Around

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