On universal approximation and error bounds for Fourier Neural Operators

On universal approximation and error bounds for Fourier Neural Operators

Journal of Machine Learning Research 22 (2021) 1-76 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 opera...

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Hauptverfasser: Kovachki, Nikola, Lanthaler, Samuel, Mishra, Siddhartha
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Mathematics - Numerical Analysis
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Mishra, Siddhartha
description Journal of Machine Learning Research 22 (2021) 1-76 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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title On universal approximation and error bounds for Fourier Neural Operators
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