On universal approximation and error bounds for Fourier Neural Operators

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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• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Fourier Neural Operator for Parametric Partial Differential Equations
• DeepM&Mnet: Inferring the electroconvection multiphysics fields based on operator approximation by neural networks
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Cited by in corpus (3)

• Long-term predictions of turbulence by implicit U-Net enhanced Fourier neural operator
• A comprehensive and fair comparison of two neural operators (with practical extensions) based on FAIR data
• Factorized Fourier Neural Operators