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On universal approximation and error bounds for Fourier Neural Operators
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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. Thus, FNOs are shown to efficiently approximate operators arising in a large class of PDEs.
Forward citations
Cited by 3 Pith papers
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A Neural Operator based Hybrid Microscale Model for Multiscale Simulation of Rate-Dependent Materials
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Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Space
A regularized random Fourier feature model with finite-element recovery learns PDE solution operators from noisy data, with a conditioning guarantee when the number of features scales like m log m.
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