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On universal approximation and error bounds for Fourier Neural Operators

by N. Kovachki and S. Lanthaler and S. Mishra

(Report number 2021-23)

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.

Keywords: operator learning spectral methods universal approximation error and complexity bounds Darcy flow incompressible Navier-Stokes

@Techreport{KLM21_965,
  author = {N. Kovachki and S. Lanthaler and S. Mishra},
  title = {On universal approximation and error bounds for Fourier Neural Operators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2021-23},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2021/2021-23.pdf },
  year = {2021}
}

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