Discretization Error of Fourier Neural Operators

• URL: http://arxiv.org/abs/2405.02221v1
• Date: Fri, 3 May 2024 16:28:05 GMT
• Title: Discretization Error of Fourier Neural Operators
• Authors: Samuel Lanthaler, Andrew M. Stuart, Margaret Trautner,
• Abstract summary: Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data. The Fourier Neural Operator (FNO) is a common model architecture used for operator learning.
• Score: 5.121705282248479
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data. The Fourier Neural Operator (FNO) is a common model architecture used for operator learning. The FNO combines pointwise linear and nonlinear operations in physical space with pointwise linear operations in Fourier space, leading to a parameterized map acting between function spaces. Although FNOs formally involve convolutions of functions on a continuum, in practice the computations are performed on a discretized grid, allowing efficient implementation via the FFT. In this paper, the aliasing error that results from such a discretization is quantified and algebraic rates of convergence in terms of the grid resolution are obtained as a function of the regularity of the input. Numerical experiments that validate the theory and describe model stability are performed.

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