Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operators

Unique Subedi, Ambuj Tewari

Keywords: Operator Learning, Fourier Linear Operators

TL;DR: We bound the statistical, discretization and truncation errors that occur while learning the linear core of the Fourier Neural Operator architechture.

Abstract: We investigate the problem of learning operators between function spaces, focusing on the linear part of a layer in the Fourier Neural Operator architecture. First, we identify three main errors that occur during the learning process: statistical error due to finite sample size, truncation error from finite rank approximation of the operator, and discretization error from handling functional data on a finite grid of domain points. Finally, we analyze a Discrete Fourier Transform (DFT) based least squares estimator, establishing both upper and lower bounds on the aforementioned errors.

Primary Area: learning theory

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Submission Number: 3272