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

TL;DR

This paper investigates the theoretical foundations of learning Fourier neural operators, focusing on controlling statistical, discretization, and truncation errors through analysis of a DFT-based estimator.

Contribution

It identifies and analyzes the main errors in learning Fourier operators, providing bounds for each error type in the context of operator learning.

Findings

Established bounds on statistical, truncation, and discretization errors.

Analyzed a DFT-based least squares estimator for operator learning.

Provided insights into error control in Fourier neural operator training.

Abstract

We study learning-theoretic foundations of operator learning, using the linear layer of the Fourier Neural Operator architecture as a model problem. 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.