A short tour of operator learning theory: Convergence rates, statistical limits, and open questions

TL;DR

This survey reviews recent advances in operator learning theory, focusing on convergence rates, statistical limits, and open questions at the intersection of approximation and statistical learning theories.

Contribution

It synthesizes current knowledge on error bounds, performance limits, and open problems in operator learning, highlighting the interplay between approximation and statistical perspectives.

Findings

Error bounds for neural network approximations of operators

Fundamental performance limits based on sample size and regularity

Open questions on the interplay between approximation and statistical limits

Abstract

This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and neural network approximations. Next, it illustrates fundamental performance limits in terms of sample size by adopting a minimax perspective and considering various notions of regularity beyond holomorphy. The paper ends with a discussion on the interplay between these two perspectives and related open questions.