A short tour of operator learning theory: Convergence rates, statistical limits, and open questions
It addresses theoretical foundations for operator learning, which is incremental as it synthesizes existing work.
This paper surveys recent developments in operator learning theory, reviewing error bounds for empirical risk minimization and illustrating fundamental performance limits in terms of sample size.
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.