A short tour of operator learning theory: Convergence rates, statistical limits, and open questions
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.
| Authors | Simone Brugiapaglia, Nicola Rares Franco, Nicholas H. Nelsen |
|---|---|
| Primary category | math.NA |
| Published | 28 February 2026 |
| Last updated at source | 3 July 2026 |
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