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

Simone Brugiapaglia; Nicola Rares Franco; Nicholas H. Nelsen

math.NAFeb 28, 2026

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

Authors: Simone Brugiapaglia, Nicola Rares Franco, Nicholas H. Nelsen

Organizations: Concordia University · Politecnico di Milano · Cornell University and The University of Texas at Austin

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.

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Jun 16, 2026 · cs.LGJ/K move · Enter open · S save

Generalization Guarantees for Multi-Input Neural Operator Learning in Sobolev Spaces

Yahong Yang, Zecheng Zhang, Wei Zhu, Wenjing Liao+1

School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332-0160, USA. · Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, Notre Dame, IN 46556, USA. · Department of Mathematics, Hong Kong Baptist University, FSC1202, Fong Shu Chuen Building, Hong Kong Baptist University, Kowloon Tong, Hong Kong.

We develop approximation and generalization error estimates for multi-input neural operators, with the output error measured in Sobolev norms. In contrast to standard operator-learning settings with a single input function, our framework allows multiple input functions defined on possibly different domains, with different dimensions and Sobolev regularities. The derived rates explicitly quantify the contribution of each input space to the final error bound. In particular, in the balanced regime, the approximation and generalization rates are governed by the interaction between the input dimensions, regularities, and Sobolev orders, while the dependence on the model complexity retains a

\log\log/\log

-type structure. Our analysis provides a general theoretical framework for multi-input operator learning, including Sobolev training, and is applicable to operator learning problems arising from partial differential equations and scientific computing.

Logarithmic Sobolev Inequalities Probabilistic Operator Learning Neural Operators Neural-Network Approximation Generalization Bounds Spectral Norm Trainability

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Abstract

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