Operator learning meets inverse problems: A probabilistic perspective

TL;DR

This paper reviews recent advances in operator learning for inverse problems, emphasizing probabilistic approaches, data-driven inversion, and the handling of noise, with a focus on theoretical and methodological developments.

Contribution

It provides a comprehensive survey of operator learning methods applied to inverse problems, highlighting probabilistic formulations, architecture designs, and theoretical insights.

Findings

End-to-end inverse operator learning enables direct data-to-solution mapping.

Structure-aware architectures improve point predictions and posterior estimates.

Theoretical analysis covers linear and nonlinear inverse problems.

Abstract

Operator learning offers a robust framework for approximating mappings between infinite-dimensional function spaces. It has also become a powerful tool for solving inverse problems in the computational sciences. This chapter surveys methodological and theoretical developments at the intersection of operator learning and inverse problems. It begins by summarizing the probabilistic and deterministic approaches to inverse problems, and pays special attention to emerging measure-centric formulations that treat observed data or unknown parameters as probability distributions. The discussion then turns to operator learning by covering essential components such as data generation, loss functions, and widely used architectures for representing function-to-function maps. The core of the chapter centers on the end-to-end inverse operator learning paradigm, which aims to directly map observed data to the solution of the inverse problem without requiring explicit knowledge of the forward map. It highlights the unique challenge that noise plays in this data-driven inversion setting, presents structure-aware architectures for both point predictions and posterior estimates, and surveys relevant theory for linear and nonlinear inverse problems. The chapter also discusses the estimation of priors and regularizers, where operator learning is used more selectively within classical inversion algorithms.