Optimal Learning of Linear Operators

Optimal Learning of Linear Operators

Abstract: Operator learning is an emerging paradigm at the intersection of scientific computing and machine learning. Its central objective is to learn operators between infinite-dimensional function spaces from data. Although operator learning has found broad applications across science and engineering, its theoretical foundations remain underdeveloped. In this talk, I will present recent progress on the statistical and computational limits of operator learning, focusing on several fundamental operator classes: bounded linear operators between Sobolev spaces, pseudo-differential operators central to the analysis of partial differential equations (PDEs) and mathematical physics, and Fourier integral operators associated with hyperbolic systems and wave propagation. Across these settings, a common multiscale framework yields minimax-optimal statistical rates together with computationally efficient estimators. These results contribute to a unified theory of operator learning at the interface of analysis and PDEs, computational mathematics, statistics, and data science.