Colloquium by Daniel Sanz-Alonso (University of Chicago)

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. 

Short bio: Daniel is an Associate Professor in the Department of Statistics at the University of Chicago, where he is also a member of the Committee on Computational and Applied Mathematics. Previously, he was a postdoc in the Division of Applied Mathematics and a member of the Data Science Initiative at Brown University. In 2016, he completed his PhD in Mathematics and Statistics at the University of Warwick, UK, under the supervision of Andrew Stuart and Gareth Roberts. His research interests are in inverse problems, data assimilation, and scientific machine learning. He was awarded the José Luis Rubio de Francia prize to the best Spanish mathematician under 32 by the Spanish Royal Society of Mathematics, and received an NSF CAREER award in 2023.