Over the past decade, machine learning methods have significantly transformed numerous branches of applied mathematics. While the most visible successes have been in finite-dimensional settings, the mathematical community has recently turned its attention toward the rigorous study of machine learning in infinite dimensions. At the heart of this topic is operator learning, an infinite-dimensional framework with many applications.

In this talk, I will provide an overview of this emerging field, highlighting three key applications: learning the parameter-to-solution map of partial differential equations (PDEs), learning inverse maps for ill-posed inverse problems, and developing generative models in infinite-dimensional spaces.

After a general introduction, I will focus on the underlying mathematical structures and present specific techniques accompanied by rigorous theoretical guarantees. In particular, I will discuss recent results on learning holomorphic operators—specifically those arising from the solution operators of elliptic PDEs-using kernel methods and Reproducing Kernel Hilbert Spaces (RKHS). Finally, I will present recent findings on learning regularization operators for inverse problems, as well as novel approaches to infinite-dimensional generative modeling utilizing pseudodifferential operators.