Title: Operator Learning for Partial Differential Equations 

Presented by Desmond Kofi Boateng, Computing PhD student, Data Science emphasis

Abstract: Operator learning is the branch of scientific machine learning that aims to approximate an unknown operator, often the solution to a partial differential equation (PDE), directly from data. Neural operator approaches such as Deep Operator Networks (DeepONets) and Fourier Neural Operators (FNOs) have been used to solve this problem. Recently, kernel-based operator learning has emerged as a compelling alternative that offers competitive accuracy while avoiding the complexities of deep learning.

These methods leverage the structure of reproducing kernel Hilbert spaces (RKHS), providing both theoretical guarantees and interpretability. In particular, polyharmonic spline kernels, including the well-known thin plate splines, naturally give rise to Beppo–Levi spaces, which serve as native spaces for these kernels. In this talk, I will first review the mathematical foundations of operator learning, focusing on neural operator approaches, specifically, DeepONets, FNO, and kernel methods. I will then present a parameter-free kernel approach based on polyharmonic splines that eliminates the need for costly shape parameter tuning. Numerical benchmarks on canonical PDE problems—including the Navier–Stokes, Burgers, Helmholtz, structural mechanics, Darcy flow, and advection equations. This highlights the potential of classical kernel methods as efficient and interpretable alternatives for modern operator learning.

Advisor: Dr. Grady Wright

Committee Members: Dr. Michael Perlmutter, Dr. Varun Shankar

External Examiner: Dr. Francesca Spezzano