Elliptic PDE learning is provably data-efficient

Abstract

Can one learn a solution operator associated with a differential operator from pairs of solutions and righthand sides? If so, how many pairs are required? These two questions have received significant research attention in operator learning. More precisely, given input-output pairs from an unknown elliptic PDE, we will derive a theoretically rigorous scheme for learning the associated Green's function. By exploiting the hierarchical low-rank structure of Green’s functions and randomized linear algebra, we will have a provable learning rate. Along the way, we will develop a more general theory for the randomized singular value decomposition and show how these techniques extend to parabolic and hyperbolic PDEs. This talk partially explains the success of operator networks like DeepONet in data-sparse settings.

Speaker

Alex Townsend is an Associate Professor at Cornell University in the Mathematics Department. His research is in Applied Mathematics and focuses on spectral methods, low-rank techniques, fast transforms, and theoretical aspects of deep learning. Prior to Cornell, he was an Applied Math instructor at MIT (2014-2016) and a DPhil student at the University of Oxford (2010-2014). He was awarded a SIAM CSE best paper prize in 2023, a Weiss Junior Fellowship in 2022, Simons Fellowship in 2022, an NSF CAREER in 2021, a SIGEST paper award in 2019, the SIAG/LA Early Career Prize in applicable linear algebra in 2018, and the Leslie Fox Prize in 2015.