Work

Elliptic PDE learning is provably data-efficient

Year: 2023

Type: article

Abstract: Partial differential equations (PDE) learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning techniques achieve spectacular results with limited data availability. Still, these results are empirical. Our work provides theoretical guarantees on the number of input-output training pairs required in PDE learning. Specifically, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers solution operators of three-dimensional uniformly elliptic PDEs from input-output data and achieves an exponential convergence rate of the error with respect to the size of the training dataset with an exceptionally high probability of success.

Source: Proceedings of the National Academy of Sciences

DOI: 10.1073/pnas.2303904120

Authors: Nicolas Boullé, Diana Halikias, Alex Townsend

Institutions: Isaac Newton Institute for Mathematical Sciences, University of Cambridge, Cornell University

Language: English

Topic: Model Reduction and Neural Networks

Subfield: Statistical and Nonlinear Physics

Field: Physics and Astronomy

Domain: Physical Sciences

Open Access status: hybrid

Funders: National Science Foundation, Simons Foundation

Awards: Collaborative Research: Optimal-Complexity Spectral Methods for Complex Fluids, CAREER: Computing with Rational Functions, Graduate Research Fellowship Program (GRFP)