Learning elliptic partial differential equations with randomized linear algebra

TL;DR

This paper introduces a theoretically rigorous method for learning the Green's function of elliptic PDEs in three dimensions using randomized linear algebra, achieving high accuracy with a finite number of training pairs.

Contribution

It develops the first rigorous scheme for learning PDE Green's functions exploiting hierarchical low-rank structures, extending randomized SVD to Hilbert--Schmidt operators.

Findings

Achieves near-optimal approximation of Green's functions

Requires a polynomial number of training pairs relative to the desired accuracy

Extends randomized SVD to operators in Hilbert spaces

Abstract

Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$. By exploiting the hierarchical low-rank structure of $G$, we show that one can construct an approximant to $G$ that converges almost surely and achieves a relative error of $\mathcal{O}(\Gamma_\epsilon^{-1/2}\log^3(1/\epsilon)\epsilon)$ using at most $\mathcal{O}(\epsilon^{-6}\log^4(1/\epsilon))$ input-output training pairs with high probability, for any $0<\epsilon<1$. The quantity $0<\Gamma_\epsilon\leq 1$ characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert--Schmidt operators and characterize the quality of covariance kernels for PDE learning.