Learning elliptic partial differential equations with randomized linear algebra

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 an expected relative error of $\epsilon$ using at most $\mathcal{O}(\epsilon^{-6}\log^4(1/\epsilon)/\Gamma_\epsilon)$ input-output training pairs, 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.