  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.
