  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.
