Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning associated Green's function $G$. By exploiting hierarchical low-rank structure $G$, show that one can construct approximant to $G$ converges almost surely and achieves a relative error $\mathcal{O}(\Gamma_\epsilon^{-1/2}\log^3(1/\epsilon...