Can one learn a differential operator from pairs of solutions and righthand sides? If so, how many pairs are required? These two questions have received significant research attention in differential equation learning. Given input-output pairs from an unknown elliptic partial differential equation in three dimensions, we will derive a theoretically rigorous scheme for learning the associated Green’s function G. By exploiting the hierarchical low-rank structure of Green’s functions and extending the randomized SVD algorithm to Hilbert-Schmidt operators, we will identify a learning rate associated with elliptic partial differential operators in three dimensions and bound the number of input-output training pairs required to recover a Green’s function approximately. Finally, we will present a deep learning method for learning Green’s functions and recovering properties and features of the underlying PDEs.