Diana Halikias, Data-efficient matrix recovery and operator learning

Can one learn a partial differential equation (PDE) from only input-output function pairs? If so, how many are needed? We provide theoretical guarantees on the number of input-output training pairs required to learn a 3D uniformly elliptic PDE. In particular, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers the corresponding Green's function and achieves an exponential convergence rate of the error with respect to the size of the training dataset with an exceptionally high probability of success. This work provides a theoretical explanation for the observed strong performance of recent deep learning techniques in PDE learning, even when there is limited data availability. We also discuss the importance of access to the adjoint operator in this problem, which relates closely to the role of transpose-matrix-vector products in sketching algorithms.