Seminar Details

Can one learn a partial differential equation (PDE) from only input-output function pairs? If so, how many are needed? It turns out that this is closely related to the problem of structured matrix approximation from matrix-vector products. We describe some algorithms and lower bounds on the query complexity for recovering various common structures of matrices. We then discuss the extension of these methods to infinite dimensions in the context of learning the Green's function of a uniformly elliptic PDE in 3D. 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 corresponds to the role of transpose-matrix-vector products in sketching algorithms for linear algebra.