Data-driven discovery of PDEs #

Nicolas Boullé, Alex Townsend

14:10 Tuesday in 2Q49.

Part of the IMA Lighthill-Thwaites prize session.

Abstract #

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 partial differential equation, 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 and parabolic partial differential operators and bound the number of input-output training pairs required to recover a Green’s function approximately.