Data-driven discovery and interpolation of Green's functions

To gain a deeper understanding of nature, we present a data-driven approach to mathematically model unknown physical systems, by learning a Green's function for its hidden, governing partial differential equations. The systems considered are observed as input-output pairs, by collecting physical responses under excitations drawn from a Gaussian process. Two methods are offered to learn the Green's function: 1) using the proper orthogonal decomposition modes of the system as a surrogate for the empirical eigenvectors of the Green's function and fit the eigenvalues using the data; and 2) using a generalization of randomized singular value decomposition to construct a low-rank approximation to the Green's function. These are demonstrated 1D examples: Poisson, Helmholtz, Airy, and multi-physics contexts. We also present a 2D demonstration, for the Poisson problem. Additionally, we propose a way to interpolate between Green's functions learned for different modeling contexts, by performing principled interpolation on a manifold. The interpolation is demonstrated on Airy's problem in 1D and Helmholtz problem in 2D.