Principled interpolation of Green?s functions learned from data

We present a data-driven approach to mathematically model physical systems whose governing partial differential equations are unknown, by learning their associated Green's function. The subject systems are observed by collecting input-output pairs of system responses under excitations drawn from a Gaussian process. Two methods are proposed to learn the Green's function. In the first method, we use the proper orthogonal decomposition (POD) modes of the system as a surrogate for the eigenvectors of the Green's function, and subsequently fit the eigenvalues, using data. In the second, we employ a generalization of the randomized singular value decomposition (SVD) to operators, in order to construct a low-rank approximation to the Green's function. Then, we propose a manifold interpolation scheme, for use in an offline-online setting, where offline excitationresponse data, taken at specific model parameter instances, are compressed into empirical eigenmodes. These eigenmodes are subsequently used within a manifold interpolation scheme, to uncover other suitable eigenmodes at unseen model parameters. The approximation and interpolation numerical techniques are demonstrated on several examples in one and two dimensions. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

We propose a data-driven approach to mathematically model unknown physical systems using the learned Green's function. Two methods are presented to learn the Green's function, one using the proper orthogonal decomposition (POD) modes and the other using a generalization of the randomized singular value decomposition (SVD). A manifold interpolation scheme is then proposed to uncover suitable eigenmodes at unseen model parameters. The approximation and interpolation techniques are demonstrated on examples in one and two dimensions.