Principled interpolation of Green's functions learned from data

TL;DR

This paper introduces a data-driven framework for modeling physical systems by learning their Green's functions from input-output data, employing POD and randomized SVD methods, and enabling efficient interpolation across parameters.

Contribution

It proposes two novel methods for learning Green's functions from data and a manifold interpolation scheme for parameter variation, advancing data-driven modeling of unknown PDE systems.

Findings

Effective Green's function approximation demonstrated in 1D and 2D examples.

The methods enable accurate interpolation of system responses at unseen parameters.

The approach offers a principled way to model physical systems without explicit PDE knowledge.

Abstract

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 excitation-response 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.