  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.
