chebgreen: Learning and Interpolating Continuous Empirical Green’s Functions from Data

In this work, we present a mesh-independent, data-driven library, chebgreen, to mathematically model one-dimensional systems, possessing an associated control parameter, and whose governing partial differential equation is unknown. The proposed method learns a Empirical Green's Function for the associated, but hidden, boundary value problem, in the form of a Rational Neural Network and then represent these in terms of a continuous singular value expansion, using our Python implementation of the chebfun library for two dimensions, chebpy2. We uncover the Green's function, at an unseen control parameter value, by interpolating the left and right singular functions within a suitable library, expressed as points on a manifold of Quasimatrices, while the associated singular values are interpolated with Lagrange polynomials. We conduct the interpolation through a new manifold interpolation scheme that generalizes previous manifold interpolation methods to manifolds of Quasimatrices. Furthermore, we prove that the mapping to the tangent space in the finite dimensional case may be generalized to the Quasimatrix case.