Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces

Understand

We propose a new data-driven approach for learning the fundamental solutions (Green's functions) of various linear partial differential equations (PDEs) given sample pairs of input-output functions.

• Building off the theory of functional linear regression (FLR), we estimate the best-fit Green's function and bias term of the fundamental solution in a reproducing kernel Hilbert space (RKHS) which allows us to regularize their smoothness and impose various structural constraints.
• We derive a general representer theorem for operator RKHSs to approximate the original infinite-dimensional regression problem by a finite-dimensional one, reducing the search space to a parametric class of Green's functions.
• In order to study the prediction error of our Green's function estimator, we extend prior results on FLR with scalar outputs to the case with functional outputs.