Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces

The rapid development of data-driven scientific discovery holds the promise of new and faster methods to analyze, understand, and predict various complex phenomena whose physical laws are still beyond our grasp. Of central interest to this development is the ability to solve efficiently a broad range of differential equations, and more precisely partial differential equations (PDEs) which still largely require advanced numerical techniques tailored for specific problems.

In this talk, we study what is certainly one of the most inspiring outcomes of this program: solving PDEs from input/output data. Namely, we aim to predict the solution u(x, t) of a PDE under new initial conditions u(x, 0) or external forcings f(x, t) that represents the ambient conditions of an evolving system. In other words, we propose to learn an operator that maps these inputs u(x,0), f(x, t) to output solutions u(x, t) from input/output data. While such operators can be nonlinear, we focus on linear operators which are a natural approximation to nonlinear phenomena and are, in general, more robust to model misspecification.

For many linear PDEs, the solution u(x, t) is given in closed-form by an integral equation where a kernel specific to the PDE, otherwise known as a Green’s function, is integrated against the input initial condition u(x,0) or forcing function f(x, t). We propose a new data-driven approach for learning the Green's functions of various linear PDEs, by estimating the best-fit Green's function in a reproducing kernel Hilbert space (RKHS) which allows us to regularize its smoothness and impose various structural constraints.

Finally, we derive nonasymptotic rates for the prediction error of our Green's function estimator and apply our method to several linear PDEs including the Poisson, Helmholtz, Schrödinger, Fokker-Planck, and heat equation. We highlight the method’s ability to extrapolate to more finely sampled meshes without any additional training.

The relevant preprint can be found here: https://arxiv.org/abs/2108.11580