Solving and learning nonlinear PDEs with Gaussian processes

We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), inverse problems (IPs) involving the identification of parameters in PDEs, using Gaussian processes. The proposed approach: (1) provides natural generalization collocation kernel methods to PDEs IPs; (2) has guaranteed convergence very general class comes equipped with path compute error bounds specific PDE approximations; (3) inherits state-of-the-art computational complexity linear solvers dense matrices. main idea our method is approximate solution given as maximum posteriori (MAP) estimator process conditioned on at finite number points. Although this optimization problem infinite-dimensional, it can be reduced finite-dimensional one by introducing additional variables corresponding values derivatives points; generalizes representer theorem arising regression. form quadratic objective function subject constraints; solved variant Gauss–Newton method. resulting algorithm (a) interpreted successive linearizations PDE, (b) practice found converge small iterations (2 10), wide range PDEs. Most traditional approaches IPs interleave parameter updates numerical PDE; solves both simultaneously. Experiments elliptic Burgers' equation, regularized Eikonal an IP permeability Darcy flow illustrate efficacy scope framework.