Abstract

Ricardo Baptista, Edoardo Calvello, Matthieu Darcy, Houman Owhadi, Andrew M. Stuart, and Xianjin Yang. Abstract. We consider the use of Gaussian Processes (GPs) or Neural Networks (NNs) to numerically approx-imate the solutions to nonlinear partial differential equations (PDEs) with rough forcing or sourceterms, which commonly arise as pathwise solutions to stochastic PDEs. Kernel methods have re-cently been generalized to solve nonlinear PDEs by approximating their solutions as the maximuma posteriori estimator of GPs that are conditioned to satisfy the PDE at a finite set of collocationpoints. The convergence and error guarantees of these methods, however, rely on the PDE beingdefined in a classical sense and its solution possessing sufficient regularity to belong to the associatedreproducing kernel Hilbert space. We propose a generalization of these methods to handle roughlyforced nonlinear PDEs while preserving convergence guarantees with an oversmoothing GP kernelthat is misspecified relative to the true solution’s regularity. This is achieved by conditioning aregular GP to satisfy the PDE with a modified source term in a weak sense (when integrated againsta finite number of test functions). This is equivalent to replacing the empirical L2-loss on the PDEconstraint by an empirical negative-Sobolev norm. We further show that this loss function can beused to extend physics-informed neural networks (PINNs) to stochastic equations, thereby resultingin a new NN-based variant termed Negative Sobolev Norm-PINN (NeS-PINN).