Solving Roughly Forced Nonlinear PDEs via Misspecified Kernel Methods and Neural Networks

TL;DR

This paper introduces a novel approach combining misspecified Gaussian Process kernels and neural networks to approximate solutions to rough nonlinear PDEs, extending existing methods to handle less regular solutions with convergence guarantees.

Contribution

It generalizes kernel methods and PINNs to rough PDEs by using a negative Sobolev norm-based loss, accommodating misspecified kernels and irregular solutions.

Findings

Proposes a weak conditioning approach for rough PDEs

Extends PINNs to stochastic PDEs with negative Sobolev norm loss

Provides convergence guarantees for the proposed methods

Abstract

We consider the use of Gaussian Processes (GPs) or Neural Networks (NNs) to numerically approximate the solutions to nonlinear partial differential equations (PDEs) with rough forcing or source terms, which commonly arise as pathwise solutions to stochastic PDEs. Kernel methods have recently been generalized to solve nonlinear PDEs by approximating their solutions as the maximum a posteriori estimator of GPs that are conditioned to satisfy the PDE at a finite set of collocation points. The convergence and error guarantees of these methods, however, rely on the PDE being defined in a classical sense and its solution possessing sufficient regularity to belong to the associated reproducing kernel Hilbert space. We propose a generalization of these methods to handle roughly forced nonlinear PDEs while preserving convergence guarantees with an oversmoothing GP kernel that is misspecified relative to the true solution's regularity. This is achieved by conditioning a regular GP to satisfy the PDE with a modified source term in a weak sense (when integrated against a finite number of test functions). This is equivalent to replacing the empirical $L^2$-loss on the PDE constraint by an empirical negative-Sobolev norm. We further show that this loss function can be used to extend physics-informed neural networks (PINNs) to stochastic equations, thereby resulting in a new NN-based variant termed Negative Sobolev Norm-PINN (NeS-PINN).