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

TL;DR

This work develops a kernel-based framework for solving nonlinear PDEs and SPDEs with rough forcing by replacing the empirical $L^2$ PDE residual with a weak loss based on the negative Sobolev norm $\|\cdot\|_{H^{-s}}$, enabling convergence even when the true solution lies outside the underlying RKHS. By formulating the continuum problem as an optimal recovery in a misspecified RKHS and discretizing with test functions, the authors derive a Gauss-Newton based numerical method with a representer theorem that yields a finite-dimensional solution expressed as kernel mixtures. They prove existence and convergence guarantees as the regularization parameter $\gamma$ tends to zero and the discretization parameters grow, respectively, and demonstrate the approach on rough spatial PDEs and time-dependent SPDEs, comparing with PINNs and showing improved performance when using the weak Sobolev loss. The paper also discusses extending the framework to define new solution concepts for singular PDEs/SPDEs and presents auxiliary proofs, NN architectures, and sampling schemes for stochastic inputs. Overall, the proposed Negative Sobolev Norm-PINN (NeS-PINN) and kernel methods offer robust, theoretically principled tools for challenging PDEs with irregular forcing and lay groundwork for broader applicability to singular SPDEs and kernel-misspecified priors.

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).

Figures (7)

• Figure 1: Numerical results for the one-dimensional linear elliptic PDE \ref{['eq: Poisson equation']}.
• Figure 1: Space-time white noise generated by two methods.
• Figure 2: One example solution to the two-dimensional semi-linear elliptic PDE \ref{['eq: 2d semi-linear pde']}.
• Figure 3: Numerical solutions to the two-dimensional semi-linear elliptic PDE \ref{['eq: 2d semi-linear pde']} using the kernel and NN approximations with either the negative Sobolev loss or the pointwise loss.
• Figure 4: Impact of norm selection on the recovered solution.

Theorems & Definitions (34)

• Remark 2.1
• Theorem 2.2: Existence of a Minimizer
• Proof 1
• Theorem 2.3: Convergence
• Proof 2
• Lemma 2.4: Density
• Remark 2.5: Convergence Rate for Poisson Equation
• Proposition 3.1
• Proof 3
• Remark 3.2