Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs

TL;DR

This paper provides Sobolev-space error estimates for Gaussian process and kernel methods applied to nonlinear and parametric PDEs, highlighting conditions for dimension-benign convergence and the impact of solution regularity.

Contribution

It introduces a priori Sobolev-space error bounds for kernel-based PDE solutions, connecting regularity, stability, and convergence rates, with applications to high-dimensional problems.

Findings

Error estimates depend on Sobolev regularity and stability assumptions.

Dimension-benign convergence occurs when solutions are sufficiently smooth.

Trade-off identified between solution regularity and the curse of dimensionality.

Abstract

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.