Toward Efficient Kernel-Based Solvers for Nonlinear PDEs

TL;DR

This paper presents a scalable kernel learning framework for efficiently solving nonlinear PDEs by removing the need for complex operator embedding and Gram matrix computations, enabling large-scale problem solving.

Contribution

The proposed method introduces a kernel interpolation approach that simplifies implementation and improves scalability for nonlinear PDE solvers, with proven convergence and rate analysis.

Findings

Reduces computational costs by avoiding full Gram matrix calculations.

Demonstrates effectiveness on benchmark PDEs with numerical experiments.

Achieves scalable solutions for large numbers of collocation points.

Abstract

We introduce a novel kernel learning framework toward efficiently solving nonlinear partial differential equations (PDEs). In contrast to the state-of-the-art kernel solver that embeds differential operators within kernels, posing challenges with a large number of collocation points, our approach eliminates these operators from the kernel. We model the solution using a standard kernel interpolation form and differentiate the interpolant to compute the derivatives. Our framework obviates the need for complex Gram matrix construction between solutions and their derivatives, allowing for a straightforward implementation and scalable computation. As an instance, we allocate the collocation points on a grid and adopt a product kernel, which yields a Kronecker product structure in the interpolation. This structure enables us to avoid computing the full Gram matrix, reducing costs and scaling efficiently to a large number of collocation points. We provide a proof of the convergence and rate analysis of our method under appropriate regularity assumptions. In numerical experiments, we demonstrate the advantages of our method in solving several benchmark PDEs.