Toward Efficient Kernel-Based Solvers for Nonlinear PDEs

• URL: http://arxiv.org/abs/2410.11165v3
• Date: Sun, 03 Nov 2024 04:01:46 GMT
• Title: Toward Efficient Kernel-Based Solvers for Nonlinear PDEs
• Authors: Zhitong Xu, Da Long, Yiming Xu, Guang Yang, Shandian Zhe, Houman Owhadi,
• Abstract summary: This paper introduces 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, our approach eliminates these operators from the kernel. We model the solution using a standard kernel form and differentiate the interpolant to compute the derivatives.
• Score: 19.975293084297014
• License:
• Abstract: This paper introduces 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.

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