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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.11165 |
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| _version_ | 1866909639855046656 |
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| author | Xu, Zhitong Long, Da Xu, Yiming Yang, Guang Zhe, Shandian Owhadi, Houman |
| author_facet | Xu, Zhitong Long, Da Xu, Yiming Yang, Guang Zhe, Shandian Owhadi, Houman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11165 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toward Efficient Kernel-Based Solvers for Nonlinear PDEs Xu, Zhitong Long, Da Xu, Yiming Yang, Guang Zhe, Shandian Owhadi, Houman Machine Learning 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. |
| title | Toward Efficient Kernel-Based Solvers for Nonlinear PDEs |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2410.11165 |