Toward Efficient Kernel-Based Solvers for Nonlinear PDEs

Zhitong Xu, Da Long, Yiming Xu, Guang Yang, Shandian Zhe, Houman Owhadi
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:69526-69544, 2025.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-xu25u, title = {Toward Efficient Kernel-Based Solvers for Nonlinear {PDE}s}, author = {Xu, Zhitong and Long, Da and Xu, Yiming and Yang, Guang and Zhe, Shandian and Owhadi, Houman}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {69526--69544}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/xu25u/xu25u.pdf}, url = {https://proceedings.mlr.press/v267/xu25u.html}, 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.} }
Endnote
%0 Conference Paper %T Toward Efficient Kernel-Based Solvers for Nonlinear PDEs %A Zhitong Xu %A Da Long %A Yiming Xu %A Guang Yang %A Shandian Zhe %A Houman Owhadi %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-xu25u %I PMLR %P 69526--69544 %U https://proceedings.mlr.press/v267/xu25u.html %V 267 %X 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.
APA
Xu, Z., Long, D., Xu, Y., Yang, G., Zhe, S. & Owhadi, H.. (2025). Toward Efficient Kernel-Based Solvers for Nonlinear PDEs. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:69526-69544 Available from https://proceedings.mlr.press/v267/xu25u.html.

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