KROM: Kernelized Reduced Order Modeling
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917300575141888 |
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| author | Bacho, Aras Lee, Jonghyeon Owhadi, Houman |
| author_facet | Bacho, Aras Lee, Jonghyeon Owhadi, Houman |
| contents | We propose KROM, a kernel-based reduced-order framework for fast solution of nonlinear partial differential equations. KROM formulates PDE solution as a minimum-norm (Gaussian-process) recovery problem in an RKHS, and accelerates the resulting kernel solves by sparsifying the precision matrix via sparse Cholesky factorization. A central ingredient is an empirical kernel constructed from a snapshot library of PDE solutions (generated under varying forcings, initial data, boundary data, or parameters). This snapshot-driven kernel adapts to problem-specific structure -- boundary behavior, oscillations, nonsmooth features, linear constraints, conservation and dissipation laws -- thereby reducing the dependence on hand-tuned stationary kernels. The resulting method yields an implicit reduced model: after sparsification, only a localized subset of effective degrees of freedom is used online. We report numerical results for semilinear elliptic equations, discontinuous-coefficient Darcy flow, viscous Burgers, Allen--Cahn, and two-dimensional Navier--Stokes, showing that empirical kernels can match or outperform Matérn baselines, especially in nonsmooth regimes. We also provide error bounds that separate discretization effects, snapshot-space approximation error, and sparse-Cholesky approximation error. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_00360 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | KROM: Kernelized Reduced Order Modeling Bacho, Aras Lee, Jonghyeon Owhadi, Houman Numerical Analysis Machine Learning We propose KROM, a kernel-based reduced-order framework for fast solution of nonlinear partial differential equations. KROM formulates PDE solution as a minimum-norm (Gaussian-process) recovery problem in an RKHS, and accelerates the resulting kernel solves by sparsifying the precision matrix via sparse Cholesky factorization. A central ingredient is an empirical kernel constructed from a snapshot library of PDE solutions (generated under varying forcings, initial data, boundary data, or parameters). This snapshot-driven kernel adapts to problem-specific structure -- boundary behavior, oscillations, nonsmooth features, linear constraints, conservation and dissipation laws -- thereby reducing the dependence on hand-tuned stationary kernels. The resulting method yields an implicit reduced model: after sparsification, only a localized subset of effective degrees of freedom is used online. We report numerical results for semilinear elliptic equations, discontinuous-coefficient Darcy flow, viscous Burgers, Allen--Cahn, and two-dimensional Navier--Stokes, showing that empirical kernels can match or outperform Matérn baselines, especially in nonsmooth regimes. We also provide error bounds that separate discretization effects, snapshot-space approximation error, and sparse-Cholesky approximation error. |
| title | KROM: Kernelized Reduced Order Modeling |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2603.00360 |