KROM: Kernelized Reduced Order Modeling

• URL: http://arxiv.org/abs/2603.00360v1
• Date: Fri, 27 Feb 2026 22:52:22 GMT
• Title: KROM: Kernelized Reduced Order Modeling
• Authors: Aras Bacho, Jonghyeon Lee, Houman Owhadi,
• Abstract summary: KROM formulates PDE solution as a minimum-norm (Gaussian-process) recovery problem in an RKHS.<n>A central ingredient is an empirical kernel constructed from a snapshot library of PDE solutions.
• Score: 3.988493458010939
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: 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.

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