KROM: Kernelized Reduced Order Modeling

TL;DR

KROM introduces a kernel-based reduced-order modeling framework that efficiently solves nonlinear PDEs by leveraging empirical kernels and sparse Cholesky factorization, adapting to problem-specific features for improved accuracy.

Contribution

The paper presents a novel kernelized reduced-order modeling approach that constructs empirical kernels from solution snapshots, enabling adaptive and efficient PDE solutions with theoretical error bounds.

Findings

Empirical kernels outperform Matérn kernels in nonsmooth regimes.

Sparse Cholesky factorization accelerates kernel solves significantly.

Method effectively handles complex PDEs like Navier-Stokes and Allen--Cahn.

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 Mat\'ern baselines, especially in nonsmooth regimes. We also provide error bounds that separate discretization effects, snapshot-space approximation error, and sparse-Cholesky approximation error.