KROM: Kernelized Reduced Order Modeling

2026-02-27

AI Summary

This work proposes a minimum-norm Gaussian process recovery framework in reproducing kernel Hilbert space (RKHS) to address the high computational complexity of solving nonlinear partial differential equations (PDEs) and the reliance of traditional reduced-order models on manual hyperparameter tuning. By constructing problem-adaptive empirical kernels from PDE snapshot libraries, the method automatically captures boundary behavior, nonsmooth features, and physical constraints, eliminating dependence on fixed stationary kernels. Sparse Cholesky decomposition is integrated to accelerate kernel solves, enabling implicit and localized online reduced-order modeling. On multiple benchmark nonlinear PDE problems, the proposed empirical kernels match or outperform Matérn kernels in nonsmooth regimes and come with theoretical guarantees of separable error bounds.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• nonlinear partial differential equations
• reduced-order modeling
• nonsmooth features
• parameter-dependent PDEs
• fast solution

Innovation

• kernelized reduced-order modeling
• empirical kernel
• sparse Cholesky factorization
• Gaussian process regression
• nonlinear PDEs