KROM: Kernelized Reduced Order Modeling
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.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
The algebra of Krom logic programs
This paper investigates the algebraic structure of Krom logic programs, consisting only of facts and rules with at most one body atom. We show that sequential composition endows the class of Krom programs with a natural …
Kernelized Few-Shot Object Detection With Efficient Integral Aggregation
We design a Kernelized Few-shot Object Detector by leveraging kernelized matrices computed over multiple proposal regions, which yield expressive non-linear representations whose model complexity is learned on the fl…
Few-Shot Object DetectionObjectobject-detectionObject Detection+1Algebraic characterizations of least model and uniform equivalence of propositional Krom logic programs
This research note provides algebraic characterizations of the least model, subsumption, and uniform equivalence of propositional Krom logic programs.
Accelerated Variance Reduced Stochastic Extragradient Method for Sparse Machine Learning Problems
Recently, many stochastic gradient descent algorithms with variance reduction have been proposed. Moreover, their proximal variants such as Prox-SVRG can effectively solve non-smooth problems, which makes that they are w…
BIG-bench Machine LearningFace RecognitionRobust Face RecognitionScalable Spatiotemporally Varying Coefficient Modelling with Bayesian Kernelized Tensor Regression
As a regression technique in spatial statistics, the spatiotemporally varying coefficient model (STVC) is an important tool for discovering nonstationary and interpretable response-covariate associations over both space …
regression