用数据驱动的核方法快速求解非线性微分方程，提升精度与效率。
KROM: Kernelized Reduced Order Modeling
- 基于数据生成的核函数，自动捕捉方程的边界、振荡等特性。
- 通过稀疏化精度矩阵，仅需局部自由度在线求解，速度大幅提升。
- 适用于复杂非光滑问题，适合需要快速仿真与建模的研究者。
我们提出 KROM，一种基于核的降阶框架，用于快速求解非线性偏微分方程。KROM 将 PDE 解表示为再生核希尔伯特空间（RKHS）中的最小范数（高斯过程）恢复问题，并通过稀疏乔列斯基分解加速核求解。核心是利用不同激励、初值、边值或参数下生成的解快照库构建经验核，该核能自适应捕捉问题特有结构——如边界行为、振荡、非光滑特征、线性约束及守恒/耗散定律，降低对人工调参平稳核的依赖。所得方法为隐式降阶模型：稀疏化后，仅使用局部有效自由度在线计算。在半线性椭圆方程、间断系数达西流、粘性伯格斯方程、艾伦-钱恩方程及二维纳维-斯托克斯方程上验证，经验核性能可媲美或超越马特尔核基线，尤其在非光滑情形表现更优。同时提供了误差界，分离了离散化误差、快照空间近似误差和稀疏乔列斯基近似误差。
原文摘要 · Abstract (English)
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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