KROM: Kernelized Reduced Order Modeling

TL;DR

Numerical results are reported 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.

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.

Figures (15)

• Figure 1: True solution (left) compared with the solution given by the empirical kernel (centre) and Matérn-2.5 kernel (right) for $M=64^2$ collocation points and sparsity parameter $\rho = 4$, as well as $N=5000$ solutions for constructing the empirical kernel.
• Figure 2: Decrease in relative error of solution obtained by the empirical kernel as the number of solutions $N$ (left) increases while number of collocation points $M=32^2$ and sparsity parameter $\rho = 5$ are fixed; as $\rho$ (centre) increases while $N=200, M=32^2$ are fixed; and as $M$ (right) increases for $\rho=4, N=5000$ are fixed.
• Figure 3: Pointwise relative error of the solution to the nonlinear elliptic PDE obtained by the empirical kernel for $M=32^2, \rho = 5$, $N=200$ for the empirical (left) and Matérn-2.5 (right) kernels; notice the relative error is higher where the true solution is near zero.
• Figure 4: The Darcy Flow numerical solution obtained by a finite element method (left), an empirical kernel with $N=40$ solutions (centre) and a Matérn kernel with lengthscale parameter $\theta =0.3$ (right). We have used sparsity parameter $\rho = 4, M=32^2$ collocation points and 2 Gauss-Newton iterations to solve the PDE for both kernels.
• Figure 5: Relative error of the solution to the Darcy Flow PDE obtained by the empirical kernel for $M=32^2$ collocation points as sparsity parameter $\rho$ increases while the number of solutions used to construct the empirical kernel $N=200$ stays fixed (left); and as $N$ increases for fixed $\rho=4$ (right).

Theorems & Definitions (10)

• Theorem 4.1: Empirical-kernel error estimate and convergence in $N$
• proof
• Definition 4.2: Solution manifold and Kolmogorov $N$-width
• Theorem 4.4: Sub-exponential decay of $d_N(\mathcal{M};V)$
• Remark 4.5: Implications for the empirical-kernel ROM
• Theorem A.1
• proof
• Example A.2
• Theorem A.3: Representer Theorem
• proof : Proof of Theorem \ref{['thm:empirical_kernel_rate']}