KROM: Kernelized Reduced Order Modeling

Aras Bacho; Jonghyeon Lee; Houman Owhadi

math.NAFeb 27, 2026

KROM: Kernelized Reduced Order Modeling

Authors: Aras Bacho, Jonghyeon Lee, Houman Owhadi

Organizations: Department of Computing and Mathematical Sciences, Caltech, CA, USA.

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 & tables

Panel of Figure 1 — 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.

Panel of Figure 2 — 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.

Panel of Figure 3 — 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.

Panel of Figure 4 — 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.

Panel of Figure 5 — 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).

Panel of Figure 6 — Figure 6. Pointwise relative error of the solution to the Darcy Flow PDE obtained by solving the PDE for $M=32^{2},\rho=4$ with the empirical (left) and Matérn-2.5 (right) kernels. We have used $N=40$ solutions to construct the empirical kernel to generate this error plot.

Panel of Figure 7 — Figure 7. Solving the Burgers equation with the empirical (left) and Matérn-2.5 (right) kernels, sparsity parameter $\rho=5,M=2001$ collocation points and $N=40$ solutions for the empirical kernel.

Panel of Figure 8 — Figure 8. Relative error of the solution to the Burgers equation obtained by the empirical and Matérn-2.5 kernels for $M=2001$ collocation points as sparsity parameter $\rho$ increases for $N=40$ solutions used for the empirical kernel (left); and relative error for the empirical kernel as $N$ increases as $\rho=5$ and $M=2001$ stay fixed (right).

Panel of Figure 9 — Figure 9. Results for the Allen-Cahn equation for the finite difference method (left), empirical kernel (center) and Matérn-2.5 kernel (right)

Panel of Figure 10 — Figure 10. Absolute error of the solution to the Allen-Cahn equation obtained by the empirical and Matérn kernels for $M=51^{2}$ collocation points, sparsity parameter $\rho=5$ and $N=10$ solutions to generate the empirical kernel; we illustrate the absolute error instead of the relative error as the accuracy at each point can be seen more clearly in the former than in the latter.

Panel of Figure 11 — Figure 11. Relative error of the solution to the Allen-Cahn equation obtained by the empirical and Matérn-2.5 kernels as sparsity parameter $\rho$ increases (we have used $N=10$ solutions to create the empirical kernel); and the relative error for the empirical kernel as $N$ increases as $\rho=5$ (right). We have used $M=51^{2}$ collocation points for all our experiments

Panel of Figure 12 — Figure 12. Solution of the 2D Navier-Stokes equation with a classical solver (left), empirical kernel with sparsity parameter $N=50$ solutions (centre) and Matérn-2.5 kernel with lengthscale parameter $\theta=0.3$ (right). We have used $M=32^{2}$ collocation points and sparsity parameter $\rho=5$ for both kernels.

Panel of Figure 13 — Figure 13. Relative error of the solution to 2D Navier-Stokes obtained by the empirical (left) and Matérn-2.5 (right) kernels for $M=32^{2}$ , $\rho=5,N=100$

Panel of Figure 14 — Figure 14. Relative error of the solution to the 2D Navier-Stokes obtained by the empirical kernel for $M=32^{2}$ as $\rho$ increases while $N=100$ (left) and as $N$ increases as $\rho=5$ (right).

Figure 15. Energy spectrum decay of the Navier-Stokes equation at final time $t=5$ with different solvers against $k$ , the magnitude of the wavevector.

experiment set	Semilinear elliptic	Darcy flow	Burgers’ 1D
PDE type	stationary, semilinear	stationary, nonlinear	time-dependent, advection–diffusion
domain $\Omega$	$(0,1)^{2}$	$(0,1)^{2}$	$[-1,1]$
BC / IC	$u=g=0$ on $\partial\Omega$	$p=g=0$ on $\partial\Omega$	$u(\pm 1,t)=0$ , $u(x,0)=-\sin(\pi x)$
data distribution	$f$ from GP, $\sigma=0.15$	$f$ from GP, $\sigma=0.2$	random trigonometric ICs
collocation grid $M$	$M=32^{2}$	$M=32^{2}$	$M=2001$
snapshots $N$	$N=10,20,\dots,200$ FEM solutions	$N=10,20,\dots,200$ FEM solutions	$N=10,20,\dots,80$ WENO solutions

Table 1. Summary of numerical experiments for KROM. The row data distribution describes how right-hand sides or initial conditions are sampled to generate snapshot solutions used to build the empirical kernel.

M

denotes the number of collocation points in the kernel discretization,

N

the number of snapshot solutions, and

\rho

the sparsity parameter in the sparse Cholesky approximation of

K(\phi,\phi)^{-1}

. Matérn–2.5 kernels with the indicated lengthscale

\theta

serve as baseline GP solvers.

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Multifidelity Gaussian process regression for solving nonlinear partial differential equations

Fatima-Zahrae El-Boukkouri, Josselin Garnier, Olivier Roustant

Institut de Mathématiques de Toulouse, Université de Toulouse, INSA, Toulouse, 31077, France · CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, Palaiseau, 91120, France

Surrogate Modeling PDE Solving Multifidelity Modeling Physics-Informed ML Gaussian Process Regression

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Solving nonlinear partial differential equations (PDEs) using kernel methods offers a compelling alternative to traditional numerical solvers. However, the performance of these methods strongly depends on the choice of kernel. In this work, as the available information is inherently multifidelity, we propose a kernel learning approach based on cokriging, leveraging empirical information from multifidelity simulations. In the first step, we fit a differentiable non-stationary kernel to an empirical kernel obtained from low-fidelity simulations. In the second step, we derive a high-fidelity kernel with estimated hyperparameters, and construct a corresponding high-fidelity mean using the multifidelity framework. These components can then be used within a Gaussian process framework for solving PDEs. Finally, we demonstrate the performance of the proposed physics-informed method on the Burgers' equation.

Abstract

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