# KROM: Kernelized Reduced Order Modeling

arXiv: https://arxiv.org/abs/2603.00360 · HTML: https://mlsanity.com/paper/2603.00360/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 82 references that are in the map.
- First posted on arXiv 27 Feb 2026.

## Builds on (most cited first, 20 of 20)

- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representations](https://mlsanity.com/paper/2206.02607/) (2022)
- [Sparse Cholesky factorization by Kullback-Leibler minimization](https://mlsanity.com/paper/2004.14455/) (2020)
- [Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processes](https://mlsanity.com/paper/2304.01294/) (2023)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions](https://mlsanity.com/paper/2211.06786/) (2022)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Deep Bayesian Gaussian Processes for Uncertainty Estimation in Electronic Health Records](https://mlsanity.com/paper/2003.10170/) (2020)
- [Solving High Frequency and Multi-Scale PDEs with Gaussian Processes](https://mlsanity.com/paper/2311.04465/) (2023)
- [Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots](https://mlsanity.com/paper/2311.17007/) (2023)
- [A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2401.03492/) (2024)
- [On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flows](https://mlsanity.com/paper/2001.10110/) (2020)
- [VpROM: A novel Variational AutoEncoder-boosted Reduced Order Model for the treatment of parametric dependencies in nonlinear systems](https://mlsanity.com/paper/2304.12437/) (2023)
- [Neural-Network-Augmented Projection-Based Model Order Reduction for Mitigating the Kolmogorov Barrier to Reducibility of CFD Models](https://mlsanity.com/paper/2212.08939/) (2022)
- [TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNs](https://mlsanity.com/paper/2403.03459/) (2024)
- [Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball](https://mlsanity.com/paper/2108.10517/) (2021)
- [Multi-view Kernel PCA for Time series Forecasting](https://mlsanity.com/paper/2301.09811/) (2023)
- [Decision Theoretic Bootstrapping](https://mlsanity.com/paper/2103.09982/) (2021)
- [Discrete empirical interpolation in the tensor t-product framework](https://mlsanity.com/paper/2410.14519/) (2024)

## Built on by (most cited first, 1 of 1)

- [Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennes](https://mlsanity.com/paper/2603.27936/) (2026)

## Abstract (arXiv)

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 Mat'ern baselines, especially in nonsmooth regimes. We also provide error bounds that separate discretization effects, snapshot-space approximation error, and sparse-Cholesky approximation error.