mlsanityData to 14 Oct 2026

KROM: Kernelized Reduced Order Modeling

arXiv 2603.00360 ↗First posted 27 Feb 2026math.NA cs.LG cs.NAcited by 1

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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.

Key facts. Cited by 1 paper 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

20 of 82 references.

References · 20

  1. 2019Machine Learning for Fluid Mechanicscited by 385
  2. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  3. 2022CROM: Continuous Reduced-Order Modeling of PDEs Using Implicit Neural Representationscited by 58
  4. 2020Sparse Cholesky factorization by Kullback-Leibler minimizationcited by 41
  5. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  6. 2022Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reductioncited by 30
  7. 2022Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutionscited by 29
  8. 2021Computational Graph Completioncited by 28
  9. 2020Deep Bayesian Gaussian Processes for Uncertainty Estimation in Electronic Health Recordscited by 16
  10. 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
  11. 2023Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dotscited by 8
  12. 2024A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equationscited by 8
8 more
  1. 2020On the stability of projection-based model order reduction for convection-dominated laminar and turbulent flowscited by 6
  2. 2023VpROM: A novel Variational AutoEncoder-boosted Reduced Order Model for the treatment of parametric dependencies in nonlinear systemscited by 6
  3. 2022Neural-Network-Augmented Projection-Based Model Order Reduction for Mitigating the Kolmogorov Barrier to Reducibility of CFD Modelscited by 5
  4. 2024TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNscited by 5
  5. 2021Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ballcited by 2
  6. 2023Multi-view Kernel PCA for Time series Forecastingcited by 2
  7. 2021Decision Theoretic Bootstrappingcited by 1
  8. 2024Discrete empirical interpolation in the tensor t-product frameworkcited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 1.

1 paper.

  1. 2026Deflation-PINNs: Learning Multiple Solutions for PDEs and Landau-de Gennescited by 0