mlsanityData to 7 Oct 2026

Physics-informed kernel learning

arXiv 2409.13786 ↗First posted 20 Sep 2024stat.ML cs.LG math.STcited by 11

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Abstract (arXiv)

Physics-informed machine learning typically integrates physical priors into the learning process by minimizing a loss function that includes both a data-driven term and a partial differential equation (PDE) regularization. Building on the formulation of the problem as a kernel regression task, we use Fourier methods to approximate the associated kernel, and propose a tractable estimator that minimizes the physics-informed risk function. We refer to this approach as physics-informed kernel learning (PIKL). This framework provides theoretical guarantees, enabling the quantification of the physical prior's impact on convergence speed. We demonstrate the numerical performance of the PIKL estimator through simulations, both in the context of hybrid modeling and in solving PDEs. In particular, we show that PIKL can outperform physics-informed neural networks in terms of both accuracy and computation time. Additionally, we identify cases where PIKL surpasses traditional PDE solvers, particularly in scenarios with noisy boundary conditions.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 14 of its 39 references that are in the map. First posted on arXiv 20 Sep 2024.

Builds on

14 of 39 references.

References · 14

  1. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  2. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  3. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  4. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  5. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  6. 2024Challenges in Training PINNs: A Loss Landscape Perspectivecited by 109
  7. 2024Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equationscited by 87
  8. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  9. 2024The Challenges of the Nonlinear Regime for Physics-Informed Neural Networkscited by 21
  10. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  11. 2022Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networkscited by 19
  12. 2020Physics Informed Deep Kernel Learningcited by 12
2 more
  1. 2024Physics-informed machine learning as a kernel methodcited by 11
  2. 2023Investigations on convergence behaviour of Physics Informed Neural Networks across spectral ranges and derivative orderscited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 5; 2026: 6.

11 papers.

  1. 2026SerpentFlow: Generative Unpaired Domain Alignment via Shared-Structure Decompositioncited by 4
  2. 2025Uncertainty-Aware Diagnostics for Physics-Informed Machine Learningcited by 3
  3. 2025Learning Theory for Kernel Bilevel Optimizationcited by 2
  4. 2025Understanding Generalization in Physics Informed Models through Affine Variety Dimensionscited by 1
  5. 2025Error analysis for the deep Kolmogorov methodcited by 0
  6. 2025Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Methodcited by 0
  7. 2026Knowledge-Informed Kernel State Reconstruction from Heterogeneous Partial Observationscited by 0
  8. 2026Multifidelity Gaussian process regression for solving nonlinear partial differential equationscited by 0
  9. 2026A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learningcited by 0
  10. 2026Physics Informed Random Feature Neural Networks for Solving PDEscited by 0
  11. 2026Fast Learning Rates for Physics-Informed Kernel Methodscited by 0