mlsanityData to 7 Oct 2026

A Kernel Approach for PDE Discovery and Operator Learning

arXiv 2210.08140 ↗First posted 14 Oct 2022stat.ML cs.LGcited by 7

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Abstract (arXiv)

This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel smoothing is utilized to denoise the data and approximate derivatives of the solution. This information is then used in a kernel regression model to learn the algebraic form of the PDE. The learned PDE is then used within a kernel based solver to approximate the solution of the PDE with a new source/boundary term, thereby constituting an operator learning framework. Numerical experiments compare the method to state-of-the-art algorithms and demonstrate its competitive performance.

Key facts. Cited by 7 papers in the map (data through 7 Oct 2026). Builds on 17 of its 67 references that are in the map. First posted on arXiv 14 Oct 2022.

Builds on

17 of 67 references.

References · 17

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  3. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  4. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 360
  5. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  6. 2017PDE-Net: Learning PDEs from Datacited by 304
  7. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  8. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  9. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 139
  10. 2021Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and controlcited by 134
  11. 2018Kernel Flows: from learning kernels from data into the abysscited by 49
  12. 2017Linearly constrained Gaussian processescited by 41
5 more
  1. 2021A Probabilistic State Space Model for Joint Inference from Differential Equations and Datacited by 21
  2. 2020Gaussian Process Regression constrained by Boundary Value Problemscited by 15
  3. 2022Constraining Gaussian Processes to Systems of Linear Ordinary Differential Equationscited by 14
  4. 2021Probabilistic ODE Solutions in Millions of Dimensionscited by 13
  5. 2021Numerical methods for Mean field Games based on Gaussian Processes and Fourier Featurescited by 7

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Cited by year

Citing papers per year: 2023: 3; 2024: 2; 2025: 2.

7 papers.

  1. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  2. 2023Multi-Resolution Active Learning of Fourier Neural Operatorscited by 22
  3. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  4. 2024Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problemscited by 18
  5. 2024Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulationcited by 4
  6. 2025Data-efficient Kernel Methods for Learning Hamiltonian Systemscited by 2
  7. 2025ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spacescited by 1