mlsanityData to 14 Oct 2026

Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs

arXiv 2305.04962 ↗First posted 8 May 2023math.NA cs.NA stat.MLcited by 21

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Abstract (arXiv)

We introduce a priori Sobolev-space error estimates for the solution of nonlinear, and possibly parametric, PDEs using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

Key facts. Cited by 21 papers in the map (data through 14 Oct 2026). Builds on 28 of its 100 references that are in the map. First posted on arXiv 8 May 2023.

Builds on

28 of 100 references.

References · 28

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2017Deep Neural Networks as Gaussian Processescited by 622
  4. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  5. 2015Deep Kernel Learningcited by 445
  6. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 364
  7. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  8. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  9. 2017Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equationscited by 72
  10. 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
  11. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 57
  12. 2020A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challengescited by 52
16 more
  1. 2017Bayesian Probabilistic Numerical Methodscited by 51
  2. 2023Kernel Methods are Competitive for Operator Learningcited by 50
  3. 2018Kernel Flows: from learning kernels from data into the abysscited by 49
  4. 2020Sparse Cholesky factorization by Kullback-Leibler minimizationcited by 41
  5. 2021Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimalitycited by 40
  6. 2023Sparse Cholesky Factorization for Solving Nonlinear PDEs via Gaussian Processescited by 31
  7. 2017Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexitycited by 24
  8. 2017Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problemscited by 20
  9. 2022AutoIP: A United Framework to Integrate Physics into Gaussian Processescited by 20
  10. 2020Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimationcited by 18
  11. 2021Solving high-dimensional parabolic PDEs using the tensor train formatcited by 17
  12. 2020Do ideas have shape? Idea registration as the continuous limit of artificial neural networkscited by 16
  13. 2021Bayesian Numerical Methods for Nonlinear Partial Differential Equationscited by 15
  14. 2022One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernelscited by 15
  15. 2021Numerical methods for Mean field Games based on Gaussian Processes and Fourier Featurescited by 7
  16. 2022A Kernel Approach for PDE Discovery and Operator Learningcited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 3; 2024: 5; 2025: 6; 2026: 7.

21 papers.

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  1. 2024Physics-informed kernel learningcited by 11
  2. 2023Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemescited by 10
  3. 2024Toward Efficient Kernel-Based Solvers for Nonlinear PDEscited by 7
  4. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 7
  5. 2024Solving Partial Differential Equations with Random Feature Modelscited by 6
  6. 2023Gaussian process learning of nonlinear dynamicscited by 5
  7. 2024Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operatorcited by 5
  8. 2025Solving Nonlinear PDEs with Sparse Radial Basis Function Networkscited by 3
  9. 2025Data-efficient Kernel Methods for Learning Hamiltonian Systemscited by 2
  10. 2023PDE-constrained Gaussian process surrogate modeling with uncertain data locationscited by 1
  11. 2025Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structurescited by 1
  12. 2025Tensor Gaussian Processes: Efficient Solvers for Nonlinear PDEscited by 1
9 more
  1. 2026Sparse RBF Networks for PDEs and nonlocal equations: function space theory, operator calculus, and training algorithmscited by 1
  2. 2026Structure-Aware Variational Learning of a Class of Generalized Diffusionscited by 1
  3. 2026Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learningcited by 1
  4. 2024Reduced Order Models and Conditional Expectation -- Analysing Parametric Low-Order Approximationscited by 0
  5. 2025Solving Functional PDEs with Gaussian Processes and Applications to Functional Renormalization Group Equationscited by 0
  6. 2026Neural network surrogates with uncertainty quantification for inverse problems in partial differential equationscited by 0
  7. 2026A unified perspective of Gaussian process approximation for differential equationscited by 0
  8. 2026Physics Informed Random Feature Neural Networks for Solving PDEscited by 0
  9. 2026Fast Learning Rates for Physics-Informed Kernel Methodscited by 0