mlsanityData to 7 Oct 2026

Kernel Methods are Competitive for Operator Learning

arXiv 2304.13202 ↗First posted 26 Apr 2023stat.ML cs.LG cs.NAcited by 50

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Abstract (arXiv)

We present a general kernel-based framework for learning operators between Banach spaces along with a priori error analysis and comprehensive numerical comparisons with popular neural net (NN) approaches such as Deep Operator Net (DeepONet) [Lu et al.] and Fourier Neural Operator (FNO) [Li et al.]. We consider the setting where the input/output spaces of target operator G^dagger,:, U→ V are reproducing kernel Hilbert spaces (RKHS), the data comes in the form of partial observations φ(ui), varphi(vi) of input/output functions vi=G^dagger(ui) (i=1,ldots,N), and the measurement operators φ,:, U→ ℝn and varphi,:, V → ℝm are linear. Writing psi,:, ℝn → U and chi,:, ℝm → V for the optimal recovery maps associated with φ and varphi, we approximate G^dagger with barG=chi circ barf circ φ where barf is an optimal recovery approximation of f^dagger:=varphi circ G^dagger circ psi,:,ℝn → ℝm. We show that, even when using vanilla kernels (e.g., linear or Mat'ern), our approach is competitive in terms of cost-accuracy trade-off and either matches or beats the performance of NN methods on a majority of benchmarks. Additionally, our framework offers several advantages inherited from kernel methods: simplicity, interpretability, convergence guarantees, a priori error estimates, and Bayesian uncertainty quantification. As such, it can serve as a natural benchmark for operator learning.

Key facts. Cited by 50 papers in the map (data through 7 Oct 2026). Builds on 23 of its 94 references that are in the map. First posted on arXiv 26 Apr 2023.

Builds on

23 of 94 references.

References · 23

  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. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  4. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  5. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  6. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  7. 2022Learning Operators with Coupled Attentioncited by 88
  8. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  9. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  10. 2015Operator-valued Kernels for Learning from Functional Response Datacited by 59
  11. 2018Kernel Flows: from learning kernels from data into the abysscited by 49
  12. 2020Sparse Cholesky factorization by Kullback-Leibler minimizationcited by 40
11 more
  1. 2018SwitchNet: a neural network model for forward and inverse scattering problemscited by 35
  2. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
  3. 2021Computational Graph Completioncited by 28
  4. 2017Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexitycited by 24
  5. 2020Learning dynamical systems from data: a simple cross-validation perspectivecited by 23
  6. 2018BCR-Net: a neural network based on the nonstandard wavelet formcited by 18
  7. 2020Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimationcited by 18
  8. 2020Do ideas have shape? Idea registration as the continuous limit of artificial neural networkscited by 16
  9. 2015Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Gamescited by 15
  10. 2021Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methodscited by 15
  11. 2016Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficientscited by 3

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Citing papers per year: 2023: 5; 2024: 18; 2025: 18; 2026: 9.

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  1. 2024Operator Learning: Algorithms and Analysiscited by 84
  2. 2024Standard Gaussian Process is All You Need for High-Dimensional Bayesian Optimizationcited by 33
  3. 2023Multi-Resolution Active Learning of Fourier Neural Operatorscited by 22
  4. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  5. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  6. 2024Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problemscited by 18
  7. 2024Operator Learning with Gaussian Processescited by 18
  8. 2024Mixtures of Neural Operators Reduce Active Complexity in Operator Learningcited by 16
  9. 2025Operator Learning: A Statistical Perspectivecited by 14
  10. 2024Linearization Turns Neural Operators into Function-Valued Gaussian Processescited by 11
  11. 2024Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learningcited by 10
  12. 2023Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dotscited by 8
38 more
  1. 2024A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equationscited by 8
  2. 2024Ensemble and Mixture-of-Experts DeepONets For Operator Learningcited by 8
  3. 2024Nonparametric Control Koopman Operatorscited by 7
  4. 2025Polynomial Chaos Expansion for Operator Learningcited by 7
  5. 2023Learning Homogenization for Elliptic Operatorscited by 6
  6. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 6
  7. 2025A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theorycited by 6
  8. 2025Operator Learning at Machine Precisioncited by 6
  9. 2023Gaussian process learning of nonlinear dynamicscited by 5
  10. 2025Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximationcited by 5
  11. 2024Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulationcited by 4
  12. 2024Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantificationcited by 4
  13. 2024Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulationcited by 4
  14. 2025Cauchy Random Features for Operator Learning in Sobolev Spacecited by 3
  15. 2025Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processescited by 3
  16. 2025RONOM: Reduced-Order Neural Operator Modelingcited by 2
  17. 2025Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operatorscited by 2
  18. 2026Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flowscited by 2
  19. 2024Learning epidemic trajectories through Kernel Operator Learning: from modelling to optimal controlcited by 1
  20. 2024Spectral Truncation Kernels: Noncommutativity in C^*-algebraic Kernel Machinescited by 1
  21. 2025Learning cardiac activation and repolarization times with operator learningcited by 1
  22. 2025Graph-Based Operator Learning from Limited Data on Irregular Domainscited by 1
  23. 2025Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometrycited by 1
  24. 2025A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learningcited by 1
  25. 2025Deep Learning for Subspace Regressioncited by 1
  26. 2025Regularized Random Fourier Features and Finite Element Reconstruction for Operator Learning in Sobolev Spacecited by 1
  27. 2026MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Datacited by 1
  28. 2026HyCOP: Hybrid Composition Operators for Interpretable Learning of PDEscited by 1
  29. 2026Geometry-Aware Post-Hoc Uncertainty Quantification in Operator Learningcited by 1
  30. 2024Log-PDE Methods for Rough Signature Kernelscited by 0
  31. 2024Minimal Variance Model Aggregation: A principled, non-intrusive, and versatile integration of black box modelscited by 0
  32. 2025Infinite Neural Operators: Gaussian processes on functionscited by 0
  33. 2025Solving Functional PDEs with Gaussian Processes and Applications to Functional Renormalization Group Equationscited by 0
  34. 2026Learning solution operators of PDEs with sparse approximation methodscited by 0
  35. 2026Kernel-based Operator Learning: Error Analysis, Budget Allocation, and a Physics-Informed Extensioncited by 0
  36. 2026The Frame Kernel Method for Multiscale Operator Learningcited by 0
  37. 2026Neural means and kernel corrections for operator learningcited by 0
  38. 2026One-Step Generative Surrogate Models via Block-Triangular Joint Driftingcited by 0