# Kernel Methods are Competitive for Operator Learning

arXiv: https://arxiv.org/abs/2304.13202 · HTML: https://mlsanity.com/paper/2304.13202/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 94 references that are in the map.
- First posted on arXiv 26 Apr 2023.

## Builds on (most cited first, 23 of 23)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problems](https://mlsanity.com/paper/2101.07206/) (2020)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Kernel Flows: from learning kernels from data into the abyss](https://mlsanity.com/paper/1808.04475/) (2018)
- [Sparse Cholesky factorization by Kullback-Leibler minimization](https://mlsanity.com/paper/2004.14455/) (2020)
- [SwitchNet: a neural network model for forward and inverse scattering problems](https://mlsanity.com/paper/1810.09675/) (2018)
- [Learning elliptic partial differential equations with randomized linear algebra](https://mlsanity.com/paper/2102.00491/) (2021)
- [Computational Graph Completion](https://mlsanity.com/paper/2110.10323/) (2021)
- [Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity](https://mlsanity.com/paper/1706.02205/) (2017)
- [Learning dynamical systems from data: a simple cross-validation perspective](https://mlsanity.com/paper/2007.05074/) (2020)
- [BCR-Net: a neural network based on the nonstandard wavelet form](https://mlsanity.com/paper/1810.08754/) (2018)
- [Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation](https://mlsanity.com/paper/2005.11375/) (2020)
- [Do ideas have shape? Idea registration as the continuous limit of artificial neural networks](https://mlsanity.com/paper/2008.03920/) (2020)
- [Multigrid with rough coefficients and Multiresolution operator decomposition from Hierarchical Information Games](https://mlsanity.com/paper/1503.03467/) (2015)
- [Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methods](https://mlsanity.com/paper/2103.10935/) (2021)
- [Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients](https://mlsanity.com/paper/1606.07686/) (2016)

## Built on by (most cited first, 30 of 50)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Standard Gaussian Process is All You Need for High-Dimensional Bayesian Optimization](https://mlsanity.com/paper/2402.02746/) (2024)
- [Multi-Resolution Active Learning of Fourier Neural Operators](https://mlsanity.com/paper/2309.16971/) (2023)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [Operator Learning with Gaussian Processes](https://mlsanity.com/paper/2409.04538/) (2024)
- [Operator Learning: A Statistical Perspective](https://mlsanity.com/paper/2504.03503/) (2025)
- [Mixtures of Neural Operators Reduce Active Complexity in Operator Learning](https://mlsanity.com/paper/2404.09101/) (2024)
- [Linearization Turns Neural Operators into Function-Valued Gaussian Processes](https://mlsanity.com/paper/2406.05072/) (2024)
- [Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning](https://mlsanity.com/paper/2407.00809/) (2024)
- [Ensemble and Mixture-of-Experts DeepONets For Operator Learning](https://mlsanity.com/paper/2405.11907/) (2024)
- [A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory](https://mlsanity.com/paper/2510.25379/) (2025)
- [Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots](https://mlsanity.com/paper/2311.17007/) (2023)
- [A Gaussian Process Framework for Solving Forward and Inverse Problems Involving Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2401.03492/) (2024)
- [Polynomial Chaos Expansion for Operator Learning](https://mlsanity.com/paper/2508.20886/) (2025)
- [Nonparametric Control Koopman Operators](https://mlsanity.com/paper/2405.07312/) (2024)
- [Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysis](https://mlsanity.com/paper/2503.01036/) (2025)
- [Operator Learning at Machine Precision](https://mlsanity.com/paper/2511.19980/) (2025)
- [Learning Homogenization for Elliptic Operators](https://mlsanity.com/paper/2306.12006/) (2023)
- [Spectral Embedding via Chebyshev Bases for Robust DeepONet Approximation](https://mlsanity.com/paper/2512.09165/) (2025)
- [Gaussian process learning of nonlinear dynamics](https://mlsanity.com/paper/2312.12193/) (2023)
- [Cauchy Random Features for Operator Learning in Sobolev Space](https://mlsanity.com/paper/2503.00300/) (2025)
- [Gaussian Measures Conditioned on Nonlinear Observations: Consistency, MAP Estimators, and Simulation](https://mlsanity.com/paper/2405.13149/) (2024)
- [Alpha-VI DeepONet: A prior-robust variational Bayesian approach for enhancing DeepONets with uncertainty quantification](https://mlsanity.com/paper/2408.00681/) (2024)
- [Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation](https://mlsanity.com/paper/2410.13794/) (2024)
- [Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes](https://mlsanity.com/paper/2510.05568/) (2025)
- [Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flows](https://mlsanity.com/paper/2602.15472/) (2026)
- [RONOM: Reduced-Order Neural Operator Modeling](https://mlsanity.com/paper/2507.12814/) (2025)
- [Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operators](https://mlsanity.com/paper/2509.26371/) (2025)

## 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.