mlsanityData to 7 Oct 2026

Operator Learning: Algorithms and Analysis

arXiv 2402.15715 ↗First posted 24 Feb 2024cs.LG cs.NA math.NAcited by 84

Not on the map. Why?

Abstract (arXiv)

Operator learning refers to the application of ideas from machine learning to approximate (typically nonlinear) operators mapping between Banach spaces of functions. Such operators often arise from physical models expressed in terms of partial differential equations (PDEs). In this context, such approximate operators hold great potential as efficient surrogate models to complement traditional numerical methods in many-query tasks. Being data-driven, they also enable model discovery when a mathematical description in terms of a PDE is not available. This review focuses primarily on neural operators, built on the success of deep neural networks in the approximation of functions defined on finite dimensional Euclidean spaces. Empirically, neural operators have shown success in a variety of applications, but our theoretical understanding remains incomplete. This review article summarizes recent progress and the current state of our theoretical understanding of neural operators, focusing on an approximation theoretic point of view.

Key facts. Cited by 84 papers in the map (data through 7 Oct 2026). Builds on 78 of its 145 references that are in the map. First posted on arXiv 24 Feb 2024.

Builds on

78 of 145 references.

References · 78

  1. 2017Attention Is All You Needcited by 59,932
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  5. 2016Error bounds for approximations with deep ReLU networkscited by 567
  6. 2017Nonparametric regression using deep neural networks with ReLU activation functioncited by 431
  7. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
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  9. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  10. 2023Spherical Fourier Neural Operators: Learning Stable Dynamics on the Spherecited by 225
  11. 2022U-NO: U-shaped Neural Operatorscited by 219
  12. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
38 more
  1. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  2. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  3. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  4. 2020DiffusionNet: Discretization Agnostic Learning on Surfacescited by 163
  5. 2017Deep Functional Maps: Structured Prediction for Dense Shape Correspondencecited by 134
  6. 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 125
  7. 2019Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systemscited by 96
  8. 2017Optimal Approximation with Sparsely Connected Deep Neural Networkscited by 90
  9. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
  10. 2019The phase diagram of approximation rates for deep neural networkscited by 86
  11. 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 81
  12. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
  13. 2016SyncSpecCNN: Synchronized Spectral CNN for 3D Shape Segmentationcited by 70
  14. 2018Deep Dynamical Modeling and Control of Unsteady Fluid Flowscited by 67
  15. 2022Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spacescited by 66
  16. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  17. 2020Neural Network Approximation: Three Hidden Layers Are Enoughcited by 63
  18. 2020Representation formulas and pointwise properties for Barron functionscited by 62
  19. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  20. 2020A physics-informed operator regression framework for extracting data-driven continuum modelscited by 60
  21. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  22. 2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 50
  23. 2023Kernel Methods are Competitive for Operator Learningcited by 50
  24. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  25. 2021Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systemscited by 44
  26. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  27. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  28. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 43
  29. 2022Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learningcited by 43
  30. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  31. 2019Nonlinear Approximation via Compositionscited by 37
  32. 2020A Survey of Deep Learning for Scientific Discoverycited by 36
  33. 2021Elementary superexpressive activationscited by 35
  34. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  35. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
  36. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 29
  37. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 27
  38. 2023The Parametric Complexity of Operator Learningcited by 27

50 most cited of 78 All 78 (JSON)

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 18; 2025: 35; 2026: 31.

84 papers.

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  1. 2024A Library for Learning Neural Operatorscited by 31
  2. 2024Data Complexity Estimates for Operator Learningcited by 21
  3. 2024Discretization Error of Fourier Neural Operatorscited by 20
  4. 2024Operator Learning with Gaussian Processescited by 18
  5. 2025Operator Learning: A Statistical Perspectivecited by 14
  6. 2024Derivative-informed neural operator acceleration of geometric MCMC for infinite-dimensional Bayesian inverse problemscited by 13
  7. 2025Learning Enhanced Ensemble Filterscited by 11
  8. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  9. 2024Operator Learning of Lipschitz Operators: An Information-Theoretic Perspectivecited by 9
  10. 2025Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equationscited by 9
  11. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 8
  12. 2025Enhancing Fourier Neural Operators with Local Spatial Featurescited by 8
38 more
  1. 2024LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transportcited by 7
  2. 2024Optimal sampling for least-squares approximationcited by 6
  3. 2025Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operatorscited by 6
  4. 2025KANO: Kolmogorov-Arnold Neural Operatorcited by 6
  5. 2025Operator Learning at Machine Precisioncited by 6
  6. 2024Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equationscited by 5
  7. 2024Micrometer: Micromechanics Transformer for Predicting Mechanical Responses of Heterogeneous Materialscited by 5
  8. 2024Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operatorscited by 4
  9. 2025Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?cited by 4
  10. 2025Theory-to-Practice Gap for Neural Networks and Neural Operatorscited by 4
  11. 2025Multi-Level Monte Carlo Training of Neural Operatorscited by 4
  12. 2024Structure-Preserving Operator Learningcited by 3
  13. 2025Continuum Transformers Perform In-Context Learning by Operator Gradient Descentcited by 3
  14. 2025When do World Models Successfully Learn Dynamical Systems?cited by 3
  15. 2025Deep Neural ODE Operator Networks for PDEscited by 3
  16. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  17. 2025Toward Data-Driven Surrogates of the Solar Wind with Spherical Fourier Neural Operatorcited by 3
  18. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 3
  19. 2026A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systemscited by 3
  20. 2026SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolutioncited by 3
  21. 2024Projection Methods for Operator Learning and Universal Approximationcited by 2
  22. 2024M2M: Learning controllable Multi of experts and multi-scale operators are the Partial Differential Equations needcited by 2
  23. 2024Convergence analysis of wide shallow neural operators within the framework of Neural Tangent Kernelcited by 2
  24. 2025Locally Adaptive Conformal Inference for Operator Modelscited by 2
  25. 2025Generalized Spherical Neural Operators: Green's Function Formulationcited by 2
  26. 2025Geometric Laplace Neural Operatorcited by 2
  27. 2026A short tour of operator learning theory: Convergence rates, statistical limits, and open questionscited by 2
  28. 2026Improving Feasibility via Fast Autoencoder-Based Projectionscited by 2
  29. 2026One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operatorscited by 2
  30. 2026Topological Neural Operatorscited by 2
  31. 2024Coupled Input-Output Dimension Reduction: Application to Goal-oriented Bayesian Experimental Design and Global Sensitivity Analysiscited by 1
  32. 2025Graph-Based Operator Learning from Limited Data on Irregular Domainscited by 1
  33. 2025Learning where to learn: Training data distribution optimization for scientific machine learningcited by 1
  34. 2025Neural Measures for learning distributions of Random PDEscited by 1
  35. 2025An AI Approach for Learning the Spectrum of the Laplace-Beltrami Operatorcited by 1
  36. 2025A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learningcited by 1
  37. 2025Localized PCA-Net Neural Operators for Scalable Solution Reconstruction of Elliptic PDEscited by 1
  38. 2025Expressive Power of Implicit Models: Rich Equilibria and Test-Time Scalingcited by 1

50 most cited of 84 All 84 (JSON)