mlsanityData to 14 Oct 2026

Operator Learning: A Statistical Perspective

arXiv 2504.03503 ↗First posted 4 Apr 2025stat.ML cs.LGcited by 17

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Abstract (arXiv)

Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the solution operators of partial differential equations (PDEs). These methods can also be used to develop black-box simulators to model system behavior from experimental data, even without a known mathematical model. In this article, we begin by formalizing operator learning as a function-to-function regression problem and review some recent developments in the field. We also discuss PDE-specific operator learning, outlining strategies for incorporating physical and mathematical constraints into architecture design and training processes. Finally, we end by highlighting key future directions such as active data collection and the development of rigorous uncertainty quantification frameworks.

Key facts. Cited by 17 papers in the map (data through 14 Oct 2026). Builds on 53 of its 122 references that are in the map. First posted on arXiv 4 Apr 2025.

Builds on

53 of 122 references.

References · 53

  1. 2020Scaling Laws for Neural Language Modelscited by 6,586
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2021Scaling Vision Transformerscited by 999
  5. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  6. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  8. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  9. 2021Choose a Transformer: Fourier or Galerkincited by 343
  10. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 336
  11. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  12. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
38 more
  1. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 234
  2. 2023When Do Neural Nets Outperform Boosted Trees on Tabular Data?cited by 223
  3. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  4. 2023Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behaviorcited by 131
  5. 2023Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systemscited by 121
  6. 2022Physics-Informed Deep Neural Operator Networkscited by 113
  7. 2024Operator Learning: Algorithms and Analysiscited by 85
  8. 2022Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modelingcited by 65
  9. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 62
  10. 2015Operator-valued Kernels for Learning from Functional Response Datacited by 60
  11. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  12. 2023Kernel Methods are Competitive for Operator Learningcited by 50
  13. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  14. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 44
  15. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  16. 2024A Library for Learning Neural Operatorscited by 38
  17. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  18. 2023Plasma Surrogate Modelling using Fourier Neural Operatorscited by 29
  19. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  20. 2014Functional Regressioncited by 26
  21. 2023Elliptic PDE learning is provably data-efficientcited by 26
  22. 2024Calibrated Uncertainty Quantification for Operator Learning via Conformal Predictioncited by 26
  23. 2024Data Complexity Estimates for Operator Learningcited by 26
  24. 2022Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEscited by 25
  25. 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
  26. 2016Random Fourier Features for Operator-Valued Kernelscited by 22
  27. 2023Multi-Resolution Active Learning of Fourier Neural Operatorscited by 22
  28. 2024Discretization Error of Fourier Neural Operatorscited by 22
  29. 2024Active Learning for Neural PDE Solverscited by 22
  30. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  31. 2022A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurementscited by 20
  32. 2023IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learningcited by 18
  33. 2024Operator Learning with Gaussian Processescited by 18
  34. 2022Bounding the Rademacher Complexity of Fourier neural operatorscited by 16
  35. 2016Operator-Valued Bochner Theorem, Fourier Feature Maps for Operator-Valued Kernels, and Vector-Valued Learningcited by 15
  36. 2019Learning Schatten--von Neumann Operatorscited by 7
  37. 2022Towards Size-Independent Generalization Bounds for Deep Operator Netscited by 6
  38. 2023DiffHybrid-UQ: Uncertainty Quantification for Differentiable Hybrid Neural Modelingcited by 5

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 6; 2026: 11.

17 papers.

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  1. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 12
  2. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 5
  3. 2026A short tour of operator learning theory: Convergence rates, statistical limits, and open questionscited by 4
  4. 2025Continuum Transformers Perform In-Context Learning by Operator Gradient Descentcited by 3
  5. 2025Learning where to learn: Training data distribution optimization for scientific machine learningcited by 3
  6. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  7. 2025A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learningcited by 1
  8. 2026Is Zero-Shot Super-Resolution Possible in Operator Learning?cited by 1
  9. 2026Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regimecited by 1
  10. 2026Optimal Multiscale Learning of Linear Operatorscited by 1
  11. 2026Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfacescited by 0
  12. 2026Learning solution operators of PDEs with sparse approximation methodscited by 0
5 more
  1. 2026From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operatorscited by 0
  2. 2026Near-Optimal Learning of Gaussian Sobolev Operatorscited by 0
  3. 2026Two-Step MV-DeepONet: Probabilistic Operator Learning for Uncertainty Propagation Driven by Random Input Fieldscited by 0
  4. 2026MnemoDyn: Learning Resting State Dynamics from 40K FMRI sequencescited by 0
  5. 2026Efficient Neural Surrogates for Linear Radiation Transport on the Lattice and Hohlraum benchmarkscited by 0