mlsanityData to 14 Oct 2026

A Mathematical Guide to Operator Learning

arXiv 2312.14688 ↗First posted 22 Dec 2023math.NA cs.AI cs.LGcited by 76

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Abstract (arXiv)

Operator learning aims to discover properties of an underlying dynamical system or partial differential equation (PDE) from data. Here, we present a step-by-step guide to operator learning. We explain the types of problems and PDEs amenable to operator learning, discuss various neural network architectures, and explain how to employ numerical PDE solvers effectively. We also give advice on how to create and manage training data and conduct optimization. We offer intuition behind the various neural network architectures employed in operator learning by motivating them from the point-of-view of numerical linear algebra.

Key facts. Cited by 76 papers in the map (data through 14 Oct 2026). Builds on 70 of its 154 references that are in the map. First posted on arXiv 22 Dec 2023.

Builds on

70 of 154 references.

References · 70

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  18. 2021Choose a Transformer: Fourier or Galerkincited by 343
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 1; 2024: 22; 2025: 26; 2026: 27.

76 papers.

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  1. 2025DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic predictioncited by 21
  2. 2024Operator Learning with Gaussian Processescited by 18
  3. 2024Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEscited by 15
  4. 2024A finite element-based physics-informed operator learning framework for spatiotemporal partial differential equations on arbitrary domainscited by 15
  5. 2024CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equationscited by 14
  6. 2024Linearization Turns Neural Operators into Function-Valued Gaussian Processescited by 12
  7. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 12
  8. 2024Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flowscited by 11
  9. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 11
  10. 2024Learning the Hodgkin-Huxley Model with Operator Learning Techniquescited by 10
  11. 2025Enhancing Fourier Neural Operators with Local Spatial Featurescited by 10
  12. 2026Test-time Generalization for Physics through Neural Operator Splittingcited by 10
38 more
  1. 2024Operator learning without the adjointcited by 8
  2. 2024Optimal Convergence Rates for Neural Operatorscited by 8
  3. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 7
  4. 2025Operator Learning at Machine Precisioncited by 7
  5. 2024An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Studycited by 6
  6. 2024Optimal sampling for least-squares approximationcited by 6
  7. 2024Scalable Mechanistic Neural Networks for Differential Equations and Machine Learningcited by 5
  8. 2025Towards Sharp Minimax Risk Bounds for Operator Learningcited by 5
  9. 2025Multi-Level Monte Carlo Training of Neural Operatorscited by 4
  10. 2026A short tour of operator learning theory: Convergence rates, statistical limits, and open questionscited by 4
  11. 2026One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operatorscited by 4
  12. 2026Topological Neural Operatorscited by 4
  13. 2024Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEscited by 3
  14. 2024Green Multigrid Networkcited by 3
  15. 2024Structure-Preserving Operator Learningcited by 3
  16. 2024Learning sparsity-promoting regularizers for linear inverse problemscited by 3
  17. 2025Orthogonal greedy algorithm for linear operator learning with shallow neural networkcited by 3
  18. 2025Uncertainty propagation in feed-forward neural network modelscited by 3
  19. 2025Continuum Transformers Perform In-Context Learning by Operator Gradient Descentcited by 3
  20. 2025Learning where to learn: Training data distribution optimization for scientific machine learningcited by 3
  21. 2026Fluids You Can Trust: Property-Preserving Operator Learning for Incompressible Flowscited by 3
  22. 2023Hybrid Modeling Design Patternscited by 2
  23. 2024Projection Methods for Operator Learning and Universal Approximationcited by 2
  24. 2025Noisy PDE Training Requires Bigger PINNscited by 2
  25. 2025RONOM: Reduced-Order Neural Operator Modelingcited by 2
  26. 2025Vector-Valued Reproducing Kernel Banach Spaces for Neural Networks and Operatorscited by 2
  27. 2026Operator Learning for Families of Finite-State Mean-Field Gamescited by 2
  28. 2026Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observationscited by 2
  29. 2024Global 4D Ionospheric STEC Prediction based on DeepONet for GNSS Rayscited by 1
  30. 2024Neumann Series-based Neural Operator for Solving Inverse Medium Problemcited by 1
  31. 2025chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Datacited by 1
  32. 2025Graph-Based Operator Learning from Limited Data on Irregular Domainscited by 1
  33. 2025BEKAN: Boundary condition-guaranteed evolutionary Kolmogorov-Arnold networks with radial basis functions for solving PDE problemscited by 1
  34. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
  35. 2025Physically consistent model learning for reaction-diffusion systemscited by 1
  36. 2026Neural-HSS: Hierarchical Semi-Separable Neural PDE Solvercited by 1
  37. 2026Is Zero-Shot Super-Resolution Possible in Operator Learning?cited by 1
  38. 2026Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regimecited by 1

50 most cited of 76 All 76 (JSON)