mlsanityData to 7 Oct 2026

Elliptic PDE learning is provably data-efficient

arXiv 2302.12888 ↗First posted 24 Feb 2023cs.LG cs.AI cs.NAcited by 26

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Abstract (arXiv)

PDE learning is an emerging field that combines physics and machine learning to recover unknown physical systems from experimental data. While deep learning models traditionally require copious amounts of training data, recent PDE learning techniques achieve spectacular results with limited data availability. Still, these results are empirical. Our work provides theoretical guarantees on the number of input-output training pairs required in PDE learning. Specifically, we exploit randomized numerical linear algebra and PDE theory to derive a provably data-efficient algorithm that recovers solution operators of 3D uniformly elliptic PDEs from input-output data and achieves an exponential convergence rate of the error with respect to the size of the training dataset with an exceptionally high probability of success.

Key facts. Cited by 26 papers in the map (data through 7 Oct 2026). Builds on 18 of its 82 references that are in the map. First posted on arXiv 24 Feb 2023.

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18 of 82 references.

References · 18

  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. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  5. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  6. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  7. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  8. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  9. 2016Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statisticscited by 45
  10. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 43
  11. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
  12. 2022Learning Green's functions associated with time-dependent partial differential equationscited by 25
6 more
  1. 2017Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexitycited by 24
  2. 2023Deep Operator Learning Lessens the Curse of Dimensionality for PDEscited by 18
  3. 2017Unperturbed: spectral analysis beyond Davis-Kahancited by 16
  4. 2021A generalization of the randomized singular value decompositioncited by 6
  5. 2020Error Estimation for Sketched SVD via the Bootstrapcited by 5
  6. 2022Perturbation Analysis of Randomized SVD and its Applications to Statisticscited by 3

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 2; 2024: 10; 2025: 9; 2026: 5.

26 papers.

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  1. 2023A Mathematical Guide to Operator Learningcited by 74
  2. 2024An operator learning perspective on parameter-to-observable mapscited by 22
  3. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  4. 2025Operator Learning: A Statistical Perspectivecited by 14
  5. 2024Adversarial dynamical systems characterize when data-driven learning succeeds or failscited by 10
  6. 2024Operator learning without the adjointcited by 8
  7. 2024The Sample Complexity of Learning Lipschitz Operators with respect to Gaussian Measurescited by 8
  8. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 6
  9. 2023Operator learning for hyperbolic partial differential equationscited by 4
  10. 2024PETScML: Second-order solvers for training regression problems in Scientific Machine Learningcited by 4
  11. 2024Controlling Statistical, Discretization, and Truncation Errors in Learning Fourier Linear Operatorscited by 4
  12. 2024Conservation-informed Graph Learning for Spatiotemporal Dynamics Predictioncited by 4
14 more
  1. 2024Green Multigrid Networkcited by 3
  2. 2024Structure-Preserving Operator Learningcited by 3
  3. 2025Orthogonal greedy algorithm for linear operator learning with shallow neural networkcited by 3
  4. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  5. 2025Noisy PDE Training Requires Bigger PINNscited by 2
  6. 2025Learning where to learn: Training data distribution optimization for scientific machine learningcited by 1
  7. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
  8. 2026Neural-HSS: Hierarchical Semi-Separable Neural PDE Solvercited by 1
  9. 2026Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solverscited by 1
  10. 2025Interpretability and Generalization Bounds for Learning Spatial Physicscited by 0
  11. 2025Query Efficient Structured Matrix Learningcited by 0
  12. 2026Optimal Multiscale Learning of Linear Operatorscited by 0
  13. 2026A zero-one law for one-shot system identificationcited by 0
  14. 2026Conformal Uncertainty Quantification Guarantees for Neural Operatorscited by 0