mlsanityData to 7 Oct 2026

Learning elliptic partial differential equations with randomized linear algebra

arXiv 2102.00491 ↗First posted 31 Jan 2021math.NA cs.LG cs.NAcited by 29

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Abstract (arXiv)

Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function G. By exploiting the hierarchical low-rank structure of G, we show that one can construct an approximant to G that converges almost surely and achieves a relative error of O(Gammaε-1/2log3(1/ε)ε) using at most O(ε-6log4(1/ε)) input-output training pairs with high probability, for any 0<ε<1. The quantity 0<Gammaε≤ 1 characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert--Schmidt operators and characterize the quality of covariance kernels for PDE learning.

Key facts. Cited by 29 papers in the map (data through 7 Oct 2026). Builds on 16 of its 75 references that are in the map. First posted on arXiv 31 Jan 2021.

Builds on

16 of 75 references.

References · 16

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2016Error bounds for approximations with deep ReLU networkscited by 567
  3. 2017PDE-Net: Learning PDEs from Datacited by 304
  4. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  5. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
  6. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  7. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  8. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  9. 2018Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systemscited by 88
  10. 2020Rational neural networkscited by 75
  11. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 61
  12. 2014On the folded normal distributioncited by 31
4 more
  1. 2017Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexitycited by 24
  2. 2018Neural-net-induced Gaussian process regression for function approximation and PDE solutioncited by 22
  3. 2019Meta-learning Pseudo-differential Operators with Deep Neural Networkscited by 17
  4. 2019Stability selection enables robust learning of partial differential equations from limited noisy datacited by 7

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 2; 2023: 10; 2024: 7; 2025: 4; 2026: 3.

29 papers.

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  1. 2023Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behaviorcited by 128
  2. 2024Operator Learning: Algorithms and Analysiscited by 84
  3. 2023A Mathematical Guide to Operator Learningcited by 74
  4. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  5. 2023Kernel Methods are Competitive for Operator Learningcited by 50
  6. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  7. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 43
  8. 2023Elliptic PDE learning is provably data-efficientcited by 26
  9. 2022Learning Green's functions associated with time-dependent partial differential equationscited by 25
  10. 2022Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEscited by 24
  11. 2023Error Bounds for Learning with Vector-Valued Random Featurescited by 23
  12. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
17 more
  1. 2024Operator learning without the adjointcited by 8
  2. 2021A generalization of the randomized singular value decompositioncited by 6
  3. 2023A randomized algorithm to solve reduced rank operator regressioncited by 4
  4. 2023Operator learning for hyperbolic partial differential equationscited by 4
  5. 2024Green Multigrid Networkcited by 3
  6. 2024Structure-Preserving Operator Learningcited by 3
  7. 2024An explainable operator approximation framework under the guideline of Green's functioncited by 3
  8. 2023Score Operator Newton transportcited by 1
  9. 2023Learning Green's Function Efficiently Using Low-Rank Approximationscited by 1
  10. 2025chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Datacited by 1
  11. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
  12. 2024A unified error analysis for randomized low-rank approximation with application to data assimilationcited by 0
  13. 2025An adaptive data sampling strategy for stabilizing dynamical systems via controller inferencecited by 0
  14. 2025Query Efficient Structured Matrix Learningcited by 0
  15. 2026Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEscited by 0
  16. 2026Optimal Multiscale Learning of Linear Operatorscited by 0
  17. 2026Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solvercited by 0