mlsanityData to 14 Oct 2026

Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations

arXiv 2212.04100 ↗First posted 8 Dec 2022cs.LGcited by 5

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Abstract (arXiv)

Neural networks, especially the recent proposed neural operator models, are increasingly being used to find the solution operator of differential equations. Compared to traditional numerical solvers, they are much faster and more efficient in practical applications. However, one critical issue is that training neural operator models require large amount of ground truth data, which usually comes from the slow numerical solvers. In this paper, we propose a physics-guided data augmentation (PGDA) method to improve the accuracy and generalization of neural operator models. Training data is augmented naturally through the physical properties of differential equations such as linearity and translation. We demonstrate the advantage of PGDA on a variety of linear differential equations, showing that PGDA can improve the sample complexity and is robust to distributional shift.

Key facts. Cited by 5 papers in the map (data through 14 Oct 2026). Builds on 25 of its 35 references that are in the map. First posted on arXiv 8 Dec 2022.

Builds on

25 of 35 references.

References · 25

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2016Group Equivariant Convolutional Networkscited by 1,165
  4. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  5. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
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  7. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  8. 2019General E(2)-Equivariant Steerable CNNscited by 446
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  12. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 204
13 more
  1. 2022MIONet: Learning multiple-input operators via tensor productcited by 181
  2. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 126
  3. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 97
  4. 2021Automatic Symmetry Discovery with Lie Algebra Convolutional Networkcited by 83
  5. 2019Learning to Optimize Multigrid PDE Solverscited by 79
  6. 2019Learning Neural PDE Solvers with Convergence Guaranteescited by 77
  7. 2022Lie Point Symmetry Data Augmentation for Neural PDE Solverscited by 77
  8. 2021Physics-based Deep Learningcited by 68
  9. 2021The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?cited by 61
  10. 2019Physical Symmetries Embedded in Neural Networkscited by 40
  11. 2021Fast PDE-constrained optimization via self-supervised operator learningcited by 25
  12. 2019Deep Eikonal Solverscited by 9
  13. 2021PhyAug: Physics-Directed Data Augmentation for Deep Sensing Model Transfer in Cyber-Physical Systemscited by 5

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 3; 2026: 1.

5 papers.

  1. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 24
  2. 2025Explicit Discovery of Nonlinear Symmetries from Dynamic Datacited by 6
  3. 2025Governing Equation Discovery from Data Based on Differential Invariantscited by 4
  4. 2025Generalized Lie Symmetries in Physics-Informed Neural Operatorscited by 2
  5. 2026Optimal scenario design for climate emulationcited by 1