mlsanityData to 7 Oct 2026

Lie Point Symmetry Data Augmentation for Neural PDE Solvers

arXiv 2202.07643 ↗First posted 15 Feb 2022cs.LG cs.CVcited by 76

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Abstract (arXiv)

Neural networks are increasingly being used to solve partial differential equations (PDEs), replacing slower numerical solvers. However, a critical issue is that neural PDE solvers require high-quality ground truth data, which usually must come from the very solvers they are designed to replace. Thus, we are presented with a proverbial chicken-and-egg problem. In this paper, we present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity -- Lie point symmetry data augmentation (LPSDA). In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations, based on the Lie point symmetry group of the PDEs in question, something not possible in other application areas. We present this framework and demonstrate how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.

Key facts. Cited by 76 papers in the map (data through 7 Oct 2026). Builds on 25 of its 63 references that are in the map. First posted on arXiv 15 Feb 2022.

Builds on

25 of 63 references.

References · 25

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13 more
  1. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
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  3. 2020Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solverscited by 201
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 7; 2023: 14; 2024: 20; 2025: 20; 2026: 15.

76 papers.

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  1. 2023PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solverscited by 205
  2. 2022Towards Multi-spatiotemporal-scale Generalized PDE Modelingcited by 190
  3. 2023Scalable Transformer for PDE Surrogate Modelingcited by 123
  4. 2022Clifford Neural Layers for PDE Modelingcited by 91
  5. 2022AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier-Stokes Solutionscited by 77
  6. 2024On conditional diffusion models for PDE simulationscited by 56
  7. 2023Training neural operators to preserve invariant measures of chaotic attractorscited by 43
  8. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 42
  9. 2023Geometric Clifford Algebra Networkscited by 37
  10. 2024Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learningcited by 32
  11. 2022Guaranteed Conservation of Momentum for Learning Particle-based Fluid Dynamicscited by 30
  12. 2023Self-Supervised Learning with Lie Symmetries for Partial Differential Equationscited by 29
38 more
  1. 2023Lie Point Symmetry and Physics Informed Networkscited by 24
  2. 2024HAMLET: Graph Transformer Neural Operator for Partial Differential Equationscited by 24
  3. 2024Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groupscited by 23
  4. 2024Active Learning for Neural PDE Solverscited by 22
  5. 2023Machine Learning for Partial Differential Equationscited by 20
  6. 2024Learning Infinitesimal Generators of Continuous Symmetries from Datacited by 17
  7. 2024Mechanistic Neural Networks for Scientific Machine Learningcited by 16
  8. 2024Adversarial Learning for Neural PDE Solvers with Sparse Datacited by 16
  9. 2024Symmetry-Informed Governing Equation Discoverycited by 14
  10. 2024Strategies for Pretraining Neural Operatorscited by 14
  11. 2024Zebra: In-Context Generative Pretraining for Solving Parametric PDEscited by 14
  12. 2024Masked Autoencoders are PDE Learnerscited by 11
  13. 2024Linearization Turns Neural Operators into Function-Valued Gaussian Processescited by 11
  14. 2025Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learningcited by 11
  15. 2023Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspectivecited by 10
  16. 2024Predicting Change, Not States: An Alternate Framework for Neural PDE Surrogatescited by 8
  17. 2023Invariant preservation in machine learned PDE solvers via error correctioncited by 7
  18. 2024GEPS: Boosting Generalization in Parametric PDE Neural Solvers through Adaptive Conditioningcited by 7
  19. 2023MetaPhysiCa: OOD Robustness in Physics-informed Machine Learningcited by 6
  20. 2025Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solvingcited by 6
  21. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  22. 2022Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equationscited by 5
  23. 2024Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equationscited by 5
  24. 2024Scalable Mechanistic Neural Networks for Differential Equations and Machine Learningcited by 5
  25. 2025BARNN: A Bayesian Autoregressive and Recurrent Neural Networkcited by 5
  26. 2025Geometric and Physical Constraints Synergistically Enhance Neural PDE Surrogatescited by 5
  27. 2025Explicit Discovery of Nonlinear Symmetries from Dynamic Datacited by 5
  28. 2025Latent Mixture of Symmetries for Sample-Efficient Dynamic Learningcited by 5
  29. 2023G-Signatures: Global Graph Propagation With Randomized Signaturescited by 4
  30. 2024Learning Semilinear Neural Operators : A Unified Recursive Framework For Prediction And Data Assimilationcited by 4
  31. 2025Mechanistic PDE Networks for Discovery of Governing Equationscited by 4
  32. 2025Governing Equation Discovery from Data Based on Differential Invariantscited by 4
  33. 2025Active Learning with Selective Time-Step Acquisition for PDEscited by 4
  34. 2026Explainable AI: Learning from the Learnerscited by 4
  35. 2025MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulationcited by 3
  36. 2025Overtone: Cyclic Patch Modulation for Clean, Efficient, and Flexible Physics Emulatorscited by 3
  37. 2025Adaptive Canonicalization with Application to Invariant Anisotropic Geometric Networkscited by 3
  38. 2022A Feedforward Unitary Equivariant Neural Networkcited by 2

50 most cited of 76 All 76 (JSON)