mlsanityData to 7 Oct 2026

Approximately Equivariant Networks for Imperfectly Symmetric Dynamics

arXiv 2201.11969 ↗First posted 28 Jan 2022cs.LGcited by 95

Not on the map. Why?

Abstract (arXiv)

Incorporating symmetry as an inductive bias into neural network architecture has led to improvements in generalization, data efficiency, and physical consistency in dynamics modeling. Methods such as CNNs or equivariant neural networks use weight tying to enforce symmetries such as shift invariance or rotational equivariance. However, despite the fact that physical laws obey many symmetries, real-world dynamical data rarely conforms to strict mathematical symmetry either due to noisy or incomplete data or to symmetry breaking features in the underlying dynamical system. We explore approximately equivariant networks which are biased towards preserving symmetry but are not strictly constrained to do so. By relaxing equivariance constraints, we find that our models can outperform both baselines with no symmetry bias and baselines with overly strict symmetry in both simulated turbulence domains and real-world multi-stream jet flow.

Key facts. Cited by 95 papers in the map (data through 7 Oct 2026). Builds on 47 of its 66 references that are in the map. First posted on arXiv 28 Jan 2022.

Builds on

47 of 66 references.

References · 47

  1. 2018Neural Ordinary Differential Equationscited by 3,733
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2017Deep Setscited by 1,394
  4. 2018Relational inductive biases, deep learning, and graph networkscited by 1,341
  5. 2016Group Equivariant Convolutional Networkscited by 1,154
  6. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  7. 2013Spectral Networks and Locally Connected Networks on Graphscited by 1,097
  8. 2021E(n) Equivariant Graph Neural Networkscited by 951
  9. 2020Learning Mesh-Based Simulation with Graph Networkscited by 840
  10. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  11. 2019Hamiltonian Neural Networkscited by 702
  12. 2017Dynamic Routing Between Capsulescited by 689
35 more
  1. 2021ConViT: Improving Vision Transformers with Soft Convolutional Inductive Biasescited by 484
  2. 2019General E(2)-Equivariant Steerable CNNscited by 441
  3. 2020Lagrangian Neural Networkscited by 388
  4. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  5. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
  6. 2018Invariant and Equivariant Graph Networkscited by 339
  7. 2016Steerable CNNscited by 298
  8. 2018On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groupscited by 296
  9. 2016Harmonic Networks: Deep Translation and Rotation Equivariancecited by 293
  10. 2019Gauge Equivariant Convolutional Networks and the Icosahedral CNNcited by 262
  11. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
  12. 2014Understanding image representations by measuring their equivariance and equivalencecited by 240
  13. 2018A General Theory of Equivariant CNNs on Homogeneous Spacescited by 223
  14. 2019Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learningcited by 205
  15. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
  16. 2017Learning Steerable Filters for Rotation Equivariant CNNscited by 193
  17. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  18. 2017Deep Learning for Precipitation Nowcasting: A Benchmark and A New Modelcited by 180
  19. 2017Equivariance Through Parameter-Sharingcited by 139
  20. 2016Accelerating Eulerian Fluid Simulation With Convolutional Networkscited by 128
  21. 2020Incorporating Symmetry into Deep Dynamics Models for Improved Generalizationcited by 123
  22. 2020Forecasting Sequential Data using Consistent Koopman Autoencoderscited by 114
  23. 2019Scale-Equivariant Steerable Networkscited by 110
  24. 2020Learning Stable Deep Dynamics Modelscited by 110
  25. 2020On Learning Sets of Symmetric Elementscited by 97
  26. 2017Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledgecited by 95
  27. 2020Augmenting Physical Models with Deep Networks for Complex Dynamics Forecastingcited by 93
  28. 2016Exploiting Cyclic Symmetry in Convolutional Neural Networkscited by 91
  29. 2021Residual Pathway Priors for Soft Equivariance Constraintscited by 74
  30. 2021Equivariant Q Learning in Spatial Action Spacescited by 61
  31. 2020Revisiting Spatial Invariance with Low-Rank Local Connectivitycited by 29
  32. 2020Equivariant Learning of Stochastic Fields: Gaussian Processes and Steerable Conditional Neural Processescited by 29
  33. 2020Trajectory Prediction using Equivariant Continuous Convolutioncited by 26
  34. 2019Scale Steerable Filters for Locally Scale-Invariant Convolutional Neural Networkscited by 22
  35. 2018Rotation Equivariance and Invariance in Convolutional Neural Networkscited by 12

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 12; 2023: 20; 2024: 22; 2025: 21; 2026: 20.

95 papers.

Sort:

By area:

  1. 2022Theory for Equivariant Quantum Neural Networkscited by 60
  2. 2023Approximation-Generalization Trade-offs under (Approximate) Group Equivariancecited by 45
  3. 2023Group Equivariant Fourier Neural Operators for Partial Differential Equationscited by 42
  4. 2022Relaxing Equivariance Constraints with Non-stationary Continuous Filterscited by 41
  5. 2022Representation Theory for Geometric Quantum Machine Learningcited by 40
  6. 2023Generative Adversarial Symmetry Discoverycited by 40
  7. 2024The Importance of Being Scalable: Improving the Speed and Accuracy of Neural Network Interatomic Potentials Across Chemical Domainscited by 40
  8. 2022Explaining the Explainers in Graph Neural Networks: a Comparative Studycited by 34
  9. 2022LieGG: Studying Learned Lie Group Generatorscited by 28
  10. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  11. 2022VN-Transformer: Rotation-Equivariant Attention for Vector Neuronscited by 26
  12. 2023Learning Layer-wise Equivariances Automatically using Gradientscited by 23
38 more
  1. 2024Improving Equivariant Model Training via Constraint Relaxationcited by 22
  2. 2023Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolutioncited by 19
  3. 2024Equivariance via Minimal Frame Averaging for More Symmetries and Efficiencycited by 18
  4. 2022Data Augmentation vs. Equivariant Networks: A Theory of Generalization on Dynamics Forecastingcited by 17
  5. 2023Regularizing Towards Soft Equivariance Under Mixed Symmetriescited by 16
  6. 2023Approximately Equivariant Graph Networkscited by 16
  7. 2022Machine learning and invariant theorycited by 13
  8. 2024Relaxed Equivariance via Multitask Learningcited by 13
  9. 2025Probing Equivariance and Symmetry Breaking in Convolutional Networkscited by 12
  10. 2022Port-metriplectic neural networks: thermodynamics-informed machine learning of complex physical systemscited by 11
  11. 2022Thermodynamics of learning physical phenomenacited by 10
  12. 2023Towards fully covariant machine learningcited by 10
  13. 2023Stochastic Marginal Likelihood Gradients using Neural Tangent Kernelscited by 10
  14. 2023Almost Equivariance via Lie Algebra Convolutionscited by 10
  15. 2024On the Ability of Deep Networks to Learn Symmetries from Data: A Neural Kernel Theorycited by 10
  16. 2025Learning (Approximately) Equivariant Networks via Constrained Optimizationcited by 10
  17. 2024Equivariant Symmetry Breaking Setscited by 9
  18. 2024Morphological Symmetries in Roboticscited by 9
  19. 2024Approximate Equivariance in Reinforcement Learningcited by 9
  20. 2022Equivariance-aware Architectural Optimization of Neural Networkscited by 8
  21. 2024Approximately Equivariant Neural Processescited by 8
  22. 2025A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Modelscited by 8
  23. 2023Latent Field Discovery In Interacting Dynamical Systems With Neural Fieldscited by 7
  24. 2024Relaxed Equivariant Graph Neural Networkscited by 7
  25. 2024Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networkscited by 7
  26. 2023On discrete symmetries of robotics systems: A group-theoretic and data-driven analysiscited by 6
  27. 2023Equivariant Neural Simulators for Stochastic Spatiotemporal Dynamicscited by 6
  28. 2023E(2) Equivariant Neural Networks for Robust Galaxy Morphology Classificationcited by 6
  29. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  30. 2025Achieving Approximate Symmetry Is Exponentially Easier than Exact Symmetrycited by 6
  31. 2022Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equationscited by 5
  32. 2023Equivariance Is Not All You Need: Characterizing the Utility of Equivariant Graph Neural Networks for Particle Physics Taskscited by 5
  33. 2024Approximately-symmetric neural networks for quantum spin liquidscited by 5
  34. 2024Learning Symmetries via Weight-Sharing with Doubly Stochastic Tensorscited by 5
  35. 2025A Regularization-Guided Equivariant Approach for Image Restorationcited by 5
  36. 2025Explicit Discovery of Nonlinear Symmetries from Dynamic Datacited by 5
  37. 2025Latent Mixture of Symmetries for Sample-Efficient Dynamic Learningcited by 5
  38. 2024R2Det: Exploring Relaxed Rotation Equivariance in 2D object detectioncited by 4

50 most cited of 95 All 95 (JSON)