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Equivariant Flows: sampling configurations for multi-body systems with symmetric energies

arXiv 1910.00753 ↗First posted 2 Oct 2019stat.ML cs.LG physics.chem-phcited by 57

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Abstract (arXiv)

Flows are exact-likelihood generative neural networks that transform samples from a simple prior distribution to the samples of the probability distribution of interest. Boltzmann Generators (BG) combine flows and statistical mechanics to sample equilibrium states of strongly interacting many-body systems such as proteins with 1000 atoms. In order to scale and generalize these results, it is essential that the natural symmetries of the probability density - in physics defined by the invariances of the energy function - are built into the flow. Here we develop theoretical tools for constructing such equivariant flows and demonstrate that a BG that is equivariant with respect to rotations and particle permutations can generalize to sampling nontrivially new configurations where a nonequivariant BG cannot.

Key facts. Cited by 57 papers in the map (data through 7 Oct 2026). First posted on arXiv 2 Oct 2019.

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Citing papers per year: 2021: 11; 2022: 8; 2023: 10; 2024: 13; 2025: 11; 2026: 4.

57 papers.

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  1. 2021E(n) Equivariant Graph Neural Networkscited by 951
  2. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  3. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  4. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  5. 2021E(n) Equivariant Normalizing Flowscited by 165
  6. 2023EqMotion: Equivariant Multi-agent Motion Prediction with Invariant Interaction Reasoningcited by 97
  7. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  8. 2022Equivariant Graph Mechanics Networks with Constraintscited by 67
  9. 2021Simple GNN Regularisation for 3D Molecular Property Prediction & Beyondcited by 44
  10. 2022Unsupervised Learning of Group Invariant and Equivariant Representationscited by 38
  11. 2024Equivariant Graph Neural Operator for Modeling 3D Dynamicscited by 37
  12. 2022SchNetPack 2.0: A neural network toolbox for atomistic machine learningcited by 29
38 more
  1. 2024A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applicationscited by 28
  2. 2023SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biasescited by 27
  3. 2023Learning Probabilistic Symmetrization for Architecture Agnostic Equivariancecited by 25
  4. 2022Equivariant Graph Hierarchy-Based Neural Networkscited by 24
  5. 2022DiffMD: A Geometric Diffusion Model for Molecular Dynamics Simulationscited by 24
  6. 2023E(n) Equivariant Message Passing Simplicial Networkscited by 23
  7. 2024E(n) Equivariant Topological Neural Networkscited by 22
  8. 2024Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?cited by 18
  9. 2022Structure-preserving GANscited by 17
  10. 2021Ab-initio study of interacting fermions at finite temperature with neural canonical transformationcited by 16
  11. 2021Riemannian Convex Potential Mapscited by 16
  12. 2023Sample Complexity of Probability Divergences under Group Symmetrycited by 13
  13. 2024Geometric Trajectory Diffusion Modelscited by 13
  14. 2023Sample Complexity Bounds for Estimating Probability Divergences under Invariancescited by 11
  15. 2023Using Multiple Vector Channels Improves E(n)-Equivariant Graph Neural Networkscited by 10
  16. 2024AI-driven inverse design of materials: Past, present and futurecited by 10
  17. 2025Learning (Approximately) Equivariant Networks via Constrained Optimizationcited by 10
  18. 2023Task-Equivariant Graph Few-shot Learningcited by 8
  19. 2022Deep Variational Free Energy Approach to Dense Hydrogencited by 6
  20. 2024GeoMFormer: A General Architecture for Geometric Molecular Representation Learningcited by 6
  21. 2024DuSEGO: Dual Second-order Equivariant Graph Ordinary Differential Equationcited by 6
  22. 2025Fast and Distributed Equivariant Graph Neural Networks by Virtual Node Learningcited by 6
  23. 2025BLIPs: Bayesian Learned Interatomic Potentialscited by 6
  24. 2025Scaling flow-based approaches for topology sampling in SU(3) gauge theorycited by 6
  25. 2024A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networkscited by 5
  26. 2025Boltzmann generators for amorphous particle systemscited by 5
  27. 2023Newton-Cotes Graph Neural Networks: On the Time Evolution of Dynamic Systemscited by 3
  28. 2024Learning Adaptive Hydrodynamic Models Using Neural ODEs in Complex Conditionscited by 3
  29. 2025Dynami-CAL GraphNet: A Physics-Informed Graph Neural Network Conserving Linear and Angular Momentum for Dynamical Systemscited by 3
  30. 2024Equivariant Manifold Neural ODEs and Differential Invariantscited by 2
  31. 2025PAINET: A Principled Efficient Transformer for 3D Dynamics Modelingcited by 2
  32. 2024G-RepsNet: A Fast and General Construction of Equivariant Networks for Arbitrary Matrix Groupscited by 1
  33. 2024Relaxing Continuous Constraints of Equivariant Graph Neural Networks for Physical Dynamics Learningcited by 1
  34. 2025ATOM: A Pretrained Neural Operator for Multitask Molecular Dynamicscited by 1
  35. 2026Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning modelscited by 1
  36. 2021Out-of-Distribution Detection of Melanoma using Normalizing Flowscited by 0
  37. 2021Rotation Equivariant 3D Hand Mesh Generation from a Single RGB Imagecited by 0
  38. 2022Utility of Equivariant Message Passing in Cortical Mesh Segmentationcited by 0

50 most cited of 57 All 57 (JSON)