mlsanityData to 7 Oct 2026

E(n) Equivariant Normalizing Flows

arXiv 2105.09016 ↗First posted 19 May 2021cs.LG physics.chem-ph stat.MLcited by 165

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Cited by 53 in Diffusion models, but placed by its own abstract, which didn’t fit.

Abstract (arXiv)

This paper introduces a generative model equivariant to Euclidean symmetries: E(n) Equivariant Normalizing Flows (E-NFs). To construct E-NFs, we take the discriminative E(n) graph neural networks and integrate them as a differential equation to obtain an invertible equivariant function: a continuous-time normalizing flow. We demonstrate that E-NFs considerably outperform baselines and existing methods from the literature on particle systems such as DW4 and LJ13, and on molecules from QM9 in terms of log-likelihood. To the best of our knowledge, this is the first flow that jointly generates molecule features and positions in 3D.

Key facts. Cited by 165 papers in the map (data through 7 Oct 2026). Builds on 38 of its 45 references that are in the map. First posted on arXiv 19 May 2021.

Builds on

38 of 45 references.

References · 38

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  2. 2018Neural Ordinary Differential Equationscited by 3,733
  3. 2015Variational Inference with Normalizing Flowscited by 2,161
  4. 2016Density estimation using Real NVPcited by 2,029
  5. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  6. 2016Group Equivariant Convolutional Networkscited by 1,154
  7. 2013Spectral Networks and Locally Connected Networks on Graphscited by 1,097
  8. 2014NICE: Non-linear Independent Components Estimationcited by 1,052
  9. 2021E(n) Equivariant Graph Neural Networkscited by 951
  10. 2016Automatic chemical design using a data-driven continuous representation of moleculescited by 762
  11. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  12. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
26 more
  1. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  2. 2020Directional Message Passing for Molecular Graphscited by 556
  3. 2018GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Modelscited by 406
  4. 2018GraphVAE: Towards Generation of Small Graphs Using Variational Autoencoderscited by 375
  5. 2016Steerable CNNscited by 298
  6. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  7. 2019Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Designcited by 258
  8. 2020Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Datacited by 248
  9. 2020Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densitiescited by 248
  10. 2019Efficient Graph Generation with Graph Recurrent Attention Networkscited by 231
  11. 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
  12. 2019Symmetry-adapted generation of 3d point sets for the targeted discovery of moleculescited by 171
  13. 2018Activation Functions: Comparison of trends in Practice and Research for Deep Learningcited by 130
  14. 2019Graph Normalizing Flowscited by 103
  15. 2020LieTransformer: Equivariant self-attention for Lie Groupscited by 100
  16. 2016Exploiting Cyclic Symmetry in Convolutional Neural Networkscited by 91
  17. 2021Learning Neural Generative Dynamics for Molecular Conformation Generationcited by 87
  18. 2020Sampling using SU(N) gauge equivariant flowscited by 83
  19. 2020SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flowscited by 71
  20. 2019Equivariant Flows: sampling configurations for multi-body systems with symmetric energiescited by 57
  21. 2019Equivariant Hamiltonian Flowscited by 39
  22. 2017Continuous-Time Flows for Efficient Inference and Density Estimationcited by 21
  23. 2020Group Equivariant Generative Adversarial Networkscited by 19
  24. 2013High-Dimensional Probability Estimation with Deep Density Modelscited by 17
  25. 2020Isometric Transformation Invariant and Equivariant Graph Convolutional Networkscited by 17
  26. 2019Neural Canonical Transformation with Symplectic Flowscited by 16

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 28; 2023: 40; 2024: 40; 2025: 38; 2026: 17.

165 papers.

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  1. 2022Equivariant Diffusion for Molecule Generation in 3Dcited by 666
  2. 20233D Equivariant Diffusion for Target-Aware Molecule Generation and Affinity Predictioncited by 176
  3. 2023SE(3)-Stochastic Flow Matching for Protein Backbone Generationcited by 147
  4. 2023Equivariant flow matchingcited by 131
  5. 2022Equivariant 3D-Conditional Diffusion Models for Molecular Linker Designcited by 121
  6. 2024Iterated Denoising Energy Matching for Sampling from Boltzmann Densitiescited by 113
  7. 2022Diffusion-based Molecule Generation with Informative Prior Bridgescited by 104
  8. 2022SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generatorscited by 96
  9. 2023MiDi: Mixed Graph and 3D Denoising Diffusion for Molecule Generationcited by 96
  10. 2021Inverse design of 3d molecular structures with conditional generative neural networkscited by 93
  11. 2022Periodic Graph Transformers for Crystal Material Property Predictioncited by 87
  12. 2022Equivariant Energy-Guided SDE for Inverse Molecular Designcited by 72
38 more
  1. 2023MolDiff: Addressing the Atom-Bond Inconsistency Problem in 3D Molecule Diffusion Generationcited by 65
  2. 2023A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systemscited by 65
  3. 2021Geometric Deep Learning and Equivariant Neural Networkscited by 64
  4. 2023Language models can generate molecules, materials, and protein binding sites directly in three dimensions as XYZ, CIF, and PDB filescited by 59
  5. 2022MDM: Molecular Diffusion Model for 3D Molecule Generationcited by 58
  6. 2022DiffBP: Generative Diffusion of 3D Molecules for Target Protein Bindingcited by 58
  7. 2022Geometrically Equivariant Graph Neural Networks: A Surveycited by 55
  8. 2024Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generationcited by 53
  9. 2023Graph Generation with Diffusion Mixturecited by 52
  10. 2023Navigating the Design Space of Equivariant Diffusion-Based Generative Models for De Novo 3D Molecule Generationcited by 50
  11. 2024Generative Modeling of Molecular Dynamics Trajectoriescited by 48
  12. 2025SymmCD: Symmetry-Preserving Crystal Generation with Diffusion Modelscited by 48
  13. 2024Transferable Boltzmann Generatorscited by 43
  14. 2023MUDiff: Unified Diffusion for Complete Molecule Generationcited by 41
  15. 2023Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Spacecited by 40
  16. 2024SemlaFlow -- Efficient 3D Molecular Generation with Latent Attention and Equivariant Flow Matchingcited by 40
  17. 20233D molecule generation by denoising voxel gridscited by 38
  18. 2023SE(3) Equivariant Augmented Coupling Flowscited by 37
  19. 2023Geometry-Complete Diffusion for 3D Molecule Generation and Optimizationcited by 36
  20. 2025Scalable Equilibrium Sampling with Sequential Boltzmann Generatorscited by 36
  21. 2023Functional-Group-Based Diffusion for Pocket-Specific Molecule Generation and Elaborationcited by 30
  22. 2022Diffeomorphic Counterfactuals with Generative Modelscited by 29
  23. 2023Crystal-GFN: sampling crystals with desirable properties and constraintscited by 29
  24. 2024A Survey of Generative AI for de novo Drug Design: New Frontiers in Molecule and Protein Generationcited by 29
  25. 2024A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applicationscited by 28
  26. 2023Rigid Body Flows for Sampling Molecular Crystal Structurescited by 27
  27. 2023Learning Joint 2D & 3D Diffusion Models for Complete Molecule Generationcited by 27
  28. 2023Free-form Flows: Make Any Architecture a Normalizing Flowcited by 26
  29. 2024PPFlow: Target-aware Peptide Design with Torsional Flow Matchingcited by 26
  30. 2023Towards Symmetry-Aware Generation of Periodic Materialscited by 25
  31. 2023AI-Generated Content (AIGC) for Various Data Modalities: A Surveycited by 25
  32. 2022Equivariant Networks for Crystal Structurescited by 24
  33. 2023Coarse-to-Fine: a Hierarchical Diffusion Model for Molecule Generation in 3Dcited by 24
  34. 2024UniGEM: A Unified Approach to Generation and Property Prediction for Moleculescited by 20
  35. 2025FlowMol3: Flow Matching for 3D De Novo Small-Molecule Generationcited by 20
  36. 2022Equivariant Shape-Conditioned Generation of 3D Molecules for Ligand-Based Drug Designcited by 19
  37. 2024CBGBench: Fill in the Blank of Protein-Molecule Complex Binding Graphcited by 19
  38. 2022NVDiff: Graph Generation through the Diffusion of Node Vectorscited by 18

50 most cited of 165 All 165 (JSON)