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Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs

arXiv 2302.03655 ↗First posted 7 Feb 2023cs.LG physics.chem-ph physics.comp-phcited by 113

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Abstract (arXiv)

Graph neural networks that model 3D data, such as point clouds or atoms, are typically desired to be SO(3) equivariant, i.e., equivariant to 3D rotations. Unfortunately equivariant convolutions, which are a fundamental operation for equivariant networks, increase significantly in computational complexity as higher-order tensors are used. In this paper, we address this issue by reducing the SO(3) convolutions or tensor products to mathematically equivalent convolutions in SO(2) . This is accomplished by aligning the node embeddings' primary axis with the edge vectors, which sparsifies the tensor product and reduces the computational complexity from O(L6) to O(L3), where L is the degree of the representation. We demonstrate the potential implications of this improvement by proposing the Equivariant Spherical Channel Network (eSCN), a graph neural network utilizing our novel approach to equivariant convolutions, which achieves state-of-the-art results on the large-scale OC-20 and OC-22 datasets.

Key facts. Cited by 113 papers in the map (data through 7 Oct 2026). Builds on 34 of its 58 references that are in the map. First posted on arXiv 7 Feb 2023.

Builds on

34 of 58 references.

References · 34

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  2. 2013Rich feature hierarchies for accurate object detection and semantic segmentationcited by 2,160
  3. 2016Group Equivariant Convolutional Networkscited by 1,154
  4. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  5. 2021E(n) Equivariant Graph Neural Networkscited by 951
  6. 2017Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learningcited by 935
  7. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  8. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  9. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
  10. 2020Directional Message Passing for Molecular Graphscited by 556
  11. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 467
  12. 2018Spherical CNNscited by 452
22 more
  1. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  2. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
  3. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  4. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
  5. 2016Steerable CNNscited by 298
  6. 2016Harmonic Networks: Deep Translation and Rotation Equivariancecited by 293
  7. 2019Tackling Climate Change with Machine Learningcited by 264
  8. 2022Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphscited by 258
  9. 2020Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Moleculescited by 254
  10. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 254
  11. 2020The Open Catalyst 2020 (OC20) Dataset and Community Challengescited by 246
  12. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  13. 2022e3nn: Euclidean Neural Networkscited by 171
  14. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 168
  15. 2021Spherical Message Passing for 3D Graph Networkscited by 166
  16. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  17. 2022The Open Catalyst 2022 (OC22) Dataset and Challenges for Oxide Electrocatalystscited by 95
  18. 2021Equivariant Point Network for 3D Point Cloud Analysiscited by 81
  19. 2022GemNet-OC: Developing Graph Neural Networks for Large and Diverse Molecular Simulation Datasetscited by 61
  20. 2021Rotation Invariant Graph Neural Networks using Spin Convolutionscited by 52
  21. 2022Spherical Channels for Modeling Atomic Interactionscited by 47
  22. 2020An Introduction to Electrocatalyst Design using Machine Learning for Renewable Energy Storagecited by 45

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 12; 2024: 31; 2025: 36; 2026: 34.

113 papers.

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  1. 2023EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representationscited by 201
  2. 2024FlowMM: Generating Materials with Riemannian Flow Matchingcited by 105
  3. 2025UMA: A Family of Universal Models for Atomscited by 99
  4. 2025Learning Smooth and Expressive Interatomic Potentials for Physical Property Predictioncited by 79
  5. 2023A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systemscited by 65
  6. 2024FlowLLM: Flow Matching for Material Generation with Large Language Models as Base Distributionscited by 52
  7. 2024Open Materials 2024 (OMat24) Inorganic Materials Dataset and Modelscited by 50
  8. 2024Equivariant Frames and the Impossibility of Continuous Canonicalizationcited by 41
  9. 2024ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generationcited by 41
  10. 2024The Importance of Being Scalable: Improving the Speed and Accuracy of Neural Network Interatomic Potentials Across Chemical Domainscited by 40
  11. 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 35
  12. 2023Matbench Discovery -- A framework to evaluate machine learning crystal stability predictionscited by 34
38 more
  1. 2024Complete and Efficient Graph Transformers for Crystal Material Property Predictioncited by 34
  2. 2024A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applicationscited by 28
  3. 2024A Canonicalization Perspective on Invariant and Equivariant Learningcited by 25
  4. 2024Improving Equivariant Model Training via Constraint Relaxationcited by 22
  5. 2023Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolutioncited by 19
  6. 2025An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networkscited by 19
  7. 2023The Open DAC 2023 Dataset and Challenges for Sorbent Discovery in Direct Air Capturecited by 18
  8. 2024Generalizing Denoising to Non-Equilibrium Structures Improves Equivariant Force Fieldscited by 18
  9. 2023Uncertainty-biased molecular dynamics for learning uniformly accurate interatomic potentialscited by 16
  10. 2025The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Productscited by 14
  11. 2024Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passingcited by 13
  12. 2024Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passingcited by 13
  13. 2025Enhancing the Scalability and Applicability of Kohn-Sham Hamiltonians for Molecular Systemscited by 13
  14. 2023Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materialscited by 12
  15. 2025E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Productscited by 12
  16. 2023Accelerating Molecular Graph Neural Networks via Knowledge Distillationcited by 11
  17. 2025HIP: Hessian Interatomic Potentials without derivativescited by 11
  18. 2024AI-driven inverse design of materials: Past, present and futurecited by 10
  19. 2025Efficient and Scalable Density Functional Theory Hamiltonian Prediction through Adaptive Sparsitycited by 10
  20. 2024Steerable Transformers for Volumetric Datacited by 8
  21. 2024A Recipe for Charge Density Predictioncited by 8
  22. 2025SE(3)-Equivariant Robot Learning and Control: A Tutorial Surveycited by 8
  23. 2024Learning local equivariant representations for quantum operatorscited by 7
  24. 2025SE(3)-Equivariant Diffusion Policy in Spherical Fourier Spacecited by 7
  25. 2025The Open DAC 2025 Dataset for Sorbent Discovery in Direct Air Capturecited by 7
  26. 2025Learning from the electronic structure of molecules across the periodic tablecited by 7
  27. 2026A recipe for scalable attention-based MLIPs: unlocking long-range accuracy with all-to-all node attentioncited by 7
  28. 2026EquiformerV3: Scaling Efficient, Expressive, and General SE(3)-Equivariant Graph Attention Transformerscited by 7
  29. 2023Equivariant Single View Pose Prediction Via Induced and Restricted Representationscited by 6
  30. 2024GeoMFormer: A General Architecture for Geometric Molecular Representation Learningcited by 6
  31. 2024Learning Equivariant Non-Local Electron Density Functionalscited by 6
  32. 2025Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Framescited by 5
  33. 2025Spectral/Spatial Tensor Atomic Cluster Expansion with Universal Embeddings in Cartesian Spacecited by 5
  34. 2025Learning Inter-Atomic Potentials without Explicit Equivariancecited by 5
  35. 2024TraceGrad: a Framework Learning Expressive SO(3)-equivariant Non-linear Representations for Electronic-Structure Hamiltonian Predictioncited by 4
  36. 2024SE(3)-bi-equivariant Transformers for Point Cloud Assemblycited by 4
  37. 2025Equivariant Masked Position Prediction for Efficient Molecular Representationcited by 4
  38. 2025DistMLIP: A Distributed Inference Platform for Machine Learning Interatomic Potentialscited by 4

50 most cited of 113 All 113 (JSON)