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Spin-Weighted Spherical Harmonics Enable Complete and Scalable E(3)-Equivariant Networks

arXiv 2607.01408 ↗First posted 1 Jul 2026cs.LG cs.AIcited by 0

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Abstract (arXiv)

E(3)-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by the O(L6) complexity of the Clebsch-Gordan Tensor Product (CGTP). The recently proposed Gaunt Tensor Product (GTP) reduces the complexity but is unable to capture the antisymmetric paths, resulting in incomplete expressivity. In this work, we present SpinGTP, an approach to overcome the GTP incompleteness by generalizing from scalar functions to Spin-Weighted Spherical Harmonics (SWSH). By relying on the algebraic properties of SWSH, SpinGTP recovers the missing antisymmetric interactions while maintaining the asymptotic efficiency of GTP. It also allows for a more expressive equivariant basis that naturally accounts for the parity-odd components of tensor products. We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE-MACE-OFF, and OC20. Our results show that SpinGTP achieves accuracies comparable to full CGTP. Notably, by explicitly capturing antisymmetric paths, SpinGTP exhibits superior performance in tasks involving chiral materials and non-centrosymmetric geometries. This work provides a complete, scalable, and mathematically rigorous path toward high-order equivariance in large-scale 3D atomistic system simulations.

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References · 24

  1. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  2. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  3. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  4. 2017SchNet: A continuous-filter convolutional neural network for modeling quantum interactionscited by 569
  5. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 467
  6. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  7. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  8. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  9. 2022Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphscited by 258
  10. 2020Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Moleculescited by 254
  11. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 254
  12. 2020The Open Catalyst 2020 (OC20) Dataset and Community Challengescited by 246
12 more
  1. 2023EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representationscited by 201
  2. 2021Spherical Message Passing for 3D Graph Networkscited by 166
  3. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  4. 2023Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systemscited by 118
  5. 2023Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNscited by 113
  6. 2022The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentialscited by 100
  7. 2022ComENet: Towards Complete and Efficient Message Passing for 3D Molecular Graphscited by 69
  8. 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 35
  9. 2025The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Productscited by 14
  10. 2025E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Productscited by 12
  11. 2026Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonicscited by 4
  12. 2025Augmenting Molecular Graphs with Geometries via Machine Learning Interatomic Potentialscited by 1

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