mlsanityData to 7 Oct 2026

Weisfeiler Leman for Euclidean Equivariant Machine Learning

arXiv 2402.02484 ↗First posted 4 Feb 2024cs.LGcited by 10

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Abstract (arXiv)

The k-Weisfeiler-Leman (k-WL) graph isomorphism test hierarchy is a common method for assessing the expressive power of graph neural networks (GNNs). Recently, GNNs whose expressive power is equivalent to the 2-WL test were proven to be universal on weighted graphs which encode 3D point cloud data, yet this result is limited to invariant continuous functions on point clouds. In this paper, we extend this result in three ways: Firstly, we show that PPGN can simulate 2-WL uniformly on all point clouds with low complexity. Secondly, we show that 2-WL tests can be extended to point clouds which include both positions and velocities, a scenario often encountered in applications. Finally, we provide a general framework for proving equivariant universality and leverage it to prove that a simple modification of this invariant PPGN architecture can be used to obtain a universal equivariant architecture that can approximate all continuous equivariant functions uniformly. Building on our results, we develop our WeLNet architecture, which sets new state-of-the-art results on the N-Body dynamics task and the GEOM-QM9 molecular conformation generation task.

Key facts. Cited by 10 papers in the map (data through 7 Oct 2026). Builds on 41 of its 59 references that are in the map. First posted on arXiv 4 Feb 2024.

Builds on

41 of 59 references.

References · 41

  1. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,223
  2. 2018How Powerful are Graph Neural Networks?cited by 4,355
  3. 2021E(n) Equivariant Graph Neural Networkscited by 951
  4. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  5. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
  6. 2017SchNet: A continuous-filter convolutional neural network for modeling quantum interactionscited by 569
  7. 2022GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generationcited by 511
  8. 2019Provably Powerful Graph Networkscited by 385
  9. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  10. 2018Neural Relational Inference for Interacting Systemscited by 339
  11. 2018Invariant and Equivariant Graph Networkscited by 339
  12. 2021Vector Neurons: A General Framework for SO(3)-Equivariant Networkscited by 304
29 more
  1. 2020GEOM: Energy-annotated molecular conformations for property prediction and molecular generationcited by 238
  2. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  3. 2020The Surprising Power of Graph Neural Networks with Random Node Initializationcited by 190
  4. 2022Sign and Basis Invariant Networks for Spectral Graph Representation Learningcited by 176
  5. 2019On the equivalence between graph isomorphism testing and function approximation with GNNscited by 172
  6. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  7. 2021Learning Gradient Fields for Molecular Conformation Generationcited by 162
  8. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  9. 2021Weisfeiler and Leman go Machine Learning: The Story so farcited by 121
  10. 2020Lorentz Group Equivariant Neural Network for Particle Physicscited by 120
  11. 2021GeoMol: Torsional Geometric Generation of Molecular 3D Conformer Ensemblescited by 106
  12. 2022Equivariance with Learned Canonicalization Functionscited by 106
  13. 2019Molecular geometry prediction using a deep generative graph neural networkcited by 91
  14. 2023On the Expressive Power of Geometric Graph Neural Networkscited by 89
  15. 2021Learning Neural Generative Dynamics for Molecular Conformation Generationcited by 87
  16. 2020On the Universality of Rotation Equivariant Point Cloud Networkscited by 80
  17. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  18. 2019A Generative Model for Molecular Distance Geometrycited by 73
  19. 2021An End-to-End Framework for Molecular Conformation Generation via Bilevel Programmingcited by 59
  20. 2023Smooth, exact rotational symmetrization for deep learning on point cloudscited by 39
  21. 2023Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theoremcited by 33
  22. 2022Direct Molecular Conformation Generationcited by 30
  23. 2023Learning Probabilistic Symmetrization for Architecture Agnostic Equivariancecited by 25
  24. 2023Is Distance Matrix Enough for Geometric Deep Learning?cited by 22
  25. 2022Incompleteness of graph neural networks for points clouds in three dimensionscited by 19
  26. 2021ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Cloudscited by 15
  27. 2023Using Multiple Vector Channels Improves E(n)-Equivariant Graph Neural Networkscited by 10
  28. 2023Three iterations of (d-1)-WL test distinguish non isometric clouds of d-dimensional pointscited by 9
  29. 2023On the power of graph neural networks and the role of the activation functioncited by 7

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 5; 2026: 2.

10 papers.

  1. 2024On the Completeness of Invariant Geometric Deep Learning Modelscited by 11
  2. 2025How simple can you go? An off-the-shelf transformer approach to molecular dynamicscited by 10
  3. 2024On the Expressive Power of Sparse Geometric MPNNscited by 7
  4. 2024Multivector Neurons: Better and Faster O(n)-Equivariant Clifford Graph Neural Networkscited by 4
  5. 2025Spectral Graph Neural Networks are Incomplete on Graphs with a Simple Spectrumcited by 3
  6. 2025Universally Invariant Learning in Equivariant GNNscited by 3
  7. 2025On the (Non) Injectivity of Piecewise Linear Janossy Poolingcited by 1
  8. 2026Quantitative Approximation Rates for Group Equivariant Learningcited by 1
  9. 2025Message Passing on the Edge: Towards Scalable and Expressive GNNscited by 0
  10. 2026SymDrift: One-Shot Generative Modeling under Symmetriescited by 0