mlsanityData to 7 Oct 2026

Wigner kernels: body-ordered equivariant machine learning without a basis

arXiv 2303.04124 ↗First posted 7 Mar 2023physics.chem-ph cs.LG stat.MLcited by 7

Not on the map. Why?

Abstract (arXiv)

Machine-learning models based on a point-cloud representation of a physical object are ubiquitous in scientific applications and particularly well-suited to the atomic-scale description of molecules and materials. Among the many different approaches that have been pursued, the description of local atomic environments in terms of their neighbor densities has been used widely and very succesfully. We propose a novel density-based method which involves computing ``Wigner kernels''. These are fully equivariant and body-ordered kernels that can be computed iteratively with a cost that is independent of the radial-chemical basis and grows only linearly with the maximum body-order considered. This is in marked contrast to feature-space models, which comprise an exponentially-growing number of terms with increasing order of correlations. We present several examples of the accuracy of models based on Wigner kernels in chemical applications, for both scalar and tensorial targets, reaching state-of-the-art accuracy on the popular QM9 benchmark dataset, and we discuss the broader relevance of these ideas to equivariant geometric machine-learning.

Key facts. Cited by 7 papers in the map (data through 7 Oct 2026). Builds on 25 of its 77 references that are in the map. First posted on arXiv 7 Mar 2023.

Builds on

25 of 77 references.

References · 25

  1. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  2. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  3. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  4. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  5. 2018PointConv: Deep Convolutional Networks on 3D Point Cloudscited by 656
  6. 2019Deep Learning for 3D Point Clouds: A Surveycited by 557
  7. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 467
  8. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  9. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 360
  10. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  11. 2020Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Moleculescited by 254
  12. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 254
13 more
  1. 2022TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentialscited by 179
  2. 2020Deep Learning for LiDAR Point Clouds in Autonomous Driving: A Reviewcited by 114
  3. 2022The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentialscited by 100
  4. 2016By-passing the Kohn-Sham equations with machine learningcited by 65
  5. 2021Simple GNN Regularisation for 3D Molecular Property Prediction & Beyondcited by 44
  6. 2021Equivariant representations for molecular Hamiltonians and N-center atomic-scale propertiescited by 19
  7. 2020Predicting molecular dipole moments by combining atomic partial charges and atomic dipolescited by 18
  8. 2022Learning curves for the multi-class teacher-student perceptroncited by 17
  9. 2021Gaussian Moments as Physically Inspired Molecular Descriptors for Accurate and Scalable Machine Learning Potentialscited by 15
  10. 2022Unified theory of atom-centered representations and message-passing machine-learning schemescited by 13
  11. 2022Tensor-reduced atomic density representationscited by 10
  12. 2021Optimal radial basis for density-based atomic representationscited by 4
  13. 2022A smooth basis for atomistic machine learningcited by 4

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 0; 2025: 2; 2026: 2.

7 papers.

  1. 2023Smooth, exact rotational symmetrization for deep learning on point cloudscited by 39
  2. 2023Evaluation of the MACE Force Field Architecture: from Medicinal Chemistry to Materials Sciencecited by 25
  3. 2025Data Fusion of Deep Learned Molecular Embeddings for Property Predictioncited by 2
  4. 2025Representing spherical tensors with scalar-based machine-learning modelscited by 2
  5. 2023HydraScreen: A Generalizable Structure-Based Deep Learning Approach to Drug Discoverycited by 1
  6. 2026A Tutorial Review of Bayesian Optimization with Gaussian Processes to Accelerate Stationary Point Searchescited by 0
  7. 2026Diagnosing Spectral Ceilings in Equivariant Neural Force Fieldscited by 0