mlsanityData to 7 Oct 2026

Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials

arXiv 2609.00528 ↗First posted 1 Sep 2026cs.LG math-ph math.MPcited by 0

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Abstract (arXiv)

We prove that the Hypergraph Neural Network, an invariant architecture with 3-body message passing, is a universal approximator for potential energy surfaces. Our main contribution is a multi-layer completeness theory. We show that L layers of message passing on sparse, cutoff-based graphs achieve the same representational power as having access to the full L-hop neighborhood, provided the configurations are generic, satisfy an overlap condition and a connectivity condition. This provides the first rigorous justification for the common practice of using multi-layer message passing with a per-layer cutoff smaller than the physical interaction range, the setting used by virtually all practical graph neural network based machine-learned interatomic potentials. As immediate consequences, we show that both DPA3 and CHGNet architectures inherit universal approximation.

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References · 23

  1. 2017Deep Setscited by 1,394
  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. 2020Directional Message Passing for Molecular Graphscited by 556
  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. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  9. 2022TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentialscited by 179
  10. 2021Spherical Message Passing for 3D Graph Networkscited by 166
  11. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  12. 2019On the Limitations of Representing Functions on Setscited by 89
11 more
  1. 2023On the Expressive Power of Geometric Graph Neural Networkscited by 89
  2. 2020On the Universality of Rotation Equivariant Point Cloud Networkscited by 80
  3. 2023Smooth, exact rotational symmetrization for deep learning on point cloudscited by 39
  4. 2022AdsorbML: A Leap in Efficiency for Adsorption Energy Calculations using Generalizable Machine Learning Potentialscited by 25
  5. 2024Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?cited by 18
  6. 2024On the Completeness of Invariant Geometric Deep Learning Modelscited by 11
  7. 2023Complete Neural Networks for Complete Euclidean Graphscited by 10
  8. 2023Three iterations of (d-1)-WL test distinguish non isometric clouds of d-dimensional pointscited by 9
  9. 2024On the Expressive Power of Sparse Geometric MPNNscited by 7
  10. 2024A Galois theorem for machine learning: Functions on symmetric matrices and point clouds via lightweight invariant featurescited by 6
  11. 2024Specialising and Analysing Instruction-Tuned and Byte-Level Language Models for Organic Reaction Predictioncited by 3

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