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

Abstract

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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BibTeX RIS

BibTeX

@misc{ming2026multilayer,
  title = {{Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials}},
  author = {Pingbing Ming and Han Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.00528},
  eprint = {2609.00528},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.00528},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials
AU  - Pingbing Ming
AU  - Han Wang
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.00528
UR  - https://arxiv.org/abs/2609.00528
AB  - 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.
ER  -

Abstract

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