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

arXiv: https://arxiv.org/abs/2609.00528 · HTML: https://mlsanity.com/paper/2609.00528/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 46 references that are in the map.
- First posted on arXiv 1 Sep 2026.

## Builds on (most cited first, 23 of 23)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [On the Limitations of Representing Functions on Sets](https://mlsanity.com/paper/1901.09006/) (2019)
- [On the Expressive Power of Geometric Graph Neural Networks](https://mlsanity.com/paper/2301.09308/) (2023)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [Smooth, exact rotational symmetrization for deep learning on point clouds](https://mlsanity.com/paper/2305.19302/) (2023)
- [AdsorbML: A Leap in Efficiency for Adsorption Energy Calculations using Generalizable Machine Learning Potentials](https://mlsanity.com/paper/2211.16486/) (2022)
- [Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?](https://mlsanity.com/paper/2410.11443/) (2024)
- [On the Completeness of Invariant Geometric Deep Learning Models](https://mlsanity.com/paper/2402.04836/) (2024)
- [Complete Neural Networks for Complete Euclidean Graphs](https://mlsanity.com/paper/2301.13821/) (2023)
- [Three iterations of (d-1)-WL test distinguish non isometric clouds of d-dimensional points](https://mlsanity.com/paper/2303.12853/) (2023)
- [On the Expressive Power of Sparse Geometric MPNNs](https://mlsanity.com/paper/2407.02025/) (2024)
- [A Galois theorem for machine learning: Functions on symmetric matrices and point clouds via lightweight invariant features](https://mlsanity.com/paper/2405.08097/) (2024)
- [Specialising and Analysing Instruction-Tuned and Byte-Level Language Models for Organic Reaction Prediction](https://mlsanity.com/paper/2405.10625/) (2024)

## 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.