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.

Machine Learning (cs.LG); Mathematical Physics (math-ph); Chemical Physics (physics.chem-ph); Computational Physics (physics.comp-ph)

arXiv:2609.00528 [cs.LG]

(or arXiv:2609.00528v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.00528 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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