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

arXiv · AI, language, vision and robotics · article · Sep 1, 2026 · UTC

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 p

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