Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials
cs.LG, math-ph, math.MP, physics.chem-ph, physics.comp-ph
Submitted: 2026-09-01
Updated: 2026-09-02
License: http://creativecommons.org/licenses/by-nc-sa/4.0/
The gist: We prove that the Hypergraph Neural Network, an invariant architecture with 3-body message passing, is a universal approximator for potential energy surfaces.
Terminology
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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