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

2026-08-31

AI Summary

本文通过多层完备性理论证明了超图神经网络在表示势能面上的通用逼近能力，解释了多层消息传递机制的有效性。

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.

Problem

• Multi-Layer Message Passing
• Graph Neural Network
• Interatomic Potentials
• Universal Approximation
• Completeness Theory

Innovation

• multi-layer completeness theory
• message passing
• universal approximation
• sparse graphs
• cutoff-based