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arXiv · 2510.13169

Universally Invariant Learning in Equivariant GNNs

Jiacheng Cen·Anyi Li·Ning Lin·Tingyang Xu·Yu Rong·Deli Zhao·Zihe Wang·Wenbing Huang
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Abstract

Equivariant Graph Neural Networks (GNNs) have demonstrated significant success across various applications. To achieve completeness -- that is, the universal approximation property over the space of equivariant functions -- the network must effectively capture the intricate multi-body interactions among different nodes. Prior methods attain this via deeper architectures, augmented body orders, or increased degrees of steerable features, often at high computational cost and without polynomial-time solutions. In this work, we present a theoretically grounded framework for constructing complete equivariant GNNs that is both efficient and practical. We prove that a complete equivariant GNN can be achieved through two key components: 1) a complete scalar function, referred to as the canonical form of the geometric graph; and 2) a full-rank steerable basis set. Leveraging this finding, we propose an efficient algorithm for constructing complete equivariant GNNs based on two common models: EGNN and TFN. Empirical results demonstrate that our model demonstrates superior completeness and excellent performance with only a few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy.

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Jiacheng Cen, Anyi Li, Ning Lin, Tingyang Xu, Yu Rong, Deli Zhao, Zihe Wang, Wenbing Huang. 2025-10-15. Universally Invariant Learning in Equivariant GNNs. https://arxiv.org/abs/2510.13169

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