提出高效构建完备等变图神经网络的新方法，显著降低计算开销。
Universally Invariant Learning in Equivariant GNNs
- 基于几何图的规范形式与满秩可旋基构造完备等变网络
- 仅用少数层数即达完备性，计算成本远低于传统方法
- 适合需高精度对称性建模的分子、物理系统等场景
等变图神经网络在诸多应用中表现卓越。为实现完备性——即在等变函数空间中具备通用逼近能力——网络需有效捕捉节点间的复杂多体相互作用。以往方法通过加深网络、增加体阶或提升可旋特征度数来实现，但通常计算成本高昂且无多项式时间解法。本文提出一种理论严谨的框架，可高效构建完备等变GNN。我们证明，完备等变GNN可通过两个关键组件实现：1）称为几何图规范形式的完备标量函数；2）满秩可旋基集。基于此，我们设计了针对EGNN和TFN两种常见模型的高效算法。实验表明，所提模型在仅用少量层数的情况下即展现优异完备性与性能，显著降低计算开销，同时保持强大实用性。
原文摘要 · Abstract (English)
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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