Universally Invariant Learning in Equivariant GNNs
This addresses the computational inefficiency in achieving completeness for equivariant GNNs, which is crucial for applications in fields like chemistry and physics, though it is incremental as it builds on existing models like EGNN and TFN.
The paper tackles the problem of achieving universal approximation in equivariant Graph Neural Networks (GNNs) efficiently, proposing a framework that constructs complete equivariant GNNs with reduced computational overhead and demonstrating superior completeness and performance with few layers.
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.