Invariant models, one important class of geometric deep learning models, are capable of generating meaningful geometric representations by leveraging informative geometric features. These models are characterized by their simplicity, good experimental results and computational efficiency. However, their theoretical expressive power still remains unclear, restricting a deeper understanding of the potential of such models. In this work, we concentrate on characterizing the theoretical expressiveness of invariant models. We first rigorously bound the expressiveness of the most classical invariant model, Vanilla DisGNN (message passing neural networks incorporating distance), restricting its unidentifiable cases to be only those highly symmetric geometric graphs. To break these corner cases' symmetry, we introduce a simple yet E(3)-complete invariant design by nesting Vanilla DisGNN, named GeoNGNN. Leveraging GeoNGNN as a theoretical tool, we for the first time prove the E(3)-completeness of three well-established geometric models: DimeNet, GemNet and SphereNet. Our results fill the gap in the theoretical power of invariant models, contributing to a rigorous and comprehensive understanding of their capabilities. Experimentally, GeoNGNN exhibits good inductive bias in capturing local environments, and achieves competitive results w.r.t. complicated models relying on high-order invariant/equivariant representations while exhibiting significantly faster computational speed.

翻译：不变模型作为几何深度学习模型的重要一类，通过利用信息丰富的几何特征能够生成有意义的几何表示。这类模型具有结构简单、实验效果优异和计算效率高的特点。然而，其理论表达能力仍不明确，制约了对这类模型潜力的深入理解。本文致力于刻画不变模型的理论表达能力。我们首先严格界定了最经典的不变模型Vanilla DisGNN（融入距离的消息传递神经网络）的表达能力，将其不可区分情况严格限定为仅高度对称的几何图。为突破这些极端情况的对称性，我们通过嵌套Vanilla DisGNN引入一种简单但E(3)完备的不变设计——GeoNGNN。以GeoNGNN为理论工具，我们首次证明了三个成熟几何模型（DimeNet、GemNet和SphereNet）的E(3)完备性。研究成果填补了不变模型理论能力的空白，有助于对其能力形成严谨全面的理解。实验表明，GeoNGNN在捕捉局部环境方面具有良好的归纳偏置，相较于依赖高阶不变/等变表示的复杂模型展现出显著更快的计算速度，同时取得了具有竞争力的结果。