Graph Neural Networks (GNNs) are often used for tasks involving the 3D geometry of a given graph, such as molecular dynamics simulation. Although the distance matrix of a geometric graph contains complete geometric information, it has been demonstrated that Message Passing Neural Networks (MPNNs) are insufficient for learning this geometry. In this work, we expand on the families of counterexamples that MPNNs are unable to distinguish from their distance matrices, by constructing families of novel and symmetric geometric graphs, to better understand the inherent limitations of MPNNs. We then propose $k$-DisGNNs, which can effectively exploit the rich geometry contained in the distance matrix. We demonstrate the high expressive power of $k$-DisGNNs from three perspectives: 1. They can learn high-order geometric information that cannot be captured by MPNNs. 2. They can unify some existing well-designed geometric models. 3. They are universal function approximators from geometric graphs to scalars (when $k\geq 2$) and vectors (when $k\geq 3$). Most importantly, we establish a connection between geometric deep learning (GDL) and traditional graph representation learning (GRL), showing that those highly expressive GNN models originally designed for GRL can also be applied to GDL with impressive performance, and that existing complex, equivariant models are not the only solution. Experiments verify our theory.

翻译：图神经网络常用于涉及给定图三维几何结构的任务，如分子动力学模拟。尽管几何图的距离矩阵包含完整的几何信息，但已有研究表明消息传递神经网络（MPNNs）无法充分学习这种几何结构。为深入理解MPNNs的固有限制，本文通过构造新型对称几何图族，扩展了MPNNs无法从距离矩阵区分的反例族。进而提出$k$-DisGNNs模型，该模型能有效利用距离矩阵中蕴含的丰富几何信息。我们从三个维度证明$k$-DisGNNs的强大表达能力：1）可学习MPNNs无法捕获的高阶几何信息；2）能统一现有若干精心设计的几何模型；3）是几何图到标量（$k\geq 2$）和向量（$k\geq 3$）的通用函数逼近器。更重要的是，本文建立了几何深度学习与传统图表示学习之间的理论联系，表明原本为图表示学习设计的高表达力GNN模型同样适用于几何深度学习且性能优异，同时揭示现有复杂的等变模型并非唯一解决方案。实验结果验证了我们的理论。