Graph Neural Networks (GNNs) are often used for tasks involving the 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. We then propose $k$-DisGNNs, which can effectively exploit the rich geometry contained in the distance matrix. We demonstrate the high expressive power of our models by proving the universality of $k$-DisGNNs for distinguishing geometric graphs when $k \geq 3$, and that some existing well-designed geometric models can be unified by $k$-DisGNNs as special cases. Most importantly, we establish a connection between geometric deep learning and traditional graph representation learning, showing that those highly expressive GNN models originally designed for graph structure learning can also be applied to geometric deep learning problems with impressive performance, and that existing complex, equivariant models are not the only solution. Experimental results verify our theory.

翻译：图神经网络（GNNs）常被用于处理给定图几何结构的任务，例如分子动力学模拟。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）不足以学习该几何结构。在本工作中，我们通过构建新型对称几何图族，扩展了MPNNs无法从其距离矩阵区分的反例族。随后我们提出$k$-DisGNNs，该方法能有效利用距离矩阵中蕴含的丰富几何信息。通过证明当$k \geq 3$时$k$-DisGNNs在区分几何图方面具有通用性，并表明现有若干精心设计的几何模型可被统一视为$k$-DisGNNs的特例，我们展示了所提出模型的高表达能力。更重要的是，我们建立了几何深度学习与经典图表示学习之间的联系，证明那些最初为图结构学习设计的高表达能力GNN模型也能在几何深度学习问题中取得优异表现，且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。