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 and prove 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（GNN）常被用于处理涉及给定图几何结构的任务，例如分子动力学模拟。尽管几何图形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（MPNN）不足以学习该几何结构。在本工作中，我们通过构造新颖且对称的几何图系列，扩展了MPNN无法从其距离矩阵区分的反例族。随后，我们提出$k$-DisGNN，它能够有效利用距离矩阵中包含的丰富几何信息。我们证明了所提模型的高表达能力，并表明一些现有精心设计的几何模型可作为$k$-DisGNN的特例被统一。最重要的是，我们在几何深度学习与传统图表示学习之间建立了联系，表明那些最初为图结构学习设计的高表达能力GNN模型，同样可以应用于几何深度学习问题并取得卓越性能，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。