The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at \url{https://github.com/chaitjo/geometric-gnn-dojo}

翻译：图神经网络（GNNs）的表达能力已通过Weisfeiler-Leman（WL）图同构测试得到广泛研究。然而，标准GNNs与WL框架无法适用于嵌入欧几里得空间的几何图（如生物分子、材料及其他物理系统）。本文提出几何版本的WL测试（GWL），用于区分几何图的同时保持物理对称性：置换、旋转、反射和平移。我们利用GWL刻画了具有物理对称性不变性或等变性的几何GNN在区分几何图方面的表达能力。GWL揭示了关键设计选择对几何GNN表达能力的影响机制：（1）不变性层表达能力有限，无法区分一跳恒等的几何图；（2）等变性层通过将几何信息传播至局部邻域之外，能够区分更广泛的图类；（3）高阶张量与标量化方法可实现最大表达能力的几何GNN；（4）基于区分的GWL视角与通用逼近等价。补充实验的合成数据见 \url{https://github.com/chaitjo/geometric-gnn-dojo}