The $k$-Weisfeiler-Leman ($k$-WL) graph isomorphism test hierarchy is a common method for assessing the expressive power of graph neural networks (GNNs). Recently, GNNs whose expressive power is equivalent to the $2$-WL test were proven to be universal on weighted graphs which encode $3\mathrm{D}$ point cloud data, yet this result is limited to invariant continuous functions on point clouds. In this paper, we extend this result in three ways: Firstly, we show that PPGN can simulate $2$-WL uniformly on all point clouds with low complexity. Secondly, we show that $2$-WL tests can be extended to point clouds which include both positions and velocities, a scenario often encountered in applications. Finally, we provide a general framework for proving equivariant universality and leverage it to prove that a simple modification of this invariant PPGN architecture can be used to obtain a universal equivariant architecture that can approximate all continuous equivariant functions uniformly. Building on our results, we develop our WeLNet architecture, which sets new state-of-the-art results on the N-Body dynamics task and the GEOM-QM9 molecular conformation generation task.

翻译：$k$-Weisfeiler-Leman ($k$-WL) 图同构测试层次结构是评估图神经网络 (GNNs) 表达能力的一种常用方法。最近，表达能力等价于 $2$-WL 测试的 GNNs 被证明在编码 $3\mathrm{D}$ 点云数据的加权图上是通用的，但这一结果仅限于点云上的不变连续函数。在本文中，我们从三个方面扩展了这一结果：首先，我们证明 PPGN 可以在所有点云上以低复杂度均匀地模拟 $2$-WL。其次，我们证明 $2$-WL 测试可以扩展到同时包含位置和速度的点云，这是应用中经常遇到的情况。最后，我们提供了一个证明等变通用性的通用框架，并利用它证明，通过对这个不变的 PPGN 架构进行简单修改，即可获得一个通用的等变架构，该架构能够均匀地逼近所有连续的等变函数。基于我们的结果，我们开发了 WeLNet 架构，该架构在 N-Body 动力学任务和 GEOM-QM9 分子构象生成任务上取得了新的最先进结果。