The Weisfeiler-Lehman (WL) test is a fundamental iterative algorithm for checking the isomorphism of graphs.

The $k$-Weifeiler-Leman ($k$-WL) graph isomorphism test hierarchy is a common method for assessing the expressive power of graph neural networks (GNNs). Recently, the $2$-WL test was proven to be complete on weighted graphs which encode $3\mathrm{D}$ point cloud data. Consequently, GNNs whose expressive power is equivalent to the $2$-WL test are provably universal on point clouds. 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 $2$-WL tests can be extended to point clouds which include both positions and velocity, a scenario often encountered in applications. Secondly, we show that PPGN (Maron et al., 2019) can simulate $2$-WL uniformly on all point clouds with low complexity. Finally, we show that a simple modification of this 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 can process position-velocity pairs, compute functions fully equivariant to permutations and rigid motions, and is provably complete and universal. Remarkably, WeLNet is provably complete precisely in the setting in which it is implemented in practice. Our theoretical results are complemented by experiments showing WeLNet sets new state-of-the-art results on the N-Body dynamics task and the GEOM-QM9 molecular conformation generation task.

The Weisfeiler--Lehman (WL) test is a fundamental iterative algorithm for checking isomorphism of graphs. It has also been observed that it underlies the design of several graph neural network architectures, whose capabilities and performance can be understood in terms of the expressive power of this test. Motivated by recent developments in machine learning applications to datasets involving three-dimensional objects, we study when the WL test is {\em complete} for clouds of euclidean points represented by complete distance graphs, i.e., when it can distinguish, up to isometry, any arbitrary such cloud. Our main result states that the $(d-1)$-dimensional WL test is complete for point clouds in $d$-dimensional Euclidean space, for any $d\ge 2$, and that only three iterations of the test suffice. Our result is tight for $d = 2, 3$. We also observe that the $d$-dimensional WL test only requires one iteration to achieve completeness.