Weisfeiler Leman for Euclidean Equivariant Machine Learning
Snir Hordan Tal Amir Nadav Dym
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.