Weisfeiler Leman for Euclidean Equivariant Machine Learning
<|reference_start|>Weisfeiler Leman for Euclidean Equivariant Machine Learning: 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.<|reference_end|>
@article{hordan2024weisfeiler,
    title={Weisfeiler Leman for Euclidean Equivariant Machine Learning},
    author={Snir Hordan, Tal Amir, Nadav Dym},
    journal={arXiv preprint arXiv:2402.02484},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.02484},
    primaryClass={cs.LG}
}