This paper presents $\mathrm{E}(n)$ Equivariant Message Passing Simplicial
Networks (EMPSNs), a novel approach to learning on geometric graphs and point
clouds that is equivariant to rotations, translations, and reflections. EMPSNs
can learn high-dimensional simplex features in graphs (e.g. triangles), and use
the increase of geometric information of higher-dimensional simplices in an
$\mathrm{E}(n)$ equivariant fashion. EMPSNs simultaneously generalize
$\mathrm{E}(n)$ Equivariant Graph Neural Networks to a topologically more
elaborate counterpart and provide an approach for including geometric
information in Message Passing Simplicial Networks. The results indicate that
EMPSNs can leverage the benefits of both approaches, leading to a general
increase in performance when compared to either method. Furthermore, the
results suggest that incorporating geometric information serves as an effective
measure against over-smoothing in message passing networks, especially when
operating on high-dimensional simplicial structures. Last, we show that EMPSNs
are on par with state-of-the-art approaches for learning on geometric graphs.

本文提出了 E (n) 等变消息传递单纯形网络（EMPSNs）的新方法，可以在几何图形和点云上学习，具有旋转、平移和反射等等变性。EMPSNs 可以通过等变的方式使用高维单纯形的几何信息学习高维单纯形特征，是 E (n) 等变图神经网络的拓扑更复杂的扩展，并提供了在消息传递单纯形网络中包含几何信息的方法。结果表明，EMPSNs 可以利用两种方法的优势，相对于任一方法都有更好的性能提升。此外，结果表明，在操作高维单纯形结构时，加入几何信息可以有效防止过度平滑，适用于消息传递网络。最后，我们显示 EMPSNs 与最先进的学习几何图形的方法相当。