  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.
