  Graph neural networks excel at modeling pairwise interactions, but they
cannot flexibly accommodate higher-order interactions and features. Topological
deep learning (TDL) has emerged recently as a promising tool for addressing
this issue. TDL enables the principled modeling of arbitrary multi-way,
hierarchical higher-order interactions by operating on combinatorial
topological spaces, such as simplicial or cell complexes, instead of graphs.
However, little is known about how to leverage geometric features such as
positions and velocities for TDL. This paper introduces E(n)-Equivariant
Topological Neural Networks (ETNNs), which are E(n)-equivariant message-passing
networks operating on combinatorial complexes, formal objects unifying graphs,
hypergraphs, simplicial, path, and cell complexes. ETNNs incorporate geometric
node features while respecting rotation and translation equivariance. Moreover,
ETNNs are natively ready for settings with heterogeneous interactions. We
provide a theoretical analysis to show the improved expressiveness of ETNNs
over architectures for geometric graphs. We also show how several E(n)
equivariant variants of TDL models can be directly derived from our framework.
The broad applicability of ETNNs is demonstrated through two tasks of vastly
different nature: i) molecular property prediction on the QM9 benchmark and ii)
land-use regression for hyper-local estimation of air pollution with
multi-resolution irregular geospatial data. The experiment results indicate
that ETNNs are an effective tool for learning from diverse types of richly
structured data, highlighting the benefits of principled geometric inductive
bias.
