  Topological Neural Networks (TNNs) incorporate higher-order relational
information beyond pairwise interactions, enabling richer representations than
Graph Neural Networks (GNNs). Concurrently, topological descriptors based on
persistent homology (PH) are being increasingly employed to augment the GNNs.
We investigate the benefits of integrating these two paradigms. Specifically,
we introduce TopNets as a broad framework that subsumes and unifies various
methods in the intersection of GNNs/TNNs and PH such as (generalizations of)
RePHINE and TOGL. TopNets can also be readily adapted to handle (symmetries in)
geometric complexes, extending the scope of TNNs and PH to spatial settings.
Theoretically, we show that PH descriptors can provably enhance the
expressivity of simplicial message-passing networks. Empirically, (continuous
and E(n)-equivariant extensions of) TopNets achieve strong performance across
diverse tasks, including antibody design, molecular dynamics simulation, and
drug property prediction.
