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.

通过结合拓扑神经网络（TNNs）和持久同调（PH）的方法，构建了一种广泛的框架 TopNets，扩展了 GNNs / TNNs 和 PH 在空间设置中的应用范围，提高了简单消息传递网络的表达能力，并在抗体设计、分子动力学模拟和药物属性预测等各种任务中取得了良好的性能。

拓扑神经网络的持续性、等变性、连续性

Topological Neural Networks go Persistent, Equivariant, and Continuous

In this paper we study the properties of the homology of different geometric
filtered complexes (such as Vietoris-Rips, Cech and witness complexes) built on
top of precompact spaces. Using recent developments in the theory of
topological persistence we provide simple and natural proofs of the stability
of the persistent homology of such complexes with respect to the
Gromov--Hausdorff distance. We also exhibit a few noteworthy properties of the
homology of the Rips and Cech complexes built on top of compact spaces.

本文研究在预紧空间上建立的不同几何滤复形的同调性质（如 Vietoris-Rips，Cech 和见证复形）。利用拓扑持久性理论的最新进展，我们提供了这些复形的持久同调的稳定性的简洁而自然的证明。我们还展示了在紧空间上建立的 Rips 和 Cech 复形的一些值得注意的同调性质。