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

The irreducible complexity of natural phenomena has led Graph Neural Networks
to be employed as a standard model to perform representation learning tasks on
graph-structured data. While their capacity to capture local and global
patterns is remarkable, the implications associated with long-range and
higher-order dependencies pose considerable challenges to such models. This
work starts with a theoretical framework to reveal the impact of network's
width, depth, and graph topology on the over-squashing phenomena in
message-passing neural networks. Then, the work drifts towards, higher-order
interactions and multi-relational inductive biases via Topological Neural
Networks. Such models propagate messages through higher-dimensional structures,
providing shortcuts or additional routes for information flow. With this
construction, the underlying computational graph is no longer coupled with the
input graph structure, thus mitigating the aforementioned bottlenecks while
accounting also for higher-order interactions. Inspired by Graph Attention
Networks, two topological attention networks are proposed: Simplicial and Cell
Attention Networks. The rationale behind these architecture is to leverage the
extended notion of neighbourhoods provided by the arrangement of groups of
nodes within a simplicial or cell complex to design anisotropic aggregations
able to measure the importance of the information coming from different regions
of the domain. By doing so, they capture dependencies that conventional Graph
Neural Networks might miss. Finally, a multi-way communication scheme is
introduced with Enhanced Cellular Isomorphism Networks, which augment
topological message passing schemes to enable a direct interactions among
groups of nodes arranged in ring-like structures.

图神经网络在图结构化数据的表示学习任务中具有捕捉局部和全局模式的能力，但是长程和高阶依赖性带来了挑战。为了解决此问题，该论文提出了拓扑神经网络，并引入了基于拓扑结构的消息传递机制和多元关系归纳偏置，以更好地捕捉依赖关系和设计各向异性聚合方法。此外，还介绍了增强型细胞同构网络，通过增加拓扑消息传递机制，使环形结构中的节点组之间能够直接进行交互。

拓扑神经网络：通过高阶交互缓解图神经网络的瓶颈问题

Topological Neural Networks: Mitigating the Bottlenecks of Graph Neural  Networks via Higher-Order Interactions

This paper presents the computational challenge on topological deep learning
that was hosted within the ICML 2023 Workshop on Topology and Geometry in
Machine Learning. The competition asked participants to provide open-source
implementations of topological neural networks from the literature by
contributing to the python packages TopoNetX (data processing) and TopoModelX
(deep learning). The challenge attracted twenty-eight qualifying submissions in
its two-month duration. This paper describes the design of the challenge and
summarizes its main findings.

这篇论文概述了在 ICML 2023《拓扑与几何在机器学习中的应用》研讨会中举办的拓扑深度学习的计算挑战，并描述了挑战的设计和主要发现。