A topological neural network (TNN), which takes data from a Tychonoff
topological space instead of the usual finite dimensional space, is introduced.
As a consequence, a distributional neural network (DNN) that takes Borel
measures as data is also introduced. Combined these new neural networks
facilitate things like recognizing long range dependence, heavy tails and other
properties in stochastic process paths or like acting on belief states produced
by particle filtering or hidden Markov model algorithms. The veracity of the
TNN and DNN are then established herein by a strong universal approximation
theorem for Tychonoff spaces and its corollary for spaces of measures. These
theorems show that neural networks can arbitrarily approximate uniformly
continuous functions (with respect to the sup metric) associated with a unique
uniformity. We also provide some discussion showing that neural networks on
positive-finite measures are a generalization of the recent deep learning
notion of deep sets.

本文介绍了一种基于 Tychonoff 拓扑空间的拓扑神经网络和利用 Borel 测度作为数据的分布式神经网络，证明了它们的正确性，并说明了它们是深度学习中 deep sets 的一种一般化。

普适逼近和拓扑神经网络

Universal Approximation and the Topological Neural Network

The natural world is full of complex systems characterized by intricate
relations between their components: from social interactions between
individuals in a social network to electrostatic interactions between atoms in
a protein. Topological Deep Learning (TDL) provides a comprehensive framework
to process and extract knowledge from data associated with these systems, such
as predicting the social community to which an individual belongs or predicting
whether a protein can be a reasonable target for drug development. TDL has
demonstrated theoretical and practical advantages that hold the promise of
breaking ground in the applied sciences and beyond. However, the rapid growth
of the TDL literature has also led to a lack of unification in notation and
language across Topological Neural Network (TNN) architectures. This presents a
real obstacle for building upon existing works and for deploying TNNs to new
real-world problems. To address this issue, we provide an accessible
introduction to TDL, and compare the recently published TNNs using a unified
mathematical and graphical notation. Through an intuitive and critical review
of the emerging field of TDL, we extract valuable insights into current
challenges and exciting opportunities for future development.

Topological Deep Learning 解决复杂系统数据处理与知识提取的框架，但目前 Topological Neural Network 研究缺乏统一的符号和语言。本文提供对 TDL 可访问的介绍，并使用统一数学和图形符号比较最近发表的 TNN。通过对 TDL 领域的直观和批判性评估，提取有价值的见解，为未来发展提供激动人心的机遇。