This survey provides a comprehensive exploration of applications of Topological Data Analysis (TDA) within neural network analysis. Using TDA tools such as persistent homology and Mapper, we delve into the intricate structures and behaviors of neural networks and their datasets. We discuss different strategies to obtain topological information from data and neural networks by means of TDA. Additionally, we review how topological information can be leveraged to analyze properties of neural networks, such as their generalization capacity or expressivity. We explore practical implications of deep learning, specifically focusing on areas like adversarial detection and model selection. Our survey organizes the examined works into four broad domains: 1. Characterization of neural network architectures; 2. Analysis of decision regions and boundaries; 3. Study of internal representations, activations, and parameters; 4. Exploration of training dynamics and loss functions. Within each category, we discuss several articles, offering background information to aid in understanding the various methodologies. We conclude with a synthesis of key insights gained from our study, accompanied by a discussion of challenges and potential advancements in the field.

翻译：本综述全面探讨了拓扑数据分析（TDA）在神经网络分析中的应用。通过使用持续同调与Mapper等TDA工具，我们深入研究了神经网络及其数据集的复杂结构与行为模式。我们讨论了通过TDA从数据和神经网络中获取拓扑信息的不同策略，并系统梳理了如何利用拓扑信息分析神经网络的泛化能力与表达性等特性。在深度学习实践领域，我们重点关注对抗性检测与模型选择等方向。本综述将研究对象划分为四大范畴：1. 神经网络架构表征；2. 决策区域与决策边界分析；3. 内部表征、激活值与参数研究；4. 训练动态与损失函数探索。针对每个类别，我们结合多篇文献展开讨论，提供理解各类方法论所需的背景知识。最后，我们整合研究中的关键发现，并探讨该领域面临的挑战与潜在发展方向。