| 刘冉,吴杰,张蒙蒙.人工智能与复杂系统的拓扑理论[J].数学年刊A辑,2025,46(3):225~246 |
| 人工智能与复杂系统的拓扑理论 |
| Topology in Artificial Intelligence and Complex Systems |
| Received:April 02, 2025 Revised:September 02, 2025 |
| DOI：10.16205/j.cnki.cama.2025.0014 |
| 中文关键词: 拓扑数据分析，持久同调，胞腔层，Δ- 层，Δ- 层同调，GLMY 同调，双超图，超网络，互作复形 |
| 英文关键词:Topological data analysis, Persistent homology, Cellular sheaf, Δ sheaf, Δ-sheaf homology, GLMY homology, Super-hypergraph, Hypernetwork, IntComplex |
| 基金项目: |
| Author Name | Affiliation | | LIU Ran、 | 1 School of Mathematical Sciences, Beihang University, Beijing 100191, China
Beijing Key Laboratory of Topological Statistics and Applications for Complex Systems, Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China | | WU Jie、 | 2 School of Mathematical Sciences, Hebei Normal University, Shijiazhuang 050024, China
Beijing Key Laboratory of Topological Statistics and Applications for Complex Systems, Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China | | ZHANG Mengmeng | 3 Beijing Key Laboratory of Topological Statistics and Applications for Complex Systems, Beijing Institute of Mathematical Sciences and Applications, Beijing 101408, China |
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| 中文摘要: |
| 拓扑学为分析几何形状与复杂系统提供了整体性的研究框架，通过引入代数不变量，旨在揭示系统背后的核心拓扑结构。自 20 世纪 90 年代拓扑方法在数据科学中的应用探索历经数十年，拓扑数据分析（TDA）于 2009 年正式诞生，伴随理论与方法的不断发展，TDA 已成为数据分析中广泛应用的非线性数学工具，近年来进一步应用于机器学习、深度学习与复杂网络分析，成为捕捉复杂系统中高阶互作结构的有效数学框架。本文介绍与人工智能相关的基础拓扑理论，包括单纯同调、层同调以及新发展的应用拓扑理论，为从事拓扑应用研究的数学工作者与学生、以及寻求非线性人工智能数学工具的科研人员提供系统性的概念与方法指引 |
| 英文摘要: |
| Topology is a research subject that provides a strategic and global framework for analyzing geometric shapes and complex systems. By introducing algebraic invariants, it aims to detect the core topological structures underlying these systems. The area of Topological Data Analysis (TDA for short) was born in 2009, following decades of exploration into the application of topological methods to data science since the 1990s. With continuous theoretical and methodological advances, TDA has become a widely used nonlinear mathematical tool in data analytics. In recent years, it has been further applied to machine learning, deep learning, and complex network analysis, serving as an effective mathematical framework for capturing higher-order interaction structures in complex systems. In this article, the authors introduce the fundamental topological theories relevant to artificial intelligence, including simplicial homology, sheaf homology, and newly developed applied topology theories. The goal is to provide a conceptual and methodological guide for mathematicians and students interested in applied topology, as well as for researchers in artificial intelligence seeking nonlinear mathematical tools. |
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