Liu Jian, Shen Li, Wei Guo-Wei
Mathematical Science Research Center, Chongqing University of Technology, Chongqing 400054, China.
Department of Mathematics, Michigan State University, MI 48824, USA.
Found Data Sci. 2024 Jun;6(2):221-250. doi: 10.3934/fods.2024009.
ChatGPT represents a significant milestone in the field of artificial intelligence (AI), finding widespread applications across diverse domains. However, its effectiveness in mathematical contexts has been somewhat constrained by its susceptibility to conceptual errors. Concurrently, topological data analysis (TDA), a relatively new discipline, has garnered substantial interest in recent years. Nonetheless, the advancement of TDA is impeded by the limited understanding of computational algorithms and coding proficiency among theoreticians. This work endeavors to bridge the gap between theoretical topological concepts and their practical implementation in computational topology through the utilization of ChatGPT. We showcase how a pure theoretician, devoid of computational experience and coding skills, can effectively transform mathematical formulations and concepts into functional codes for computational topology with the assistance of ChatGPT. Our strategy outlines a productive process wherein a mathematician trains ChatGPT on pure mathematical concepts, steers ChatGPT towards generating computational topology codes, and subsequently validates the generated codes using established examples. Our specific case studies encompass the computation of Betti numbers, Laplacian matrices, and Dirac matrices for simplicial complexes, as well as the persistence of various homologies and Laplacians. Furthermore, we explore the application of ChatGPT in computing recently developed topological theories for hypergraphs and digraphs, as well as the persistent harmonic space, which has not been computed in the literature, to the best of our knowledge. This work serves as an initial step towards effectively transforming pure mathematical theories into practical computational tools, with the ultimate goal of enabling real applications across diverse fields.
ChatGPT代表了人工智能（AI）领域的一个重要里程碑，在各个领域都有广泛应用。然而，它在数学情境中的有效性在一定程度上受到其易犯概念性错误的限制。与此同时，拓扑数据分析（TDA）作为一门相对较新的学科，近年来引起了广泛关注。尽管如此，TDA的发展受到理论学家对计算算法理解有限和编码能力不足的阻碍。这项工作致力于通过利用ChatGPT弥合理论拓扑概念与其在计算拓扑中的实际实现之间的差距。我们展示了一个没有计算经验和编码技能的纯理论学家如何在ChatGPT的帮助下有效地将数学公式和概念转化为计算拓扑的功能代码。我们的策略概述了一个富有成效的过程，即数学家在纯数学概念上训练ChatGPT，引导ChatGPT生成计算拓扑代码，然后使用既定示例验证生成的代码。我们的具体案例研究包括单纯复形的贝蒂数、拉普拉斯矩阵和狄拉克矩阵的计算，以及各种同调群和拉普拉斯算子的持久性。此外，据我们所知，我们还探索了ChatGPT在计算最近为超图和有向图开发的拓扑理论以及文献中尚未计算的持久调和空间方面的应用。这项工作是朝着将纯数学理论有效转化为实际计算工具迈出的第一步，最终目标是实现跨领域的实际应用。
