Topological data analysis and topological deep learning beyond persistent homology

Despite the tremendous success of artificial intelligence (AI) in science, engineering, and technology in the past decade, its explainability and generalizability have been a major concern. The solution to these challenges holds the future of AI. Topological deep learning (TDL), a new frontier in rational learning introduced by us in 2017, offers interpretable and generalized AI approaches. TDL utilizes topological data analysis (TDA), which is originally rooted in persistent homology, an algebraic topology technique for point cloud data. Recently, much effort has been given to the generalization of TDA to combinatoric spectral theory, differential topology, and geometric topology to tackle data on graphs, differentiable manifolds, and curves embedded in 3-space, respectively (see arXiv:2507.19504 for a review). These approaches reduce dimensionality, simplify geometric complexity, capture high-order interactions, and provide interpretable AI models in a manner that cannot be achieved through other mathematical, statistical, and physical methodologies. I will discuss compelling examples and applications which consistently demonstrate the advantages of TDL over competing methods.