Advancing Artificial Intelligence Through Mathematical Innovations
Abstract
Despite the tremendous success of artificial intelligence (AI) in science, engineering, and technology over the past decade, its explainability, generalizability, and reliability remain major concerns. Addressing these challenges will shape the future of AI. Topological deep learning (TDL), a new paradigm of rational learning introduced by us in 2017, offers interpretable and generalizable AI approaches. TDL utilizes topological data analysis (TDA), which is rooted in persistent homology, a technique from algebraic topology. However, persistent homology has several limitations, including a lack of localization, restriction to point-cloud data, and an inability to represent non-topological information. To address these challenges, we generalized TDA to combinatorial spectral theory (e.g., topological Laplacians and Dirac operators), differential topology (e.g., de Rham–Hodge theory), and geometric topology (e.g., Khovanov homology) to handle data on graphs, differentiable manifolds, and curves embedded in 3-space, respectively (see Artificial Intelligence Review 59, 58 (2026) for a review). To further advance AI through modern mathematical theories, we introduced commutative algebra as a new frontier in data science and machine learning.
Guowei Wei received his Ph.D. from the University of British Columbia and is currently a Professor and GRA Eminent Scholar at the University of Georgia. He is a Fellow of SIAM and AIBME and has mentored more than 150 research students, postdoctoral researchers, and visiting scholars. His research focuses on mathematical approaches to bioscience and artificial intelligence (AI). His group has developed mathematical AI approaches, including topological deep learning (TDL), manifold topological learning (MTL), and commutative algebra learning (CAL), which combine mathematical structures with AI for the analysis of complex real-world data. These approaches have been applied to problems in computational drug design and molecular bioscience, including the D3R Grand Challenges. His recent work has also combined topological learning, genotyping, and computational biophysics to study SARS-CoV-2 evolution and forecast emerging viral variants.