Mathematical Foundations of Artificial Intelligence Basics of Manifold Theory (Momiao Xiong)（Z-Library）

Author： Momiao Xiong

The Book is the 1st vol. in a 2 part series. Together, they establish a unifying mathematical framework based on smooth manifold theory and Riemannian geometry・essential tools for representing, analyzing, and integrating the growing complexity of modern artificial intelligence (AI) systems and scientific models. Differential geometry now plays a central role across AI, biology, physics, and medicine. From deep learning, generative modeling, and manifold learning to reasoning algorithms and physical AI, manifolds offer a coherent geometric language that bridges theory and practice. This volume introduces key concepts・topological and smooth manifolds, Riemannian metrics, differential forms, Lie derivatives, and statistical geometry・alongside illustrative applications to data science, genomics, drug discovery, and AI‑driven systems. This book combines rigor and intuition, integrating formal theory, computational methods, and interdisciplinary ideas, and is ideal for most professionals across a wide range of scientific fields. Key Features • Unifies core manifold concepts to support integrated thinking across disciplines • Treats manifolds as natural geometric domains for data representation in AI and the sciences • Bridges abstract theory with practical algorithms and real‑world applications • Develops Lie derivative aware graphical neural networks for adaptive‑AI and molecular property rediction • Develops Lie derivative enhanced reaction‑diffusion equations for disease gene identification and treatment design • Develops probabilistic modeling and information geometry for modern learning systems • Applies geometric insight to AI fields, including generative models, graph learning, and reasoning • Applies the Gauss map and Chen-Gauss-Bonnet theorem to physical AI incorporating geometric constraints for robotics and tumor cell location and range identification • Features step‑by‑step examples, case studies, and visual explanations to support understanding