MFAI: Mathematical Foundations of Artificial Intelligence via Transport Information Geometry

This project develops mathematical foundations and new algorithms for deep learning–based optimization through a novel integration of optimal transport, information geometry, mean-field control, and spectral graph theory. Modern machine learning methods often lack a rigorous geometric framework which can provide critical insight for understanding optimization, generalization, and sampling in high-dimensional parameter spaces. This project aims to advance the reliability of AI systems by introducing a transport–information metric framework for neural networks that combines Wasserstein geometry, information-theoretic divergences, and spectral properties of information matrices that will be used to establish stability, optimization, and generalization properties. It also advances interdisciplinary research at the interface of mathematics, computer science, and engineering through graduate and postdoctoral training, workshops, outreach activities, and the development of open-source software. The research scope consists of three interconnected thrusts. First, it develops new transport–information loss functionals and metric structures on neural network parameter spaces, with a focus on information matrices, spectral graph representations, and geodesic convexity. Second, it designs optimization algorithms based on transport–information gradient flows, including Wasserstein natural gradient and proximal methods, with applications to supervised learning and scientific computing problems such as Fokker–Planck equations and reaction–diffusion systems. Third, it constructs discrete sampling algorithms on graphs using Olivier and entropic Ricci curvatures derived from Wasserstein metrics, enabling efficient Markov kernels with provable convergence guarantees for applications including large language models and time-reversible diffusion processes. By connecting transport theory, information geometry, mean-field control, and spectral graph methods, the project establishes a unified framework for understanding and improving modern machine learning algorithms. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.