Project Grant 2601952

The National Science Foundation Division of Mathematical Sciences awarded the University of South Carolina $700,000 on September 1, 2026, under the Mathematical and Physical Sciences program (CFDA 47.049) to develop mathematical foundations and algorithms for deep learning-based optimization through integration of optimal transport, information geometry, mean-field control, and spectral graph theory.

The project addresses the absence of rigorous geometric frameworks for understanding optimization, generalization, and sampling in high-dimensional neural network parameter spaces. Research encompasses three interconnected thrusts: developing transport-information loss functionals and metric structures on neural network parameter spaces using information matrices and spectral graph representations; designing 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 and reaction-diffusion equations; and constructing discrete sampling algorithms on graphs using Ricci curvatures derived from Wasserstein metrics to enable efficient Markov kernels with convergence guarantees for large language models and time-reversible diffusion processes. The project advances interdisciplinary research through graduate and postdoctoral training, workshops, outreach activities, and open-source software development.

Performance occurs at Columbia, South Carolina, with a period of performance from September 1, 2026, through August 31, 2029.