From Proximal Divergences to Hamilton-Jacobi Regularity: Mathematical Principles for Robust AI Architectures and Inference

Artificial intelligence systems that generate new images, text, molecules, and scientific data have advanced rapidly, and they are now used across science, engineering, and beyond, from medicine, chemistry, and the physical sciences to finance, manufacturing, and other areas of business and industry. Current AI methods are still only partly understood and can behave unpredictably, often making stability and robustness crucial for their safe and effective use. This project addresses these challenges by developing a mathematical understanding that aims to make such systems more reliable and support continued progress in science, economic prosperity, and technologies that society increasingly depends on. The investigators will establish new mathematical principles to explain why successful AI algorithms work, identify the conditions under which they can fail, and guide their design to ensure dependability. The project also supports education and workforce development by training graduate and postdoctoral researchers at the interface of mathematics and artificial intelligence and by creating new undergraduate and graduate courses on generative AI, reinforcement learning, and the mathematics of machine learning. The mathematical foundations of this project are drawn from optimal transport, optimal control, and information theory, directly informing the design and analysis of modern AI algorithms. The central observation is that many modern generative algorithms, including flow-based models, diffusion models, and the transformer architectures behind large language models, can be understood as regularized transformations of probability distributions whose continuous-time limits are governed by equations from optimal control theory, known as Hamilton-Jacobi-Bellman equations. Building on this unifying viewpoint, the investigators will pursue three connected directions: new algorithms for transforming one probability distribution into another with provable stability guarantees; a control-theoretic framework that explains and guides the design of transformer architectures and clarifies the trade-off between their controllability and robustness; and risk-aware methods, drawing on tools from finance and distributionally robust optimization, that enable generative models to capture extreme events and heavy-tailed behavior. By focusing on the mathematical structure of these AI methods, the project will provide rigorous stability and robustness guarantees, explain and improve the design of leading AI architectures, and establish a principled foundation for the next generation of dependable artificial intelligence. 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.