Collaborative Research: MFAI: Stochastic Analysis of Reinforcement Learning and Generative Models

Artificial intelligence (AI) systems increasingly make decisions, generate data, and support scientific discovery in settings where uncertainty, time, and safety constraints matter. This research project will build mathematical foundations for two central branches of artificial intelligence: reinforcement learning, where an agent learns to make decisions from experience, and generative AI, where models create new data that resemble real data. Current systems can perform impressively, but their behavior is often hard to understand or guarantee when environments evolve continuously over time, when decisions must be made sequentially, or when generated data must satisfy physical, economic, or scientific constraints. This project will develop mathematical principles for more reliable, efficient, and controllable AI. The work will advance fundamental science and support the national interest by strengthening the foundations of artificial intelligence, training students and early-career researchers in mathematics and machine learning, and enabling future tools for domains such as healthcare, transportation, energy systems, finance, and scientific computing. The project is aligned with national priorities in artificial intelligence and translational science by turning mathematical advances into principles and algorithms that can support reliable AI-enabled scientific and engineering applications. The project will use stochastic analysis, stochastic control, probability theory, and optimization to study three tightly connected directions. First, it will develop theory and algorithms for continuous-time reinforcement learning in continuous state-action spaces, including adaptive exploration and exploitation methods, policy optimization, and transfer learning for controlled diffusion processes. Second, it will establish stochastic foundations for diffusion-based generative models in structured settings, including guidance under hard constraints, sampling for inverse problems without retraining large models, time-series generation, and stability analysis using adapted probability metrics. Third, it will develop mathematical frameworks for the interaction between reinforcement learning and generative models, including continuous-time reinforcement-learning methods for aligning diffusion models with reward functions and diffusion-based policy representations for high-dimensional decision problems. The project will produce rigorous guarantees, efficient algorithms, and new connections among stochastic differential equations, Hamilton-Jacobi-Bellman equations, Doob h-transforms, policy-gradient methods, Wasserstein-type stability, and generative modeling. 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.