Mathematical Foundations of Learning Complex Stochastic Systems

Many systems in science, engineering, biology, and finance evolve over time in the presence of randomness. These systems often involve interacting components operating on different time scales, while important aspects of their behavior may not be directly observable from available data. This project develops mathematical foundations for machine learning for such stochastic systems, enabling efficient model reduction, providing a principled basis for assessing when learning algorithms can be trusted, improving the reliability of data-driven models, and supporting decision-making under uncertainty. The outcomes are expected to benefit a broad range of scientific disciplines that rely on stochastic modeling and data-driven decision making. The project will also provide research opportunities to train students in theoretical, computational, and data-science skills for careers in an increasingly data-driven world. The project investigates some fundamental problems at the interface of statistical learning theory and stochastic analysis. Part I develops likelihood-based methods for learning multiscale stochastic differential equations (SDEs) from observations of the slow component alone using conditional normalizing flows and Bayesian nonparametric methods based on normalized Lévy process priors. Theoretical analysis of the resulting estimators will provide principled estimation and uncertainty quantification of averaged drift functions for high-dimensional multiscale SDEs. Part II develops a statistical framework for offline reinforcement learning and inverse reinforcement learning for controlled SDEs with unknown dynamics. Large deviation theory will be used to derive exponentially accurate confidence bounds for long-run cost functionals, and optimizing these bounds over the policy space will yield optimistic and pessimistic policies with rigorous performance guarantees. The project will advance the mathematical foundations of learning and decision making for complex stochastic systems. 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.