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Explore the mathematical foundations of robust machine learning through Hamilton-Jacobi equations and mean-field games in this 42-minute conference talk. Discover how these mathematical frameworks provide a unifying approach for analyzing, designing, and improving the robustness of generative models, with particular focus on flow-based and diffusion-based models including continuous-time normalizing flows, score-based models, and Wasserstein gradient flows. Learn how major classes of generative models naturally emerge from mean-field games under different particle dynamics and cost functions, and understand how the forward-backward PDE structure enables development of faster, more data-efficient algorithms while providing new analytical tools for robustness analysis. Examine uncertainty quantification through a proven Wasserstein uncertainty propagation theorem demonstrating that score-based generative models maintain robustness against multiple error sources including discretization, score estimation, and model form uncertainty. Investigate the parallel framework for Transformer architectures where training is formulated as an optimal control problem with depth as time and loss as terminal cost, leading to the OT-Transformer architecture that incorporates optimal transport regularization for provable improvements in generalization, robustness, and efficiency. Gain insights into how the regularity theory of Hamilton-Jacobi equations provides theoretical guarantees and practical understanding of stability for both generative models and Transformer architectures, presented by Markos Katsoulakis from the University of Massachusetts, Amherst at IPAM's Scientific Machine Learning Workshop.
