Title

Hamilton-Jacobi Equations, Mean-Field Games, and Uncertainty Quantification for Robust Machine Learning

Abstract

Hamilton-Jacobi (HJ) equations and Mean-Field Games (MFGs) provide a natural mathematical language that unites ideas from stochastic control, optimal transport, and information theory for analyzing, designing, and improving the robustness of many modern machine learning models. We show how fundamental classes of generative models, including continuous-time normalizing flows and score-based diffusion models, emerge intrinsically from MFG formulations under different particle dynamics, cost functionals, information-theoretic divergences, and probability metrics, with analogies and connections to Wasserstein gradient-flows. The forward-backward PDE structure of MFGs offers both analytical insights and informs the development of faster, data-efficient and robust algorithms. In particular, the regularity theory of HJ equations, combined with model-uncertainty quantification, provides provable performance and robustness guarantees for generative models and complex neural architectures such as transformers. Our theoretical analysis is complemented by extensive numerical validations, with diverse examples from applied mathematics and widely used ML benchmarks.