AI and the World's Hard Problems · Mathematics · arXiv · 2026-10-21
Learning Generative Dynamics with Soft Path-Law Constraints: A McKean-Vlasov FBSDE Approach
The authors introduce a unified framework for learning stochastic dynamics from distributional observations and discretely observed trajectories. It combines soft law constraints, which regularize fitting to empirical distributions, with path-dependent supervision, formulating the problem as a discrete path-dependent McKean-Vlasov control problem whose optimality system is a McKean-Vlasov forward-backward SDE. Experiments on synthetic transport, latent face transport, and human motion validate the framework, and a two-state AMASS experiment under path-law supervision captures both low-to-high and high-to-low transitions that marginal supervision alone cannot distinguish.