Standard approaches to learning generative dynamics from samples often rely on matching marginal distributions at each time step. That can leave the temporal structure underdetermined: different stochastic processes can produce identical marginals. The authors of this paper address the gap by adding path-dependent supervision, so the model is trained not only on where samples are at each moment, but on how they travel between observed points.
The proposed framework is a discrete path-dependent McKean-Vlasov control problem, with a single objective that penalizes mismatches in endpoint laws, intermediate marginal laws, and joint sampled-path laws. After deriving the optimality system as a McKean-Vlasov forward-backward stochastic differential equation, the authors simplify to a reduced FBSDE for quadratic control cost and constant diffusion. They also establish local well-posedness for a Maximum Mean Discrepancy formulation of the path-law term, and implement a sample-based neural solver with causal representations.
Experiments cover synthetic transport, latent face transport, and human motion. The most striking result is a two-state AMASS experiment: with path-law supervision, the model learns both low-to-high and high-to-low transitions, while marginal supervision alone cannot distinguish them. This makes the method a promising step for applications where directionality and transition dynamics matter more than static snapshot accuracy.