Learning Generative Dynamics with Soft Path-Law Constraints: A McKean-Vlasov FBSDE Approach

We introduce a unified framework for learning stochastic dynamics from distributional observations and discretely observed trajectories. The key contribution is to combine soft law constraints, which regularize fitting to empirical distributions, with path-dependent supervision, which captures temporal structure that marginal distributions alone cannot identify. We formulate the problem as a discrete path-dependent McKean–Vlasov control problem, penalizing endpoint, intermediate marginal, and joint sampled-path laws within a single objective. Its optimality system takes the form of a McKean–Vlasov forward-backward stochastic differential equation (FBSDE), in which the marginal terms act continuously in time while the path-law term instead induces conditional adjoint jumps at the observation dates. We derive the reduced FBSDE for quadratic control cost and constant diffusion, establish local well-posedness for a Maximum Mean Discrepancy (MMD) path-law formulation, and develop a sample-based neural solver with causal representations. Experiments on synthetic transport, latent face transport, and human motion validate the framework. In particular, a two-state AMASS experiment under path-law supervision captures both low-to-high and high-to-low transitions, which marginal supervision alone cannot distinguish.