Learning Generative Dynamics with Soft Path-Law Constraints: A McKean-Vlasov FBSDE Approach
Abstract
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.
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BibTeX
@misc{boustany2026learning,
title = {{Learning Generative Dynamics with Soft Path-Law Constraints: A McKean-Vlasov FBSDE Approach}},
author = {Samer El Boustany and Samy Mekkaoui and Yadh Hafsi and Alexandre Alouadi and Huyên Pham},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.08928},
eprint = {2605.08928},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.08928},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Learning Generative Dynamics with Soft Path-Law Constraints: A McKean-Vlasov FBSDE Approach
AU - Samer El Boustany
AU - Samy Mekkaoui
AU - Yadh Hafsi
AU - Alexandre Alouadi
AU - Huyên Pham
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2605.08928
UR - https://arxiv.org/abs/2605.08928
AB - 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.
ER -