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

arXiv: https://arxiv.org/abs/2605.08928 · HTML: https://mlsanity.com/paper/2605.08928/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 36 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 22 of 22)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
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- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [AMASS: Archive of Motion Capture as Surface Shapes](https://mlsanity.com/paper/1904.03278/) (2019)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
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- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Differential Properties of Sinkhorn Approximation for Learning with Wasserstein Distance](https://mlsanity.com/paper/1805.11897/) (2018)
- [Light and Optimal Schr"odinger Bridge Matching](https://mlsanity.com/paper/2402.03207/) (2024)
- [A mean-field games laboratory for generative modeling](https://mlsanity.com/paper/2304.13534/) (2023)
- [Numerical resolution of McKean-Vlasov FBSDEs using neural networks](https://mlsanity.com/paper/1909.12678/) (2019)
- [Dynamic programming for optimal control of stochastic McKean-Vlasov dynamics](https://mlsanity.com/paper/1604.04057/) (2016)
- [Mean-field neural networks-based algorithms for McKean-Vlasov control problems *](https://mlsanity.com/paper/2212.11518/) (2022)
- [LightSBB-M: Bridging Schr"odinger and Bass for Generative Diffusion Modeling](https://mlsanity.com/paper/2601.19312/) (2026)
- [Schr"odinger bridge for generative AI: Soft-constrained formulation and convergence analysis](https://mlsanity.com/paper/2510.11829/) (2025)
- [Schr"odinger bridge with transport relaxation](https://mlsanity.com/paper/2602.08118/) (2026)

## Abstract (arXiv)

We propose a generative framework for learning stochastic dynamics from endpoint and intermediate distributional observations. The method formulates generation as a McKean-Vlasov control problem in which terminal and time-marginal laws are enforced through soft energy constraints. The associated optimality system is a forward-backward stochastic differential equation (FBSDE) whose backward component receives a continuous drift induced by the marginal law penalties. This provides a principled alternative to hard interpolation or optimal transport maps between observed distributions: the model learns a stochastic path law whose dynamics remain globally coupled through the mean-field objective. We derive the reduced FBSDE system for quadratic control cost and constant diffusion, connecting terminal and marginal law flat derivatives to score-like training signals. The resulting neural solver is evaluated on low-dimensional distributional benchmarks, where it recovers smooth stochastic paths matching prescribed marginal laws. In a higher-dimensional ALAE latent space, endpoint supervision is used as a qualitative stress test for transporting non-smiling faces toward smiling ones in a pretrained representation. We then use articulated human motion as a structured high-dimensional case study on a curated AMASS low-to-high position dataset, using SMPL-H pose sequences and reduced pose representations. The experiments show that soft marginal law constraints can produce coherent stochastic trajectories whose intermediate distributions follow the observed evolution of human motion. The code is available at https://github.com/murex/deep-mkv-gen/tree/main.