# Deterministic Decomposition of Stochastic Generative Dynamics

arXiv: https://arxiv.org/abs/2605.08794 · HTML: https://mlsanity.com/paper/2605.08794/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 47 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 30 of 34)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [AudioLDM: Text-to-Audio Generation with Latent Diffusion Models](https://mlsanity.com/paper/2301.12503/) (2023)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets](https://mlsanity.com/paper/1707.08819/) (2017)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Stable Audio Open](https://mlsanity.com/paper/2407.14358/) (2024)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [The probability flow ODE is provably fast](https://mlsanity.com/paper/2305.11798/) (2023)
- [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024)
- [Minimizing Trajectory Curvature of ODE-based Generative Models](https://mlsanity.com/paper/2301.12003/) (2023)
- [Simulation-free Schr"odinger bridges via score and flow matching](https://mlsanity.com/paper/2307.03672/) (2023)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [Bidirectional Normalizing Flow: From Data to Noise and Back](https://mlsanity.com/paper/2512.10953/) (2025)
- [Efficient Generative Modeling beyond Memoryless Diffusion via Adjoint Schr"odinger Bridge Matching](https://mlsanity.com/paper/2602.15396/) (2026)

## Abstract (arXiv)

Modern generative models can be understood as probability transport from a simple base distribution to a target data distribution. Deterministic transport models offer tractable velocity-field parameterizations, whereas stochastic generative models capture richer density evolution through drift and diffusion. Yet when stochastic dynamics are described through deterministic velocity fields, the effects of drift and diffusion are often compressed into a single effective field, obscuring the distinct roles of deterministic evolution and stochastic fluctuation. In this work, we show that the deterministic field (bt) of a stochastic generative process admits a natural transport--osmotic decomposition that separates deterministic transport from stochastic, diffusion-induced effects: (bt = ut + dt), where (ut) governs marginal probability transport and (dt) captures an osmotic effect induced by diffusion and determined by the marginal score. Based on this decomposition, we propose Bridge Matching, a flow-based framework for learning decomposed generative dynamics through both marginal and conditional formulations. In generative modeling experiments, we recombine the learned components as (bt = ut + λd dt), showing that the proposed decomposition enables interpretable and controllable sampling by adjusting the osmotic contribution in probability transport.