Deterministic Decomposition of Stochastic Generative Dynamics

TL;DR

This paper introduces a decomposition of stochastic generative dynamics into deterministic transport and diffusion effects, enabling interpretable and controllable sampling in flow-based models.

Contribution

It presents a natural transport--osmotic decomposition of stochastic dynamics and a new framework, Bridge Matching, for learning and manipulating these components.

Findings

Decomposition separates deterministic and stochastic effects in generative models.

Adjusting osmotic contribution allows controllable sampling.

Recombination of components improves interpretability in generative processes.

Abstract

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 \(b_t\) of a stochastic generative process admits a natural transport--osmotic decomposition that separates deterministic transport from stochastic, diffusion-induced effects: \(b_t = u_t + d_t\), where \(u_t\) governs marginal probability transport and \(d_t\) 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 \(b_t = u_t + \lambda_d d_t\), showing that the proposed decomposition enables interpretable and controllable sampling by adjusting the osmotic contribution in probability transport.