Deterministic Decomposition of Stochastic Generative Dynamics

Naoya Takeishi; Xingyu Song; Yuan Mei

arxiv: 2605.08794 · v2 · pith:HAF4YSJ2new · submitted 2026-05-09 · 💻 cs.LG · cs.AI

Deterministic Decomposition of Stochastic Generative Dynamics

Xingyu Song , Yuan Mei , Naoya Takeishi This is my paper

Pith reviewed 2026-05-20 23:14 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords generative modelsstochastic differential equationsvelocity fieldsdecompositionosmotic effectsbridge matchingscore functionsprobability transport

0 comments

The pith

Stochastic generative models admit a decomposition of their deterministic velocity field into marginal transport plus an osmotic term set by the score.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the single effective field b_t used to describe stochastic generative dynamics splits naturally into two parts: u_t that carries the marginal probability mass from base to data distribution, and d_t that encodes the spreading effect caused by diffusion and fixed by the marginal score. This separation is derived directly from the underlying stochastic differential equation without extra assumptions. If the split holds, then scaling the osmotic part independently during sampling should let users dial the amount of stochastic influence while keeping the transport fixed. The authors introduce Bridge Matching to learn the two components jointly from both unconditional and conditional trajectory data. Experiments recombine them as b_t equals u_t plus lambda times d_t to demonstrate controllable generation.

Core claim

The deterministic field b_t of a stochastic generative process admits a natural transport-osmotic decomposition 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.

What carries the argument

The transport-osmotic decomposition b_t = u_t + d_t, separating marginal transport velocity from score-determined osmotic effect.

If this is right

Recombining the learned components with a tunable coefficient on the osmotic term produces controllable sampling that varies stochastic contribution without retraining.
Bridge Matching learns the decomposed dynamics through both marginal and conditional path formulations.
The split makes the distinct roles of deterministic drift and diffusion-induced fluctuation explicit in the velocity field.
The decomposition applies to existing stochastic generative models by extracting the osmotic term from the marginal score.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same split could be applied to conditional generation tasks by replacing the marginal score with a conditional one.
Hybrid models might emerge that use only the transport component for fast deterministic sampling while adding controlled osmotic strength for diversity.
The decomposition suggests a route to regularize generative training by penalizing the osmotic component separately from the transport component.

Load-bearing premise

The stochastic generative process can be represented by a deterministic velocity field whose decomposition into marginal transport and score-determined osmotic terms is well-defined and learnable.

What would settle it

Derive the explicit forms of u_t and d_t from a known SDE such as Brownian motion, recompute their sum, and check whether it recovers the original effective field b_t for non-trivial marginal densities.

Figures

Figures reproduced from arXiv: 2605.08794 by Naoya Takeishi, Xingyu Song, Yuan Mei.

**Figure 2.** Figure 2: MMD2 (lower is better) between generated and target samples on 2D transport tasks, for different source–target pairs. Horizontal axis is the recombination weight λd. As in Conditional Flow Matching (Lipman et al., 2023; Tong et al., 2023), training on these conditional targets yields marginal vector fields in expectation, with the conditional score recovering the marginal score through E[∇x log pt(x | z) |… view at source ↗

**Figure 3.** Figure 3: Marginal evolution comparison on a 2D Gaussian-to-checkerboard task. This visualization [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 7.** Figure 7: ODE sampling evolution on the Gaussian-to-moons transport task. We compare CFM [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗

**Figure 8.** Figure 8: ODE sampling evolution on the Gaussian-to-mixture transport task. We compare CFM [PITH_FULL_IMAGE:figures/full_fig_p033_8.png] view at source ↗

**Figure 9.** Figure 9: ODE sampling evolution on the Gaussian-to-checkerboard transport task. We compare [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗

**Figure 10.** Figure 10: ODE sampling evolution on the moons-to-checkerboard transport task. We compare [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗

**Figure 11.** Figure 11: ODE sampling evolution on the moons-to-mixture transport task. We compare CFM [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗

**Figure 12.** Figure 12: ODE sampling evolution on the checkerboard-to-mixture transport task. We compare [PITH_FULL_IMAGE:figures/full_fig_p034_12.png] view at source ↗

**Figure 13.** Figure 13: Additional 2D ablation on the osmotic recombination weight [PITH_FULL_IMAGE:figures/full_fig_p036_13.png] view at source ↗

**Figure 14.** Figure 14: Additional generated samples on CIFAR-10 from the checkpoint at epoch [PITH_FULL_IMAGE:figures/full_fig_p036_14.png] view at source ↗

**Figure 15.** Figure 15: Additional generated samples on ImageNet-32 from the checkpoint at epoch [PITH_FULL_IMAGE:figures/full_fig_p037_15.png] view at source ↗

**Figure 16.** Figure 16: Additional generated samples on ImageNet-64 from the checkpoint at epoch [PITH_FULL_IMAGE:figures/full_fig_p038_16.png] view at source ↗

**Figure 17.** Figure 17: Additional sampling-time controllability results obtained by varying the osmotic recom [PITH_FULL_IMAGE:figures/full_fig_p039_17.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper splits the effective velocity field in stochastic generative models into a marginal transport term and a score-driven osmotic term, then uses that split for a Bridge Matching training scheme that lets you dial the diffusion contribution up or down at sampling time.

read the letter

The core move is to write the deterministic field b_t that appears in stochastic generative SDEs as b_t = u_t + d_t, where u_t handles the usual probability transport and d_t is an osmotic correction fixed by the marginal score. They then train both pieces with a flow-matching style objective they call Bridge Matching and show that recombining them as b_t = u_t + λ_d d_t gives a knob for how much stochasticity you keep during sampling. That control is the practical payoff and the experiments suggest it works without retraining the whole model from scratch. The decomposition itself is a direct consequence of the Fokker-Planck equation once you separate the drift and diffusion contributions, so the algebra is not especially surprising once you see it, but the framing as an explicit, learnable split is new enough to be useful inside the flow and diffusion literature. The experiments are the strongest part: they demonstrate that varying λ_d produces visibly different sample diversity and quality on standard image benchmarks, which is the kind of controllable handle people actually want. The main soft spot is that the abstract and the stress-test note both leave the regularity conditions implicit. For the osmotic term to be well-defined you need the marginal density to be twice differentiable and the diffusion coefficient to satisfy standard ellipticity; if those fail the split is no longer unique or even well-posed. The paper presumably states these somewhere, but without seeing the derivation steps or any counter-example checks it is hard to judge how fragile the claim is in practice. The citation pattern looks standard for the area and does not appear to hide prior results. This is aimed at researchers already working on score-based or flow-matching models who want finer control over the stochastic component without switching frameworks. A reader who cares about the internal structure of generative SDEs will find the decomposition worth reading; the rest of the community will mainly care about the sampling knob. It is solid enough on the experimental side and the idea is clean enough that it deserves a serious referee rather than a desk reject, though the review should focus on whether the regularity assumptions are stated clearly and whether the decomposition remains stable when the score is only approximately learned.

Referee Report

2 major / 1 minor

Summary. The paper claims that any deterministic velocity field b_t arising from a stochastic generative process admits a transport-osmotic decomposition b_t = u_t + d_t, where u_t governs marginal probability transport and d_t is an osmotic term induced by diffusion and determined by the marginal score. It introduces the Bridge Matching framework to learn these components via marginal and conditional formulations, and demonstrates that recombining them as b_t = u_t + λ_d d_t enables interpretable, controllable sampling in generative modeling experiments.

Significance. If the decomposition is rigorously justified and the learning framework is shown to be stable, the work could provide a useful lens for separating deterministic transport from diffusion effects in score-based and flow-based generative models, potentially aiding interpretability and control without requiring new architectures.

major comments (2)

[Abstract] Abstract and introduction: the central identity b_t = u_t + d_t is stated at a high level but no derivation from the Fokker-Planck equation, no explicit definition of the osmotic term d_t in terms of the score, and no statement of the required regularity conditions (C^2 marginal density, elliptic diffusion coefficient, unique strong solution) are supplied. Without these, it is impossible to verify whether the claimed separation is operational or merely formal, directly affecting the soundness of the Bridge Matching proposal.
[Abstract] The abstract asserts that d_t is 'determined by the marginal score,' yet the reader's note and absence of any error analysis or verification steps leave open whether this term is independent of fitted quantities or reduces to a re-expression of the learned score; this circularity risk must be addressed with a concrete derivation or counter-example check.

minor comments (1)

Notation for the velocity fields u_t, d_t, and b_t should be introduced with explicit equations early in the manuscript rather than relying on the abstract alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed comments, which help strengthen the mathematical presentation of our work. We address each major comment below and outline the revisions we will make to the manuscript.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the central identity b_t = u_t + d_t is stated at a high level but no derivation from the Fokker-Planck equation, no explicit definition of the osmotic term d_t in terms of the score, and no statement of the required regularity conditions (C^2 marginal density, elliptic diffusion coefficient, unique strong solution) are supplied. Without these, it is impossible to verify whether the claimed separation is operational or merely formal, directly affecting the soundness of the Bridge Matching proposal.

Authors: We agree that the abstract and introduction present the decomposition at a high level for brevity. The full derivation from the Fokker-Planck equation is provided in Section 2 of the manuscript, where we start from the stochastic differential equation dx = b_t(x) dt + sigma_t dW and derive the decomposition by separating the probability current into transport and osmotic parts. Specifically, the osmotic term is defined as d_t = (sigma_t^2 / 2) nabla_x log p_t(x), which arises directly from the diffusion term in the Fokker-Planck operator. To make this more prominent, in the revised manuscript we will include a short derivation sketch in the introduction and explicitly state the regularity conditions: the marginal density p_t is twice continuously differentiable, the diffusion coefficient sigma_t is elliptic (bounded away from zero), and the SDE has a unique strong solution under standard Lipschitz assumptions. These conditions ensure the decomposition is well-defined and operational for the Bridge Matching framework. revision: yes
Referee: [Abstract] The abstract asserts that d_t is 'determined by the marginal score,' yet the reader's note and absence of any error analysis or verification steps leave open whether this term is independent of fitted quantities or reduces to a re-expression of the learned score; this circularity risk must be addressed with a concrete derivation or counter-example check.

Authors: We appreciate this concern regarding potential circularity. The term d_t depends only on the marginal density p_t through its score nabla log p_t, which is a property of the marginal distribution at each time t and does not depend on the particular realization of the stochastic process or the fitted velocity field b_t. In the Bridge Matching framework, the marginal score is estimated independently via denoising score matching on samples from the marginal distribution, while u_t is learned through conditional bridge matching that enforces the transport component. To explicitly address this, we will add a concrete derivation in Section 3 showing the independence, along with a verification on a synthetic example (e.g., a Gaussian process where the exact score and decomposition are known analytically). We will also include a brief error analysis discussing the propagation of score estimation errors into the osmotic term. This demonstrates that d_t is not a re-expression of the learned score but is computed from the marginal. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain.

full rationale

The claimed transport-osmotic decomposition b_t = u_t + d_t is presented as a mathematical identity derived from the Fokker-Planck equation applied to the underlying SDE, with d_t expressed via the marginal score. This is an analytic separation of terms rather than a self-definition, fitted input renamed as prediction, or reduction to self-citation. No load-bearing step in the abstract or described framework reduces the result to its inputs by construction; the decomposition supplies an independent interpretive lens, while learning and recombination steps operate downstream. The paper remains self-contained against external benchmarks for the core identity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The decomposition implicitly relies on the existence of a well-defined marginal score and the validity of the underlying stochastic differential equation representation.

pith-pipeline@v0.9.0 · 5729 in / 1197 out tokens · 33775 ms · 2026-05-20T23:14:24.479588+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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 (Definition 1, Prop. 1)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

forward and backward drifts coupled through marginal score ∇logπt (Eq. 10)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 5 internal anchors

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A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets

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Scaling Rectified Flow Transformers for High-Resolution Image Synthesis

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The dashed horizontal lines indicate the corresponding CFM baselines, while solid curves show the BM variants

For all BM variants, we fix λu = 1.0 and vary λd ∈ {0.0,0.5,1.0,1.5} . The dashed horizontal lines indicate the corresponding CFM baselines, while solid curves show the BM variants. G.3 Additional image generation samples We provide additional generated samples from the image-generation experiments. The samples are generated from the final evaluated check...

work page 2024
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and the PyTorch Flow Matching implementation of Shihara (2023) as implementation references. These resources provide standard utilities for Flow Matching training, sampling, and evaluation, while our supplementary code adds the proposed Bridge Matching objectives, the transport–osmotic decomposition, and the sampling-time recombination of learned fields. ...

work page 2023

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A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets

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work page internal anchor Pith review Pith/arXiv arXiv

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Scaling Rectified Flow Transformers for High-Resolution Image Synthesis

Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, Dustin Podell, Tim Dockhorn, Zion En- glish, Kyle Lacey, Alex Goodwin, Yannik Marek, and Robin Rombach. Scaling rectified flow transformers for high-resolution image synthesis.arXiv preprint arXiv:2403.03206,

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Zach Evans, Julian D. Parker, CJ Carr, Zack Zukowski, Josiah Taylor, and Jordi Pons. Stable audio open.arXiv preprint arXiv:2407.14358,

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Han Gao, Sebastian Kaltenbach, and Petros Koumoutsakos

Accessed: 2026-05-06. Han Gao, Sebastian Kaltenbach, and Petros Koumoutsakos. Generative learning for forecasting the dynamics of high-dimensional complex systems.Nature Communications, 15(1):8904,

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Mean Flows for One-step Generative Modeling

doi: 10.1038/s41467-024-53165-w. Zhengyang Geng, Mingyang Deng, Xingjian Bai, J. Zico Kolter, and Kaiming He. Mean flows for one-step generative modeling.arXiv preprint arXiv:2505.13447,

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1038/s41467-024-53165-w

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11 Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine

doi: 10.1016/j.jmva.2025.105486. 11 Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion- based generative models. InAdvances in Neural Information Processing Systems,

work page doi:10.1016/j.jmva.2025.105486 2025

[8] [8]

Yaron Lipman, Marton Havasi, Peter Holderrieth, Neta Shaul, Matt Le, Brian Karrer, Ricky T. Q. Chen, David Lopez-Paz, Heli Ben-Hamu, and Itai Gat. Flow matching guide and code.arXiv preprint arXiv:2412.06264,

work page internal anchor Pith review Pith/arXiv arXiv

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Plumbley

Haohe Liu, Zehua Chen, Yi Yuan, Xinhao Mei, Xubo Liu, Danilo Mandic, Wenwu Wang, and Mark D. Plumbley. AudioLDM: Text-to-audio generation with latent diffusion models. InInternational Conference on Machine Learning, pages 21450–21474, 2023a. Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Rectified flow. InInternational Conference on L...

work page arXiv

[10] [10]

Jeongwoo Shin, Jinhwan Sul, Joonseok Lee, Jaewong Choi, and Jaemoo Choi

Accessed: 2026-05-06. Jeongwoo Shin, Jinhwan Sul, Joonseok Lee, Jaewong Choi, and Jaemoo Choi. Efficient generative modeling beyond memoryless diffusion via adjoint schrödinger bridge matching.arXiv preprint arXiv:2602.15396,

work page arXiv 2026

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Incorporating pre-trained diffusion models in solving the schrödinger bridge problem.arXiv preprint arXiv:2508.18095,

Zhicong Tang, Tiankai Hang, Shuyang Gu, Dong Chen, and Baining Guo. Incorporating pre-trained diffusion models in solving the schrödinger bridge problem.arXiv preprint arXiv:2508.18095,

work page arXiv

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Improving and generalizing flow-based generative models with minibatch optimal transport

Alexander Tong et al. Improving and generalizing flow-based generative models with minibatch optimal transport.arXiv preprint arXiv:2302.00482,

work page internal anchor Pith review Pith/arXiv arXiv

[13] [13]

The dashed horizontal lines indicate the corresponding CFM baselines, while solid curves show the BM variants

For all BM variants, we fix λu = 1.0 and vary λd ∈ {0.0,0.5,1.0,1.5} . The dashed horizontal lines indicate the corresponding CFM baselines, while solid curves show the BM variants. G.3 Additional image generation samples We provide additional generated samples from the image-generation experiments. The samples are generated from the final evaluated check...

work page 2024

[14] [14]

and the PyTorch Flow Matching implementation of Shihara (2023) as implementation references. These resources provide standard utilities for Flow Matching training, sampling, and evaluation, while our supplementary code adds the proposed Bridge Matching objectives, the transport–osmotic decomposition, and the sampling-time recombination of learned fields. ...

work page 2023