Mean Velocity Matching: 重新思考扩散模型中的生成动力学

# Mean Velocity Matching: Rethinking Generative Dynamics in Diffusion Models
Source: [https://arxiv.org/html/2609.25444](https://arxiv.org/html/2609.25444)
Changjie Cao††thanks:Corresponding author\.Affiliation:Chengdu University of Technology, Chengdu, ChinaZhihua ZhangAffiliation:Peking University, Beijing, ChinaBingli LiuAffiliation:Chengdu University of Technology, Chengdu, ChinaZongjie CaoAffiliation:University of Electronic Science and Technology of China, Chengdu, ChinaZongyong CuiAffiliation:University of Electronic Science and Technology of China, Chengdu, ChinaYing YangAffiliation:Chengdu University of Technology, Chengdu, China

###### Abstract

This work studies prediction parameterization for stochastic generative dynamics in diffusion models\. Existing velocity\-based generative models provide the simplicity of learning a single transport field, but their standard formulation is deterministic, whereas stochastic extensions generally require additional score information or an intermediate velocity\-to\-score reconstruction\. To retain single\-field prediction while directly supporting stochastic reverse dynamics, this paper introduces Mean Velocity Matching \(MVM\)\. MVM constructs a Gaussian perturbation process for which the conditional expectation of a restoration\-oriented velocity,\(x0−xt\)/t\(x\_\{0\}\-x\_\{t\}\)/t, directly forms the reverse\-SDE drift\. Consequently, a single learned field is sufficient to parameterize the stochastic reverse process without separately estimating or reconstructing the score\. Because direct regression of this velocity becomes unbounded neart=0t=0, MVM further introduces at\\sqrt\{t\}\-scaled parameterization that preserves the reverse dynamics while yielding a bounded training target\. The same learned field also induces a deterministic probability\-flow ODE, enabling stochastic and deterministic sampling to be studied within a unified formulation\. Experiments with Transformer\-based generative models achieve an FID of1\.931\.93at 65 NFE on ImageNet32×3232\\times 32and2\.072\.07at 90 NFE on ImageNet256×256256\\times 256\. Controlled SDE–ODE comparisons further show that the ODE performs better under very low NFE, whereas the stochastic reverse process achieves lower FID when sufficient function evaluations are available\. These results demonstrate that MVM provides a direct single\-field parameterization of stochastic reverse dynamics while maintaining competitive generation quality\.

## 1INTRODUCTION

Diffusion models have achieved strong performance in image synthesis\([Dhariwal and Nichol, 2021](https://arxiv.org/html/2609.25444#bib.bib1);[Ho et al\., 2020](https://arxiv.org/html/2609.25444#bib.bib2)\), video generation\([Ho et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib3)\), three\-dimensional content generation\([Poole et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib4)\), and large\-scale text\-to\-image synthesis\([Rombach et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib16);[Saharia et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib17);[Nichol et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib20)\)\. They combine a gradual perturbation process with a learned reverse process that transports tractable noise back to the data distribution, and this view extends naturally from discrete\-time chains to continuous\-time stochastic differential equations \(SDEs\)\([Song et al\., 2021b](https://arxiv.org/html/2609.25444#bib.bib5)\)\. Transformer backbones such as DiT\([Peebles and Xie, 2023](https://arxiv.org/html/2609.25444#bib.bib33)\)further show that diffusion models scale effectively with model capacity and computation\.

![[Uncaptioned image]](https://arxiv.org/html/2609.25444v1/jgt.png)

Figure 1: Overview of Mean Velocity Matching\.The forward process continuously transforms data into Gaussian noise, whereas the learned reverse\-time SDE transports noise back to the data distribution through mean\-velocity prediction\.

A central design choice in diffusion models is the variable predicted by the neural network\. DDPM\([Ho et al\., 2020](https://arxiv.org/html/2609.25444#bib.bib2)\)adopts perturbation\-noise prediction and reported that directx0x\_\{0\}\-prediction produced worse sample quality in its early experiments\. Later work showed, however, that the preferred parameterization can depend strongly on the sampling regime\. In progressive distillation, for example, the impliedx0x\_\{0\}estimate fromϵ\\epsilon\-prediction becomes ill\-conditioned as the signal\-to\-noise ratio approaches zero, motivating direct data prediction and diffusionvv\-prediction\([Salimans and Ho, 2022](https://arxiv.org/html/2609.25444#bib.bib22)\)\. The latter combines clean\-data and noise information in a single target,

vt=αt​ϵ−σt​x0\.v\_\{t\}=\\alpha\_\{t\}\\epsilon\-\\sigma\_\{t\}x\_\{0\}\.\(1\)These observations suggest that a useful prediction target may benefit from representing data\- and noise\-related information jointly rather than committing exclusively to either endpoint\.

Flow\-based generative models provide a related perspective\. Flow Matching\([Lipman et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib7)\)and Rectified Flow\([Liu et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib9)\)learn a single velocity field that transports probability between endpoint distributions\. Under commonly used linear interpolations between data and Gaussian noise, the conditional velocity is determined jointly by the two endpoints\. Once learned, the marginal velocity directly defines a deterministic ordinary differential equation,

d​x=vθ​\(x,t\)​d​t,dx=v\_\{\\theta\}\(x,t\)\\,dt,\(2\)so generation requires only one learned field\.

Deterministic transport, however, is not the only possible reverse dynamics\. Stochastic Interpolants\([Albergo et al\., 2025](https://arxiv.org/html/2609.25444#bib.bib8)\)show that velocity fields, scores, deterministic flows, and stochastic diffusions can be related within a common framework\. Stochastic samplers can also be constructed from pretrained deterministic flow models\. In a general construction, the SDE drift combines the learned flow field with score information; for Gaussian flow models, the score can be recovered analytically from the learned velocity, avoiding a separately trained score network\([Singh and Fischer, 2024](https://arxiv.org/html/2609.25444#bib.bib40)\)\. Nevertheless, the stochastic drift in such constructions is obtained by augmenting the transport velocity with, or reconstructing it through, score information\.

This paper investigates a different possibility:*can a stochastic reverse process retain the simplicity of single\-field velocity prediction while avoiding an intermediate velocity\-to\-score reconstruction?*To address this question, the present study introduces*Mean Velocity Matching*\(MVM\), a continuous\-time generative framework in which one learned field directly parameterizes the reverse\-SDE drift\.

MVM constructs the Gaussian perturbation path

xt=\(1−tT\)​x0\+tT​ϵ,ϵ∼𝒩⁡\(0,I\),x\_\{t\}=\\left\(1\-\\frac\{t\}\{T\}\\right\)x\_\{0\}\+\\sqrt\{\\frac\{t\}\{T\}\}\\,\\epsilon,\\qquad\\epsilon\\sim\\mathcal\{N\}\(0,I\),\(3\)which continuously maps the data distribution to standard Gaussian noise at the terminal timeTT\. For a paired forward sample\(x0,xt\)\(x\_\{0\},x\_\{t\}\), define

v⁡\(xt,t,x0\)=x0−xtt\.v\(x\_\{t\},t;x\_\{0\}\)=\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}\.\(4\)By deriving the reverse probability evolution of this process, the present study shows that the marginal reverse drift is

v∗\(xt,t\)=𝔼\[x0−xtt\|xt\]\.v^\{\*\}\(x\_\{t\},t\)=\\mathbb\{E\}\\left\[\\left\.\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}\\right\|x\_\{t\}\\right\]\.\(5\)Consequently, the reverse stochastic dynamics take the simple form

d​x=v∗​\(xt,t\)​d​τ\+1T​d​w\.dx=v^\{\*\}\(x\_\{t\},t\)\\,d\\tau\+\\sqrt\{\\frac\{1\}\{T\}\}\\,dw\.\(6\)The key property is that a single predicted field determines the reverse\-SDE drift directly, without requiring a separately learned score or an intermediate velocity\-to\-score conversion during generation\. The conditional drift is tractable to learn because squared\-error regression on the sample\-wise target\(x0−xt\)/t\(x\_\{0\}\-x\_\{t\}\)/thas the same population minimizer as regression to its conditional expectation\. Direct regression becomes ill\-conditioned neart=0t=0, where the target grows as𝒪\(t−1/2\)\\mathcal\{O\}\(t^\{\-1/2\}\)\. MVM therefore uses the scaled parameterization

u⁡\(xt,t\)=t​v​\(xt,t\)=x0−xtt=tT​x0−ϵT,u\(x\_\{t\},t\)=\\sqrt\{t\}\\,v\(x\_\{t\},t\)=\\frac\{x\_\{0\}\-x\_\{t\}\}\{\\sqrt\{t\}\}=\\frac\{\\sqrt\{t\}\}\{T\}x\_\{0\}\-\\frac\{\\epsilon\}\{\\sqrt\{T\}\},\(7\)which remains bounded near the data endpoint\. During sampling, the reverse drift is recovered byvθ​\(xt,t\)=uθ​\(xt,t\)/tv\_\{\\theta\}\(x\_\{t\},t\)=u\_\{\\theta\}\(x\_\{t\},t\)/\\sqrt\{t\}\. The same learned field also admits a deterministic probability\-flow ODE with the same marginal distributions, enabling stochastic and deterministic sampling within a common formulation\.

The main contributions of this paper are as follows\.

- •The present study introduces MVM, in which a single learned velocity\-like field directly parameterizes the reverse\-SDE drift, without requiring a separately learned score field or an intermediate velocity\-to\-score reconstruction during generation\.
- •The paper derives a Gaussian perturbation process for which the marginal reverse drift is exactly𝔼⁡\[\(x0−xt\)/t∣xt\]\\mathbb\{E\}\[\(x\_\{0\}\-x\_\{t\}\)/t\\mid x\_\{t\}\], allowing the stochastic reverse dynamics to be learned by ordinary quadratic regression on paired forward samples\.
- •At\\sqrt\{t\}\-scaled parameterization is introduced to remove the𝒪\(t−1/2\)\\mathcal\{O\}\(t^\{\-1/2\}\)divergence of the direct regression target neart=0t=0while preserving the corresponding reverse dynamics\.
- •MVM is instantiated with Transformer\-based generative models and evaluated on class\-conditional ImageNet at32×3232\\times 32and256×256256\\times 256\. The stochastic reverse SDE and its deterministic probability\-flow ODE are further compared under matched numerical budgets\.

## 2RELATED WORK

### 2\.1NOISE PREDICTION PARADIGMS

Noise prediction is a standard diffusion parameterization and is closely related to score matching\([Hyvärinen, 2005](https://arxiv.org/html/2609.25444#bib.bib10)\), which estimates log\-density gradients without evaluating normalized densities\. Score\-based generative models extend this idea to progressively perturbed distributions\([Song and Ermon, 2019](https://arxiv.org/html/2609.25444#bib.bib11);[Song and Ermon, 2020](https://arxiv.org/html/2609.25444#bib.bib12)\)\.

DDPM\([Ho et al\., 2020](https://arxiv.org/html/2609.25444#bib.bib2)\)uses a simple and effective noise parameterization\. For a Gaussian perturbation process

xt=αt​x0\+σt​ϵ,ϵ∼𝒩⁡\(0,I\),x\_\{t\}=\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsilon,\\qquad\\epsilon\\sim\\mathcal\{N\}\(0,I\),\(8\)the network predictsϵ\\epsilon, which is analytically related to the score and therefore provides a convenient representation of reverse denoising dynamics\.

Continuous\-time formulations generalize diffusion chains to SDEs\([Song et al\., 2021b](https://arxiv.org/html/2609.25444#bib.bib5)\)and associate the reverse SDE with a probability\-flow ODE sharing the same time\-dependent marginals\.

Subsequent work improves training and sampling while retaining noise/score targets\. Improved DDPM\([Nichol and Dhariwal, 2021](https://arxiv.org/html/2609.25444#bib.bib14)\)and Variational Diffusion Models\([Kingma et al\., 2021](https://arxiv.org/html/2609.25444#bib.bib15)\)refine objectives and variances, while Karras et al\.\([Karras et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib6)\)highlight the importance of preconditioning, noise\-level sampling, loss weighting, and numerical solvers\.

For sampling, DDIM\([Song et al\., 2021a](https://arxiv.org/html/2609.25444#bib.bib13)\)introduces deterministic implicit generation and DPM\-Solver\([Lu et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib19)\)reduces network evaluations with high\-order solvers\. Classifier\-free guidance\([Ho and Salimans, 2021](https://arxiv.org/html/2609.25444#bib.bib18)\)controls the fidelity–diversity trade\-off and is widely used in large\-scale image\-generation systems\([Rombach et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib16);[Saharia et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib17);[Nichol et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib20)\)\.

Because equivalent parameterizations can induce different loss weightings and conditioning, clean\-data and mixed signal–noise targets remain important alternatives\.

### 2\.2DATA PREDICTION PARADIGMS

Another class of methods predicts the clean sample\. Althoughx0x\_\{0\}andϵ\\epsilonare algebraically related under Gaussian perturbations, their regression errors are weighted differently across noise levels\.

Clean\-data prediction is natural for generalized corruption\. Cold Diffusion\([Bansal et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib21)\)uses deterministic degradations such as blur, masking, and pixelation, while Soft Diffusion\([Daras et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib25)\)extends to linear corruption operators and predicts a clean image whose corrupted version matches the observation\.

Related ideas appear in discrete diffusion: Structured Denoising Diffusion Models\([Austin et al\., 2021](https://arxiv.org/html/2609.25444#bib.bib23)\)predict clean discrete variables, and Vector Quantized Diffusion\([Gu et al\., 2022](https://arxiv.org/html/2609.25444#bib.bib24)\)operates in a discrete latent space for text\-to\-image generation\.

Consistency Models\([Song et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib26)\)likewise map states on the same probability\-flow trajectory toward a common endpoint, enabling generation with few network evaluations\. The relative effectiveness of data and noise prediction depends on the perturbation process and training weighting; neither target is universally superior, and each induces a different optimization geometry\.

### 2\.3VELOCITY PREDICTION PARADIGMS

A third family combines signal and noise\. Salimans and Ho\([Salimans and Ho, 2022](https://arxiv.org/html/2609.25444#bib.bib22)\)introducedvv\-prediction for progressive distillation\. Under a variance\-preserving parameterization

xt=αt​x0\+σt​ϵ,αt2\+σt2=1,x\_\{t\}=\\alpha\_\{t\}x\_\{0\}\+\\sigma\_\{t\}\\epsilon,\\qquad\\alpha\_\{t\}^\{2\}\+\\sigma\_\{t\}^\{2\}=1,\(9\)the corresponding target can be written asvt=αt​ϵ−σt​x0v\_\{t\}=\\alpha\_\{t\}\\epsilon\-\\sigma\_\{t\}x\_\{0\}\.

This target combines clean\-data and noise information in one regression variable and improves numerical stability for progressive distillation\.

A conceptually different family models probability transport through neural vector fields\. Continuous normalizing flows and neural ODEs\([Chen et al\., 2018](https://arxiv.org/html/2609.25444#bib.bib27);[Grathwohl et al\., 2019](https://arxiv.org/html/2609.25444#bib.bib28)\)use deterministic dynamics, while Flow Matching\([Lipman et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib7)\)learns marginal vector fields along prescribed conditional probability paths\.

Rectified Flow\([Liu et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib9)\)progressively straightens transport paths to reduce discretization error and permit coarser integration\. Optimal\-transport extensions further improve path geometry\([Tong et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib29);[Pooladian et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib30)\), and rectified\-flow Transformers scale well to high\-resolution image synthesis\([Esser et al\., 2024](https://arxiv.org/html/2609.25444#bib.bib31)\)\.

Deterministic flows and stochastic diffusions can describe the same density evolution\. Score\-based SDEs\([Song et al\., 2021b](https://arxiv.org/html/2609.25444#bib.bib5)\)connect reverse\-time SDEs with probability\-flow ODEs, while Stochastic Interpolants\([Albergo et al\., 2025](https://arxiv.org/html/2609.25444#bib.bib8)\)relate transport equations, forward/backward Fokker–Planck equations, velocity fields, scores, and stochastic drifts in one framework\.

MVM differs from both diffusionvv\-prediction and the instantaneous vector fields of Flow Matching\. For statextx\_\{t\}and sourcex0x\_\{0\}, the present study defines

vmean​\(xt,t,x0\)=x0−xtt\.v\_\{\\mathrm\{mean\}\}\(x\_\{t\},t;x\_\{0\}\)=\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}\.\(10\)This is a finite\-time average restoration velocity rather than an instantaneous path derivative\.

For the stochastic process considered in this paper, the conditional expectation

v∗\(xt,t\)=𝔼\[x0−xtt\|xt\]v^\{\*\}\(x\_\{t\},t\)=\\mathbb\{E\}\\left\[\\left\.\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}\\right\|x\_\{t\}\\right\]\(11\)is exactly the marginal reverse drift\. Thus, the reverse SDE can be learned from one sample\-wise quadratic target, distinct from diffusionvv\-prediction, deterministic Flow Matching, and general stochastic interpolants\.

## 3Mean Velocity Matching

This section derives MVM from probability\-density evolution\. Proofs of Theorems 1–6 are given in the appendix\.

### 3\.1Single\-Image Scenario

#### 3\.1\.1Forward Process

For a single samplex0x\_\{0\}, the initial density is a Diracδ\\deltafunction centered atx0x\_\{0\}\. The forward process should transform this density into a standard normal distribution att=Tt=T\. This evolution is analyzed through its Fourier transform\.

ℱ⁡\(p⁡\(x,t\)\)=p^​\(k,t\)=∫−∞\+∞p⁡\(x,t\)​ei​k​x​𝑑x\\mathcal\{F\}\\left\(p\\left\(x,t\\right\)\\right\)=\\hat\{p\}\\left\(k,t\\right\)=\\int\_\{\-\\infty\}^\{\+\\infty\}p\\left\(x,t\\right\)e^\{ikx\}\\,dx\(12\)
Here,ppis the density,kkthe angular frequency,ℱ\\mathcal\{F\}the Fourier transform, andp^​\(k,t\)\\hat\{p\}\(k,t\)the characteristic function\.

Theorem 1:If the probability density function is aδ\\deltafunction, its characteristic function is

p^​\(k\)=ei​k​x0\.\\hat\{p\}\(k\)=e^\{ikx\_\{0\}\}\.\(13\)
Theorem 2:If the probability density function is a normal distribution with meanμ\\muand varianceσ2\\sigma^\{2\}, its characteristic function is

p^​\(k\)=ei​k​μ−12​σ2​k2\.\\hat\{p\}\(k\)=e^\{ik\\mu\-\\frac\{1\}\{2\}\\sigma^\{2\}k^\{2\}\}\.\(14\)
Theorem 3:If the probability density function is a standard normal distribution, its characteristic function is

p^​\(k\)=e−k22\.\\hat\{p\}\(k\)=e^\{\-\\frac\{k^\{2\}\}\{2\}\}\.\(15\)
Assume that the evolution of the characteristic function over time follows the characteristic function transformation formula below\.

p^​\(k,t\)=p^​\(k,0\)​e−λ⁡\(k\)​t\\hat\{p\}\(k,t\)=\\hat\{p\}\(k,0\)e^\{\-\\lambda\(k\)t\}\(16\)By Theorems 1 and 3, the endpoint conditions arep^​\(k,0\)=ei​k​x0\\hat\{p\}\(k,0\)=e^\{ikx\_\{0\}\}andp^\(k,T\)=e−k2/2\\hat\{p\}\(k,T\)=e^\{\-k^\{2\}/2\}, which determine

λ⁡\(k\)=k22​T\+i​k​x0T\.\\lambda\(k\)=\\frac\{k^\{2\}\}\{2T\}\+\\frac\{ikx\_\{0\}\}\{T\}\.\(17\)Substitutingλ⁡\(k\)\\lambda\(k\)and comparing the result with the characteristic function in Theorem 2 gives

ln⁡p^​\(k,t\)\\displaystyle\\ln\\hat\{p\}\(k,t\)=i​k​x0​\(1−tT\)−k2​t2​T=i​k​μt−12​σt2​k2,\\displaystyle=ikx\_\{0\}\\left\(1\-\\frac\{t\}\{T\}\\right\)\-\\frac\{k^\{2\}t\}\{2T\}=ik\\mu\_\{t\}\-\\frac\{1\}\{2\}\\sigma\_\{t\}^\{2\}k^\{2\},\(18\)μt\\displaystyle\\mu\_\{t\}=x0\(1−tT\),σt2=tT\.\\displaystyle=x\_\{0\}\\left\(1\-\\frac\{t\}\{T\}\\right\),\\qquad\\sigma\_\{t\}^\{2\}=\\frac\{t\}\{T\}\.Consequently, the forward process is

xt=\(1−tT\)​x0\+tT​ε,ε∼𝒩⁡\(0,1\)\.x\_\{t\}=\\left\(1\-\\frac\{t\}\{T\}\\right\)x\_\{0\}\+\\sqrt\{\\frac\{t\}\{T\}\}\\,\\varepsilon,\\qquad\\varepsilon\\sim\\mathcal\{N\}\(0,1\)\.\(19\)

#### 3\.1\.2Reverse Process

The forward characteristic function is:

ln⁡p^​\(k,t\)=i​k​x0​\(1−tT\)−k2​t2​T\\ln\\hat\{p\}\(k,t\)=ikx\_\{0\}\\left\(1\-\\frac\{t\}\{T\}\\right\)\-\\frac\{k^\{2\}t\}\{2T\}\(20\)Let reverse time beτ=T−t\\tau=T\-t\.

ln⁡p^​\(k,τ\)=i​k​x0​\(1−T−τT\)−k2​\(T−τ\)2​T=i​k​x0​\(τT\)−k22\+k2​τ2​T\\ln\\hat\{p\}\(k,\\tau\)=ikx\_\{0\}\\left\(1\-\\frac\{T\-\\tau\}\{T\}\\right\)\-\\frac\{k^\{2\}\(T\-\\tau\)\}\{2T\}=ikx\_\{0\}\\left\(\\frac\{\\tau\}\{T\}\\right\)\-\\frac\{k^\{2\}\}\{2\}\+\\frac\{k^\{2\}\\tau\}\{2T\}\(21\)Differentiating with respect toτ\\tauand using∂τp^=p^​∂τln⁡p^\\partial\_\{\\tau\}\\hat\{p\}=\\hat\{p\}\\,\\partial\_\{\\tau\}\\ln\\hat\{p\}gives

∂p^∂τ\\displaystyle\\frac\{\\partial\\hat\{p\}\}\{\\partial\\tau\}=p^​\(i​k​x0T\+k22​T\)\\displaystyle=\\hat\{p\}\\left\(ik\\frac\{x\_\{0\}\}\{T\}\+\\frac\{k^\{2\}\}\{2T\}\\right\)=i​k​x0T​p^⏟A\+k2T​p^⏟B−k22​T​p^⏟C\.\\displaystyle=\\underbrace\{ik\\frac\{x\_\{0\}\}\{T\}\\hat\{p\}\}\_\{A\}\+\\underbrace\{\\frac\{k^\{2\}\}\{T\}\\hat\{p\}\}\_\{B\}\-\\underbrace\{\\frac\{k^\{2\}\}\{2T\}\\hat\{p\}\}\_\{C\}\.\(22\)
Usingℱ−1\(ikp^\)=−∂xp\\mathcal\{F\}^\{\-1\}\(ik\\hat\{p\}\)=\-\\partial\_\{x\}pandℱ−1​\(−k2​p^\)=∂x2p\\mathcal\{F\}^\{\-1\}\(\-k^\{2\}\\hat\{p\}\)=\\partial\_\{x\}^\{2\}p, the three terms becomeA=−∂x\[\(x0/T\)p\]A=\-\\partial\_\{x\}\[\(x\_\{0\}/T\)p\],B=−\(1/T\)∂x2pB=\-\(1/T\)\\partial\_\{x\}^\{2\}p, andC=\(1/2​T\)​∂x2pC=\(1/2T\)\\partial\_\{x\}^\{2\}p\.

∴∂p∂τ=−∂∂x​\(x0T​p\)−1T​∂2p∂x2\+12​T​∂2p∂x2\\therefore\\frac\{\\partial p\}\{\\partial\\tau\}=\-\\frac\{\\partial\}\{\\partial x\}\\left\(\\frac\{x\_\{0\}\}\{T\}p\\right\)\-\\frac\{1\}\{T\}\\frac\{\\partial^\{2\}p\}\{\\partial x^\{2\}\}\+\\frac\{1\}\{2T\}\\frac\{\\partial^\{2\}p\}\{\\partial x^\{2\}\}\(23\)
Since∂xp=p​∇x​ln⁡p\\partial\_\{x\}p=p\\nabla\_\{x\}\\ln p, theBBterm can be absorbed into the drift:

∂p∂τ=−∂∂x​\[\(x0T\+1T​∇x​ln⁡p\)​p\]\+12​∂2∂x2​\(1T​p\)\\frac\{\\partial p\}\{\\partial\\tau\}=\-\\frac\{\\partial\}\{\\partial x\}\\left\[\\left\(\\frac\{x\_\{0\}\}\{T\}\+\\frac\{1\}\{T\}\\nabla\_\{x\}\\ln p\\right\)p\\right\]\+\\frac\{1\}\{2\}\\frac\{\\partial^\{2\}\}\{\\partial x^\{2\}\}\\left\(\\frac\{1\}\{T\}p\\right\)\(24\)
Theorem: Fokker\-Planck Equation\([Risken, 1989](https://arxiv.org/html/2609.25444#bib.bib32)\)

For an SDEd​x=f⁡\(x,t\)​d​t\+g⁡\(t\)​d​wdx=f\(x,t\)dt\+g\(t\)dw, the Fokker–Planck equation is:

∂p∂t=−∂∂x​\[f⁡\(x,t\)​p\]\+12​∂2∂x2​\[g​\(t\)2​p\]\\frac\{\\partial p\}\{\\partial t\}=\-\\frac\{\\partial\}\{\\partial x\}\\left\[f\(x,t\)p\\right\]\+\\frac\{1\}\{2\}\\frac\{\\partial^\{2\}\}\{\\partial x^\{2\}\}\\left\[g\(t\)^\{2\}p\\right\]\(25\)
Thusf=x0T\+1T​∇x​ln⁡pf=\\frac\{x\_\{0\}\}\{T\}\+\\frac\{1\}\{T\}\\nabla\_\{x\}\\ln pandg=1/Tg=\\sqrt\{1/T\}\. Substituting the score identity into the SDE gives

d​x\\displaystyle dx=\(x0T\+1T​∇x​ln​p\)​d​τ\+1T​d​w,∇x​ln​p=−εt/T\\displaystyle=\\left\(\\frac\{x\_\{0\}\}\{T\}\+\\frac\{1\}\{T\}\\nabla\_\{x\}\\ln p\\right\)d\\tau\+\\sqrt\{\\frac\{1\}\{T\}\}\\,dw,\\qquad\\nabla\_\{x\}\\ln p=\-\\frac\{\\varepsilon\}\{\\sqrt\{t/T\}\}\(26\)=\(x0T−εt​T\)​d​τ\+1T​d​w\.\\displaystyle=\\left\(\\frac\{x\_\{0\}\}\{T\}\-\\frac\{\\varepsilon\}\{\\sqrt\{tT\}\}\\right\)d\\tau\+\\sqrt\{\\frac\{1\}\{T\}\}\\,dw\.
From the forward process,x0T−εt​T=x0−xtt\\frac\{x\_\{0\}\}\{T\}\-\\frac\{\\varepsilon\}\{\\sqrt\{tT\}\}=\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}\.

Definingv⁡\(xt,t\)=x0−xttv\(x\_\{t\},t\)=\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}gives the reverse SDE:

d​x=v⁡\(xt,t\)​d​τ\+1T​d​wdx=v\(x\_\{t\},t\)d\\tau\+\\sqrt\{\\frac\{1\}\{T\}\}dw\(27\)

### 3\.2Multi\-Image Scenario

#### 3\.2\.1Forward Process

Let the probability density forNNimages be:

p⁡\(x0\)=1N​∑j=1Nδ⁡\(x0−x0\(j\)\)p\(x\_\{0\}\)=\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}\\delta\\left\(x\_\{0\}\-x\_\{0\}^\{\(j\)\}\\right\)\(28\)
wherexi\(1\),xi\(2\),…,xi\(N\)x\_\{i\}^\{\(1\)\},x\_\{i\}^\{\(2\)\},\\dots,x\_\{i\}^\{\(N\)\},i∈\[0,1\]i\\in\[0,1\], denote the images att=it=i\.

For each sample, use the same forward process:

xt\(k\)=\(1−tT\)​x0\(k\)\+tT​εx\_\{t\}^\{\(k\)\}=\\left\(1\-\\frac\{t\}\{T\}\\right\)x\_\{0\}^\{\(k\)\}\+\\sqrt\{\\frac\{t\}\{T\}\}\\varepsilon\(29\)
Whent=Tt=T,xT\(k\)∼𝒩⁡\(0,1\)x\_\{T\}^\{\(k\)\}\\sim\\mathcal\{N\}\(0,1\)\.

The marginal density is:

p⁡\(xt\)\\displaystyle p\(x\_\{t\}\)=∫−∞\+∞p⁡\(xt∣x0\)​\(1N​∑j=1Nδ⁡\(x0−x0\(j\)\)\)​d​x0\\displaystyle=\\int\_\{\-\\infty\}^\{\+\\infty\}p\(x\_\{t\}\\mid x\_\{0\}\)\\left\(\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}\\delta\\left\(x\_\{0\}\-x\_\{0\}^\{\(j\)\}\\right\)\\right\)dx\_\{0\}=1N​∑j=1Np⁡\(xt∣x0\(j\)\)\\displaystyle=\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}p\\left\(x\_\{t\}\\mid x\_\{0\}^\{\(j\)\}\\right\)\(30\)
Each conditional component satisfiesp⁡\(xt∣x0\(j\)\)∼𝒩⁡\(\(1−t/T\)​x0\(j\),t/T\)p\(x\_\{t\}\\mid x\_\{0\}^\{\(j\)\}\)\\sim\\mathcal\{N\}\(\(1\-t/T\)x\_\{0\}^\{\(j\)\},t/T\)and becomes𝒩⁡\(0,1\)\\mathcal\{N\}\(0,1\)att=Tt=T; therefore the mixturep⁡\(xT\)p\(x\_\{T\}\)is also standard normal\.

Thus the same forward process maps the full data distribution to a standard normal distribution\.

#### 3\.2\.2Reverse Process

Theorem 4:If the reverse process for a single image is formulated as

d​x=v⁡\(xt,t\)​d​τ\+1T​d​wdx=v\(x\_\{t\},t\)d\\tau\+\\sqrt\{\\frac\{1\}\{T\}\}dw\(31\)then the reverse process for multiple images is given by

d​x=v∗​\(xt,t\)​d​τ\+1T​d​wdx=v^\{\*\}\(x\_\{t\},t\)d\\tau\+\\sqrt\{\\frac\{1\}\{T\}\}dw\(32\)where the respective velocity fields are defined as

v⁡\(xt,t\)=x0−xtt,v∗​\(xt,t\)=𝔼x0\|xt​\[x0−xtt\]v\(x\_\{t\},t\)=\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\},\\quad v^\{\*\}\(x\_\{t\},t\)=\\mathbb\{E\}\_\{x\_\{0\}\\mid x\_\{t\}\}\\left\[\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}\\right\]\(33\)
Theorem 5:Let the loss functionL⁡\(θ\)L\(\\theta\)be defined as

L⁡\(θ\)=𝔼x0∼d​a​t​a,xt\|x0​\[‖vθ​\(xt\)−x0−xtt‖2\]L\(\\theta\)=\\mathbb\{E\}\_\{x\_\{0\}\\sim data,x\_\{t\}\\mid x\_\{0\}\}\\left\[\\left\\\|v\_\{\\theta\}\(x\_\{t\}\)\-\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}\\right\\\|^\{2\}\\right\]\(34\)
and the loss functionL∗​\(θ\)L^\{\*\}\(\\theta\)be defined as

L∗​\(θ\)=𝔼x0∼d​a​t​a,xt\|x0​\[‖vθ​\(xt\)−𝔼x0\|xt​\[x0−xtt\]‖2\]L^\{\*\}\(\\theta\)=\\mathbb\{E\}\_\{x\_\{0\}\\sim data,x\_\{t\}\\mid x\_\{0\}\}\\left\[\\left\\\|v\_\{\\theta\}\(x\_\{t\}\)\-\\mathbb\{E\}\_\{x\_\{0\}\\mid x\_\{t\}\}\\left\[\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}\\right\]\\right\\\|^\{2\}\\right\]\(35\)
Then, optimizingL⁡\(θ\)L\(\\theta\)is mathematically equivalent to optimizingL∗​\(θ\)L^\{\*\}\(\\theta\)\.

By Theorem 5, it is sufficient to optimizeL⁡\(θ\)L\(\\theta\), yielding the sample\-wise reverse\-process loss:

‖vθ​\(xt,t\)−v⁡\(xt,t\)‖2\\left\\\|v\_\{\\theta\}\(x\_\{t\},t\)\-v\(x\_\{t\},t\)\\right\\\|^\{2\}\(36\)
Although Theorem 5 establishes the equivalence of the two regression objectives, directly using the mean\-velocity target may lead to numerical instability neart=0t=0\. From the forward process,

v⁡\(xt,t\)\\displaystyle v\(x\_\{t\},t\)=x0−xtt\\displaystyle=\\frac\{x\_\{0\}\-x\_\{t\}\}\{t\}=x0T−εt​T\.\\displaystyle=\\frac\{x\_\{0\}\}\{T\}\-\\frac\{\\varepsilon\}\{\\sqrt\{tT\}\}\.\(37\)Therefore, the magnitude of the direct regression target grows as𝒪\(t−1/2\)\\mathcal\{O\}\(t^\{\-1/2\}\)and becomes unbounded whenttapproaches zero\.

To obtain a bounded training target, this paper introduces the scaled mean velocity

u⁡\(xt,t\)\\displaystyle u\(x\_\{t\},t\)=t​v​\(xt,t\)\\displaystyle=\\sqrt\{t\}\\,v\(x\_\{t\},t\)=x0−xtt\\displaystyle=\\frac\{x\_\{0\}\-x\_\{t\}\}\{\\sqrt\{t\}\}=tT​x0−εT\.\\displaystyle=\\frac\{\\sqrt\{t\}\}\{T\}x\_\{0\}\-\\frac\{\\varepsilon\}\{\\sqrt\{T\}\}\.\(38\)limt→0u⁡\(xt,t\)=−εT,\\lim\_\{t\\rightarrow 0\}u\(x\_\{t\},t\)=\-\\frac\{\\varepsilon\}\{\\sqrt\{T\}\},\(39\)which remains finite\. The practical training objective is therefore defined as

L~​\(θ\)=𝔼x0∼pdata,xt\|x0​\[‖uθ​\(xt,t\)−x0−xtt‖2\]\.\\widetilde\{L\}\(\\theta\)=\\mathbb\{E\}\_\{x\_\{0\}\\sim p\_\{\\mathrm\{data\}\},\\,x\_\{t\}\\mid x\_\{0\}\}\\left\[\\left\\\|u\_\{\\theta\}\(x\_\{t\},t\)\-\\frac\{x\_\{0\}\-x\_\{t\}\}\{\\sqrt\{t\}\}\\right\\\|^\{2\}\\right\]\.\(40\)During sampling, the mean\-velocity field is recovered by

vθ​\(xt,t\)=uθ​\(xt,t\)t\.v\_\{\\theta\}\(x\_\{t\},t\)=\\frac\{u\_\{\\theta\}\(x\_\{t\},t\)\}\{\\sqrt\{t\}\}\.\(41\)Accordingly, the reverse SDE used for generation is

d​x=uθ​\(xt,t\)t​d​τ\+1T​d​w\.dx=\\frac\{u\_\{\\theta\}\(x\_\{t\},t\)\}\{\\sqrt\{t\}\}\\,d\\tau\+\\sqrt\{\\frac\{1\}\{T\}\}\\,dw\.\(42\)

### 3\.3Deterministic ODE Formulation

In addition to the stochastic reverse process, MVM admits a deterministic probability\-flow ODE with the same marginal distributions\([Song et al\., 2021b](https://arxiv.org/html/2609.25444#bib.bib5)\)\.

Theorem 6:Letpt​\(x\)p\_\{t\}\(x\)denote the marginal density of the MVM forward process\. The probability\-flow ODE corresponding to the reverse SDE is

d​x=\[v∗​\(xt,t\)−12​T​∇xt​log⁡pt​\(xt\)\]​d​τ\.dx=\\left\[v^\{\*\}\(x\_\{t\},t\)\-\\frac\{1\}\{2T\}\\nabla\_\{x\_\{t\}\}\\log p\_\{t\}\(x\_\{t\}\)\\right\]d\\tau\.\(43\)For the MVM perturbation path, the score and mean velocity satisfy

∇xt​log​pt​\(xt\)=\(T−t\)​v∗​\(xt,t\)−xt\.\\nabla\_\{x\_\{t\}\}\\log p\_\{t\}\(x\_\{t\}\)=\(T\-t\)v^\{\*\}\(x\_\{t\},t\)\-x\_\{t\}\.\(44\)Consequently, the deterministic reverse process can be written as

d​x=\[T\+t2​T​v∗​\(xt,t\)\+xt2​T\]​d​τ\.dx=\\left\[\\frac\{T\+t\}\{2T\}v^\{\*\}\(x\_\{t\},t\)\+\\frac\{x\_\{t\}\}\{2T\}\\right\]d\\tau\.\(45\)Since the network predictsuθ​\(xt,t\)=t​v∗​\(xt,t\)u\_\{\\theta\}\(x\_\{t\},t\)=\\sqrt\{t\}\\,v^\{\*\}\(x\_\{t\},t\), the practical ODE is

d​x=\[T\+t2​T​t​uθ​\(xt,t\)\+xt2​T\]​d​τ\.dx=\\left\[\\frac\{T\+t\}\{2T\\sqrt\{t\}\}u\_\{\\theta\}\(x\_\{t\},t\)\+\\frac\{x\_\{t\}\}\{2T\}\\right\]d\\tau\.\(46\)

## 4Experiments

The experiments are conducted on class\-conditional ImageNet at resolutions of32×3232\\times 32and256×256256\\times 256\([Deng et al\., 2009](https://arxiv.org/html/2609.25444#bib.bib34)\)\. ImageNet\-3232is used to evaluate the basic generation performance of MVM, while ImageNet\-256256examines its ability to generate higher\-resolution images\. For ImageNet\-3232, the reverse process is parameterized by a pixel\-space DiT\([Peebles and Xie, 2023](https://arxiv.org/html/2609.25444#bib.bib33)\), with REPA introduced to accelerate training\([Yu et al\., 2025](https://arxiv.org/html/2609.25444#bib.bib35)\)\. For ImageNet\-256256, an RAE with a frozen DINOv3 encoder reduces the spatial resolution before a DDT\-style Transformer performs latent generation\([Zheng et al\., 2025](https://arxiv.org/html/2609.25444#bib.bib36);[Siméoni et al\., 2025](https://arxiv.org/html/2609.25444#bib.bib37)\); REPA is also employed\. SA\-Solver is used for sampling\([Xue et al\., 2023](https://arxiv.org/html/2609.25444#bib.bib38)\)\. Classifier\-free guidance is used in the ImageNet\-3232SDE–ODE comparison, while REPA internal guidance is employed for the reported ImageNet\-256256result\. The first experiment evaluates generation quality using FID\([Heusel et al\., 2017](https://arxiv.org/html/2609.25444#bib.bib39)\)\. The second compares the FID–NFE curves of the reverse SDE and its ODE counterpart on ImageNet\-3232\([Song et al\., 2021b](https://arxiv.org/html/2609.25444#bib.bib5)\)\.

### 4\.1Main Generation Results

This paper first evaluates the generation quality of Mean Velocity Matching \(MVM\) on ImageNet at two representative resolutions, namely32×3232\\times 32and256×256256\\times 256\. Following common practice in generative modeling, Fréchet Inception Distance \(FID\) is adopted as the primary metric, where a lower value indicates that the generated distribution is closer to the real data distribution\. The number of function evaluations \(NFE\) is additionally reported to characterize the computational cost of sampling\. All FID values of MVM are computed from50,00050\{,\}000generated samples\.

The compared methods cover representative diffusion\-, score\-, flow\-, GAN\-, latent\-diffusion\-, and Transformer\-based generative models\. The results are summarized in Tables[2](https://arxiv.org/html/2609.25444#S4.T2)and[2](https://arxiv.org/html/2609.25444#S4.T2)\.

Table 1:Generation results on ImageNet32×3232\\times 32\. Lower FID and NFE are better\.MethodFID↓\\downarrowNFE↓\\downarrowDDPM6\.99262Score Matching5\.68178ScoreFlow14\.14195FM w/ Diffusion6\.37193FM w/ OT5\.021222\.2765MVM w/ ODE\(CFG=1\.20, GI=\[0\.60,0\.80\]\)2\.04652\.2665MVM w/ SDE\(CFG=1\.20, GI=\[0\.60,0\.80\]\)1\.9365
Table 2:Generation results on ImageNet256×256256\\times 256\. Lower FID and NFE are better\.MethodFID↓\\downarrowNFE↓\\downarrowADM10\.94250StyleGAN\-XL2\.30\-DiT\-XL/2\-G2\.27250BigGAN\-deep6\.95\-LDM\-4\-G3\.602502\.4090MVM w/ ODE\(IG=1\.45, GI=\[0,1\]\)2\.16902\.2890MVM w/ SDE\(IG=1\.45, GI=\[0,1\]\)2\.0790

On ImageNet32×3232\\times 32, MVM achieves an FID of1\.931\.93with6565function evaluations using reverse\-SDE sampling\. As shown in Table[2](https://arxiv.org/html/2609.25444#S4.T2), the proposed method substantially improves upon the reported diffusion\-, score\-, and flow\-based baselines while requiring fewer function evaluations\.

The present study further evaluates MVM on the more challenging class\-conditional ImageNet256×256256\\times 256benchmark\. As shown in Table[2](https://arxiv.org/html/2609.25444#S4.T2), MVM obtains an FID of2\.072\.07using9090function evaluations\. This result demonstrates that the proposed mean\-velocity formulation remains effective when scaling from low\-resolution generation to higher\-resolution class\-conditional image synthesis\.

### 4\.2Comparison of SDE and ODE Sampling

Figure 2:FID–NFE curves on ImageNet32×3232\\times 32\. Lower FID is better\.The second experiment compares the stochastic reverse SDE with the deterministic probability\-flow ODE on ImageNet32×3232\\times 32\. Both samplers use the same epoch\-200 EMA checkpoint, initial Gaussian seeds, class labels, time\-discretization rule, predictor–corrector orders, classifier\-free\-guidance configuration, number of generated images, and matched NFE\. SA\-Solver uses Adams predictor–corrector orders P3/C4, CFG scale1\.201\.20, and a guidance interval of\[0\.60,0\.80\]\[0\.60,0\.80\]measured in log\-SNR progress\. The ODE usesτ⁡\(t\)=0\\tau\(t\)=0, whereas the SDE usesτ⁡\(t\)=ρ​\(1−t\)/\(1\+t\)\\tau\(t\)=\\rho\\sqrt\{\(1\-t\)/\(1\+t\)\}withρ=0\.75\\rho=0\.75\.

At NFE values of1313and2626, the probability\-flow ODE obtains FID values of3\.3563\.356and2\.1972\.197, respectively, outperforming the reverse SDE values of6\.1486\.148and2\.5792\.579\. The ordering reverses as the computational budget increases\. At NFE values of3939,5252, and6565, the reverse SDE achieves FID values of2\.0392\.039,1\.9421\.942, and1\.9351\.935, respectively, whereas the ODE obtains2\.0882\.088,2\.0252\.025, and2\.0342\.034\. As shown in Figure[2](https://arxiv.org/html/2609.25444#S4.F2), the ODE is preferable under a strict low\-NFE budget, while the stochastic reverse process yields lower FID from NFE=39=39onward\. The best observed SDE result is1\.93491\.9349at NFE=65=65, whereas the best ODE result is2\.02532\.0253at NFE=52=52\. Therefore, sufficient numerical resolution is particularly important for the stochastic reverse dynamics, which ultimately attain better generation quality than the corresponding deterministic flow\.

## 5Conclusion

This paper introduces Mean Velocity Matching \(MVM\), a generative framework that constructs a tractable diffusion path through Fourier transform and learns the reverse dynamics using a single mean\-velocity field\. The resulting formulation supports both stochastic reverse\-SDE and deterministic probability\-flow ODE sampling while preserving the same forward process\. Experiments on ImageNet32×3232\\times 32and256×256256\\times 256demonstrate strong generation quality, achieving FID values of1\.931\.93and2\.072\.07, respectively\. The SDE–ODE comparison further shows that the ODE is advantageous at very low NFE, whereas the SDE attains lower FID when more function evaluations are available\. These results establish MVM as an effective and scalable framework for image generation\.

## AI Use Statement

Generative AI tools were used to assist with literature discovery and to improve the clarity and readability of the manuscript\. All references identified with AI assistance were independently verified by the authors, and all AI\-assisted edits were reviewed by the authors, who take full responsibility for the final content\.

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## Appendix AProofs of Theorems

This appendix contains the complete proofs of Theorems 1–6 stated in the main text\. Throughout the appendix, the Fourier transform is defined by

p^​\(k\)=ℱ⁡\[p\]​\(k\)=∫−∞\+∞p⁡\(x\)​ei​k​x​𝑑x\.\\widehat\{p\}\(k\)=\\mathcal\{F\}\[p\]\(k\)=\\int\_\{\-\\infty\}^\{\+\\infty\}p\(x\)e^\{ikx\}\\,dx\.\(47\)All differentiations under the integral sign and integrations by parts are understood under the standard integrability and smoothness conditions\. These conditions hold for the Gaussian transition densities used by MVM for everyt\>0t\>0\.

### A\.1Proof of Theorem 1

For a deterministic samplex0x\_\{0\}, the associated probability measure is the Dirac measureδ⁡\(x−x0\)\\delta\(x\-x\_\{0\}\)\. Applying the defining sifting property of the Dirac delta to Equation equation[47](https://arxiv.org/html/2609.25444#A1.E47)gives

p^​\(k\)\\displaystyle\\widehat\{p\}\(k\)=∫−∞\+∞δ⁡\(x−x0\)​ei​k​x​𝑑x\\displaystyle=\\int\_\{\-\\infty\}^\{\+\\infty\}\\delta\(x\-x\_\{0\}\)e^\{ikx\}\\,dx\(48\)=ei​k​x0\.\\displaystyle=e^\{ikx\_\{0\}\}\.Hence, the characteristic function of a point mass atx0x\_\{0\}isei​k​x0e^\{ikx\_\{0\}\}\. This proves Theorem 1\.□\\square

### A\.2Proof of Theorem 2

LetX∼𝒩⁡\(μ,σ2\)X\\sim\\mathcal\{N\}\(\\mu,\\sigma^\{2\}\)and write

X=μ\+σ​Z,Z∼𝒩⁡\(0,1\)\.X=\\mu\+\\sigma Z,\\qquad Z\\sim\\mathcal\{N\}\(0,1\)\.\(49\)It is sufficient first to obtain the characteristic function ofZZ\. Denote its density byϕ\(z\)=\(2π\)−1/2e−z2/2\\phi\(z\)=\(2\\pi\)^\{\-1/2\}e^\{\-z^\{2\}/2\}and define

I⁡\(a\)=𝔼⁡\[ei​a​Z\]=∫−∞\+∞ei​a​z​ϕ​\(z\)​𝑑z\.I\(a\)=\\mathbb\{E\}\[e^\{iaZ\}\]=\\int\_\{\-\\infty\}^\{\+\\infty\}e^\{iaz\}\\phi\(z\)\\,dz\.\(50\)Because\|z\|​ϕ​\(z\)\|z\|\\phi\(z\)is integrable, differentiation under the integral sign is valid\. Sinceϕ′​\(z\)=−z​ϕ​\(z\)\\phi^\{\\prime\}\(z\)=\-z\\phi\(z\), integration by parts yields

I′​\(a\)\\displaystyle I^\{\\prime\}\(a\)=i​∫−∞\+∞z​ei​a​z​ϕ​\(z\)​𝑑z\\displaystyle=i\\int\_\{\-\\infty\}^\{\+\\infty\}ze^\{iaz\}\\phi\(z\)\\,dz\(51\)=−i∫−∞\+∞ei​a​zϕ′\(z\)dz\\displaystyle=\-i\\int\_\{\-\\infty\}^\{\+\\infty\}e^\{iaz\}\\phi^\{\\prime\}\(z\)\\,dz=−i⁡\(\[ei​a​z​ϕ​\(z\)\]−∞\+∞−i​a​∫−∞\+∞ei​a​z​ϕ​\(z\)​dz\)\\displaystyle=\-i\\left\(\\left\[e^\{iaz\}\\phi\(z\)\\right\]\_\{\-\\infty\}^\{\+\\infty\}\-ia\\int\_\{\-\\infty\}^\{\+\\infty\}e^\{iaz\}\\phi\(z\)\\,dz\\right\)=−a​I​\(a\)\.\\displaystyle=\-aI\(a\)\.The boundary term in Equation equation[51](https://arxiv.org/html/2609.25444#A1.E51)vanishes becauseϕ⁡\(z\)\\phi\(z\)decays exponentially\. Moreover,

I⁡\(0\)=∫−∞\+∞ϕ⁡\(z\)​𝑑z=1\.I\(0\)=\\int\_\{\-\\infty\}^\{\+\\infty\}\\phi\(z\)\\,dz=1\.\(52\)Solving the ordinary differential equation in Equation equation[51](https://arxiv.org/html/2609.25444#A1.E51)subject to Equation equation[52](https://arxiv.org/html/2609.25444#A1.E52)gives

I\(a\)=e−a2/2\.I\(a\)=e^\{\-a^\{2\}/2\}\.\(53\)Consequently, the characteristic function ofXXis

p^​\(k\)\\displaystyle\\widehat\{p\}\(k\)=𝔼⁡\[ei​k​X\]\\displaystyle=\\mathbb\{E\}\[e^\{ikX\}\]\(54\)=𝔼⁡\[ei​k​\(μ\+σ​Z\)\]\\displaystyle=\\mathbb\{E\}\[e^\{ik\(\\mu\+\\sigma Z\)\}\]=ei​k​μ​I​\(k​σ\)\\displaystyle=e^\{ik\\mu\}I\(k\\sigma\)=exp⁡\(i​k​μ−12​σ2​k2\)\.\\displaystyle=\\exp\\\!\\left\(ik\\mu\-\\frac\{1\}\{2\}\\sigma^\{2\}k^\{2\}\\right\)\.This proves Theorem 2\.□\\square

### A\.3Proof of Theorem 3

A standard normal distribution is the special caseμ=0\\mu=0andσ2=1\\sigma^\{2\}=1of Theorem 2\. Substitution into Equation equation[54](https://arxiv.org/html/2609.25444#A1.E54)gives

p^\(k\)=exp\(ik⋅0−12k2\)=e−k2/2\.\\widehat\{p\}\(k\)=\\exp\\\!\\left\(ik\\cdot 0\-\\frac\{1\}\{2\}k^\{2\}\\right\)=e^\{\-k^\{2\}/2\}\.\(55\)This proves Theorem 3\.□\\square

### A\.4Proof of Theorem 4

Letπ⁡\(d​x0\)\\pi\(dx\_\{0\}\)denote the data distribution\. For a fixed clean samplex0x\_\{0\}, letqt​\(x∣x0\)q\_\{t\}\(x\\mid x\_\{0\}\)be the forward conditional density, and introduce reverse time and the corresponding conditional density by

τ=T−t,rτ​\(x∣x0\)=qT−τ​\(x∣x0\)\.\\tau=T\-t,\\qquad r\_\{\\tau\}\(x\\mid x\_\{0\}\)=q\_\{T\-\\tau\}\(x\\mid x\_\{0\}\)\.\(56\)The single\-image reverse process assumed in Theorem 4 is

d​Xτ=v⁡\(Xτ,t,X0\)​d​τ\+1T​d​W¯τ,v⁡\(x,t,x0\)=x0−xt,t=T−τ\.dX\_\{\\tau\}=v\(X\_\{\\tau\},t;X\_\{0\}\)\\,d\\tau\+\\frac\{1\}\{\\sqrt\{T\}\}\\,d\\overline\{W\}\_\{\\tau\},\\qquad v\(x,t;x\_\{0\}\)=\\frac\{x\_\{0\}\-x\}\{t\},\\qquad t=T\-\\tau\.\(57\)Therefore, for every fixedx0x\_\{0\}, its conditional reverse density satisfies the Fokker–Planck equation

∂τrτ\(x∣x0\)=−∂x\[v\(x,t;x0\)rτ\(x∣x0\)\]\+12​T∂x2rτ\(x∣x0\)\.\\partial\_\{\\tau\}r\_\{\\tau\}\(x\\mid x\_\{0\}\)=\-\\partial\_\{x\}\\\!\\left\[v\(x,t;x\_\{0\}\)r\_\{\\tau\}\(x\\mid x\_\{0\}\)\\right\]\+\\frac\{1\}\{2T\}\\partial\_\{x\}^\{2\}r\_\{\\tau\}\(x\\mid x\_\{0\}\)\.\(58\)The marginal density in reverse time is obtained by averaging the conditional density over the data distribution:

rτ​\(x\)=∫rτ​\(x∣x0\)​π​\(d​x0\)\.r\_\{\\tau\}\(x\)=\\int r\_\{\\tau\}\(x\\mid x\_\{0\}\)\\,\\pi\(dx\_\{0\}\)\.\(59\)Differentiating Equation equation[59](https://arxiv.org/html/2609.25444#A1.E59)with respect toτ\\tau, substituting Equation equation[58](https://arxiv.org/html/2609.25444#A1.E58), and interchanging differentiation and integration gives

∂τrτ​\(x\)\\displaystyle\\partial\_\{\\tau\}r\_\{\\tau\}\(x\)=∫∂τrτ​\(x∣x0\)​π​\(d​x0\)\\displaystyle=\\int\\partial\_\{\\tau\}r\_\{\\tau\}\(x\\mid x\_\{0\}\)\\,\\pi\(dx\_\{0\}\)\(60\)=−∂x\[∫v\(x,t;x0\)rτ\(x∣x0\)π\(dx0\)\]\+12​T∂x2∫rτ\(x∣x0\)π\(dx0\)\\displaystyle=\-\\partial\_\{x\}\\\!\\left\[\\int v\(x,t;x\_\{0\}\)r\_\{\\tau\}\(x\\mid x\_\{0\}\)\\,\\pi\(dx\_\{0\}\)\\right\]\+\\frac\{1\}\{2T\}\\partial\_\{x\}^\{2\}\\int r\_\{\\tau\}\(x\\mid x\_\{0\}\)\\,\\pi\(dx\_\{0\}\)=−∂x\[∫v\(x,t;x0\)rτ\(x∣x0\)π\(dx0\)\]\+12​T∂x2rτ\(x\)\.\\displaystyle=\-\\partial\_\{x\}\\\!\\left\[\\int v\(x,t;x\_\{0\}\)r\_\{\\tau\}\(x\\mid x\_\{0\}\)\\,\\pi\(dx\_\{0\}\)\\right\]\+\\frac\{1\}\{2T\}\\partial\_\{x\}^\{2\}r\_\{\\tau\}\(x\)\.Wheneverrτ​\(x\)\>0r\_\{\\tau\}\(x\)\>0, Bayes’ rule gives the posterior distribution of the clean sample conditioned on the current state:

π⁡\(d​x0∣x,τ\)=rτ​\(x∣x0\)​π​\(d​x0\)rτ​\(x\)\.\\pi\(dx\_\{0\}\\mid x,\\tau\)=\\frac\{r\_\{\\tau\}\(x\\mid x\_\{0\}\)\\,\\pi\(dx\_\{0\}\)\}\{r\_\{\\tau\}\(x\)\}\.\(61\)Using Equation equation[61](https://arxiv.org/html/2609.25444#A1.E61), the drift integral in Equation equation[60](https://arxiv.org/html/2609.25444#A1.E60)can be rewritten as

∫v⁡\(x,t,x0\)​rτ​\(x∣x0\)​π​\(d​x0\)\\displaystyle\\int v\(x,t;x\_\{0\}\)r\_\{\\tau\}\(x\\mid x\_\{0\}\)\\,\\pi\(dx\_\{0\}\)=rτ​\(x\)​∫v⁡\(x,t,x0\)​π​\(d​x0∣x,τ\)\\displaystyle=r\_\{\\tau\}\(x\)\\int v\(x,t;x\_\{0\}\)\\,\\pi\(dx\_\{0\}\\mid x,\\tau\)\(62\)=rτ\(x\)𝔼\[v\(x,t;X0\)\|Xt=x\]\.\\displaystyle=r\_\{\\tau\}\(x\)\\,\\mathbb\{E\}\\\!\\left\[\\left\.v\(x,t;X\_\{0\}\)\\right\|X\_\{t\}=x\\right\]\.Define the marginal reverse drift as

v∗​\(x,t\)\\displaystyle v^\{\*\}\(x,t\):=𝔼\[v\(x,t;X0\)\|Xt=x\]\\displaystyle:=\\mathbb\{E\}\\\!\\left\[\\left\.v\(x,t;X\_\{0\}\)\\right\|X\_\{t\}=x\\right\]\(63\)=𝔼\[X0−xt\|Xt=x\]\.\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\left\.\\frac\{X\_\{0\}\-x\}\{t\}\\right\|X\_\{t\}=x\\right\]\.Substituting Equations equation[62](https://arxiv.org/html/2609.25444#A1.E62)and equation[63](https://arxiv.org/html/2609.25444#A1.E63)into Equation equation[60](https://arxiv.org/html/2609.25444#A1.E60)yields

∂τrτ\(x\)=−∂x\[v∗\(x,t\)rτ\(x\)\]\+12​T∂x2rτ\(x\)\.\\partial\_\{\\tau\}r\_\{\\tau\}\(x\)=\-\\partial\_\{x\}\\\!\\left\[v^\{\*\}\(x,t\)r\_\{\\tau\}\(x\)\\right\]\+\\frac\{1\}\{2T\}\\partial\_\{x\}^\{2\}r\_\{\\tau\}\(x\)\.\(64\)Equation equation[64](https://arxiv.org/html/2609.25444#A1.E64)is precisely the Fokker–Planck equation associated with

d​Xτ=v∗​\(Xτ,t\)​d​τ\+1T​d​W¯τ,t=T−τ\.dX\_\{\\tau\}=v^\{\*\}\(X\_\{\\tau\},t\)\\,d\\tau\+\\frac\{1\}\{\\sqrt\{T\}\}\\,d\\overline\{W\}\_\{\\tau\},\\qquad t=T\-\\tau\.\(65\)Thus, marginalizing the single\-image reverse dynamics replaces the sample\-dependent velocity by its posterior conditional expectation while leaving the diffusion coefficient unchanged\. This proves Theorem 4\.□\\square

### A\.5Proof of Theorem 5

For a fixedt\>0t\>0, define the observable sample\-wise target and its conditional mean by

Yt:=X0−Xtt,mt​\(Xt\):=𝔼⁡\[Yt∣Xt\]=v∗​\(Xt,t\)\.Y\_\{t\}:=\\frac\{X\_\{0\}\-X\_\{t\}\}\{t\},\\qquad m\_\{t\}\(X\_\{t\}\):=\\mathbb\{E\}\[Y\_\{t\}\\mid X\_\{t\}\]=v^\{\*\}\(X\_\{t\},t\)\.\(66\)With this notation, the two population losses in Theorem 5 are

L⁡\(θ\)=𝔼⁡\[‖vθ​\(Xt,t\)−Yt‖2\]L\(\\theta\)=\\mathbb\{E\}\\\!\\left\[\\left\\\|v\_\{\\theta\}\(X\_\{t\},t\)\-Y\_\{t\}\\right\\\|^\{2\}\\right\]\(67\)and

L∗​\(θ\)=𝔼⁡\[‖vθ​\(Xt,t\)−mt​\(Xt\)‖2\]\.L^\{\*\}\(\\theta\)=\\mathbb\{E\}\\\!\\left\[\\left\\\|v\_\{\\theta\}\(X\_\{t\},t\)\-m\_\{t\}\(X\_\{t\}\)\\right\\\|^\{2\}\\right\]\.\(68\)Add and subtractmt​\(Xt\)m\_\{t\}\(X\_\{t\}\)inside Equation equation[67](https://arxiv.org/html/2609.25444#A1.E67)\. Expanding the squared Euclidean norm gives

L⁡\(θ\)\\displaystyle L\(\\theta\)=𝔼⁡\[‖vθ​\(Xt,t\)−mt​\(Xt\)\+mt​\(Xt\)−Yt‖2\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\left\\\|v\_\{\\theta\}\(X\_\{t\},t\)\-m\_\{t\}\(X\_\{t\}\)\+m\_\{t\}\(X\_\{t\}\)\-Y\_\{t\}\\right\\\|^\{2\}\\right\]\(69\)=𝔼⁡\[‖vθ​\(Xt,t\)−mt​\(Xt\)‖2\]\+𝔼⁡\[‖Yt−mt​\(Xt\)‖2\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\left\\\|v\_\{\\theta\}\(X\_\{t\},t\)\-m\_\{t\}\(X\_\{t\}\)\\right\\\|^\{2\}\\right\]\+\\mathbb\{E\}\\\!\\left\[\\left\\\|Y\_\{t\}\-m\_\{t\}\(X\_\{t\}\)\\right\\\|^\{2\}\\right\]\+2​𝔼​\[\(vθ​\(Xt,t\)−mt​\(Xt\)\)⊤​\(mt​\(Xt\)−Yt\)\]\.\\displaystyle\+2\\mathbb\{E\}\\\!\\left\[\\bigl\(v\_\{\\theta\}\(X\_\{t\},t\)\-m\_\{t\}\(X\_\{t\}\)\\bigr\)^\{\\top\}\\bigl\(m\_\{t\}\(X\_\{t\}\)\-Y\_\{t\}\\bigr\)\\right\]\.The final cross\-term in Equation equation[69](https://arxiv.org/html/2609.25444#A1.E69)is zero\. Indeed, the tower property of conditional expectation and the definition ofmt​\(Xt\)m\_\{t\}\(X\_\{t\}\)imply

𝔼⁡\[\(vθ​\(Xt,t\)−mt​\(Xt\)\)⊤​\(mt​\(Xt\)−Yt\)\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\bigl\(v\_\{\\theta\}\(X\_\{t\},t\)\-m\_\{t\}\(X\_\{t\}\)\\bigr\)^\{\\top\}\\bigl\(m\_\{t\}\(X\_\{t\}\)\-Y\_\{t\}\\bigr\)\\right\]\(70\)=𝔼\[\(vθ\(Xt,t\)−mt\(Xt\)\)⊤𝔼\[mt\(Xt\)−Yt\|Xt\]\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\bigl\(v\_\{\\theta\}\(X\_\{t\},t\)\-m\_\{t\}\(X\_\{t\}\)\\bigr\)^\{\\top\}\\mathbb\{E\}\\\!\\left\[\\left\.m\_\{t\}\(X\_\{t\}\)\-Y\_\{t\}\\right\|X\_\{t\}\\right\]\\right\]=𝔼⁡\[\(vθ​\(Xt,t\)−mt​\(Xt\)\)⊤​\(mt​\(Xt\)−𝔼⁡\[Yt∣Xt\]\)\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\bigl\(v\_\{\\theta\}\(X\_\{t\},t\)\-m\_\{t\}\(X\_\{t\}\)\\bigr\)^\{\\top\}\\bigl\(m\_\{t\}\(X\_\{t\}\)\-\\mathbb\{E\}\[Y\_\{t\}\\mid X\_\{t\}\]\\bigr\)\\right\]=0\.\\displaystyle=0\.Combining Equations equation[68](https://arxiv.org/html/2609.25444#A1.E68), equation[69](https://arxiv.org/html/2609.25444#A1.E69), and equation[70](https://arxiv.org/html/2609.25444#A1.E70)gives the conditional\-expectation Pythagorean identity

L⁡\(θ\)=L∗​\(θ\)\+𝔼⁡\[‖Yt−mt​\(Xt\)‖2\]⏟Ct\.L\(\\theta\)=L^\{\*\}\(\\theta\)\+\\underbrace\{\\mathbb\{E\}\\\!\\left\[\\left\\\|Y\_\{t\}\-m\_\{t\}\(X\_\{t\}\)\\right\\\|^\{2\}\\right\]\}\_\{C\_\{t\}\}\.\(71\)The termCtC\_\{t\}is independent ofθ\\theta\. Therefore,

arg​minθ⁡L​\(θ\)=arg​minθ⁡L∗​\(θ\),∇θL​\(θ\)=∇θL∗​\(θ\)\\operatorname\*\{arg\\,min\}\_\{\\theta\}L\(\\theta\)=\\operatorname\*\{arg\\,min\}\_\{\\theta\}L^\{\*\}\(\\theta\),\\qquad\\nabla\_\{\\theta\}L\(\\theta\)=\\nabla\_\{\\theta\}L^\{\*\}\(\\theta\)\(72\)whenever the gradients exist\. Hence, regression to the sample\-wise target\(X0−Xt\)/t\(X\_\{0\}\-X\_\{t\}\)/tand regression to its conditional mean have identical population minimizers\.

If training also samplesttfrom a distributionρ\\rho, the same argument is applied after conditioning on the joint input\(Xt,t\)\(X\_\{t\},t\)\. In that case,

𝔼⁡\[‖vθ​\(Xt,t\)−Yt‖2\]\\displaystyle\\mathbb\{E\}\\\!\\left\[\\left\\\|v\_\{\\theta\}\(X\_\{t\},t\)\-Y\_\{t\}\\right\\\|^\{2\}\\right\]\(73\)=𝔼\[‖vθ\(Xt,t\)−𝔼\[Yt∣Xt,t\]‖2\]\+𝔼\[‖Yt−𝔼\[Yt∣Xt,t\]‖2\]\.\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\left\\\|v\_\{\\theta\}\(X\_\{t\},t\)\-\\mathbb\{E\}\[Y\_\{t\}\\mid X\_\{t\},t\]\\right\\\|^\{2\}\\right\]\+\\mathbb\{E\}\\\!\\left\[\\left\\\|Y\_\{t\}\-\\mathbb\{E\}\[Y\_\{t\}\\mid X\_\{t\},t\]\\right\\\|^\{2\}\\right\]\.The second term in Equation equation[73](https://arxiv.org/html/2609.25444#A1.E73)is again independent ofθ\\theta, so the equivalence also holds for the time\-averaged training objective\. This proves Theorem 5\.□\\square

### A\.6Proof of Theorem 6

For the MVM forward process, the conditional density is

pt​\(x∣x0\)=𝒩⁡\(x,\(1−tT\)​x0,tT​I\),0<t≤T\.p\_\{t\}\(x\\mid x\_\{0\}\)=\\mathcal\{N\}\\\!\\left\(x;\\left\(1\-\\frac\{t\}\{T\}\\right\)x\_\{0\},\\frac\{t\}\{T\}I\\right\),\\qquad 0<t\\leq T\.\(74\)Here,xxdenotes a spatial variable, and the reverse time isτ=T−t\\tau=T\-t\. By Theorem 4, the marginal reverse SDE is

d​x=v∗​\(xt,t\)​d​τ\+1T​d​W¯τ\.dx=v^\{\*\}\(x\_\{t\},t\)\\,d\\tau\+\\frac\{1\}\{\\sqrt\{T\}\}\\,d\\overline\{W\}\_\{\\tau\}\.\(75\)Its marginal density satisfies the Fokker–Planck equation

∂τpt\(x\)=−∇x⋅\[v∗\(x,t\)pt\(x\)\]\+12​TΔxpt\(x\),\\partial\_\{\\tau\}p\_\{t\}\(x\)=\-\\nabla\_\{x\}\\\!\\cdot\\\!\\left\[v^\{\*\}\(x,t\)p\_\{t\}\(x\)\\right\]\+\\frac\{1\}\{2T\}\\Delta\_\{x\}p\_\{t\}\(x\),\(76\)where∂τpt​\(x\)\\partial\_\{\\tau\}p\_\{t\}\(x\)means differentiation ofpT−τ​\(x\)p\_\{T\-\\tau\}\(x\)withxxfixed\.

Using∇xpt=pt​∇x​log⁡pt\\nabla\_\{x\}p\_\{t\}=p\_\{t\}\\nabla\_\{x\}\\log p\_\{t\}, this equation becomes

∂τpt​\(x\)\\displaystyle\\partial\_\{\\tau\}p\_\{t\}\(x\)=−∇x⋅\[v∗\(x,t\)pt\(x\)\]\+12​T∇x⋅\[pt\(x\)∇xlogpt\(x\)\]\\displaystyle=\-\\nabla\_\{x\}\\\!\\cdot\\\!\\left\[v^\{\*\}\(x,t\)p\_\{t\}\(x\)\\right\]\+\\frac\{1\}\{2T\}\\nabla\_\{x\}\\\!\\cdot\\left\[p\_\{t\}\(x\)\\nabla\_\{x\}\\log p\_\{t\}\(x\)\\right\]\(77\)=−∇x⋅\[\(v∗\(x,t\)−12​T∇xlogpt\(x\)\)pt\(x\)\]\.\\displaystyle=\-\\nabla\_\{x\}\\\!\\cdot\\left\[\\left\(v^\{\*\}\(x,t\)\-\\frac\{1\}\{2T\}\\nabla\_\{x\}\\log p\_\{t\}\(x\)\\right\)p\_\{t\}\(x\)\\right\]\.This is the continuity equation associated with the probability\-flow ODE

d​x=\[v∗​\(xt,t\)−12​T​∇xt​log⁡pt​\(xt\)\]​d​τ\.dx=\\left\[v^\{\*\}\(x\_\{t\},t\)\-\\frac\{1\}\{2T\}\\nabla\_\{x\_\{t\}\}\\log p\_\{t\}\(x\_\{t\}\)\\right\]d\\tau\.\(78\)Thus, the prescribed marginal density pathptp\_\{t\}satisfies the density equation of this deterministic flow, with the same Gaussian initializationpT=𝒩⁡\(0,I\)p\_\{T\}=\\mathcal\{N\}\(0,I\)\.

It remains to express the score using the mean velocity\. The conditional Gaussian score is

∇x​log​pt​\(x∣x0\)=−Tt​\[x−\(1−tT\)​x0\]\.\\nabla\_\{x\}\\log p\_\{t\}\(x\\mid x\_\{0\}\)=\-\\frac\{T\}\{t\}\\left\[x\-\\left\(1\-\\frac\{t\}\{T\}\\right\)x\_\{0\}\\right\]\.\(79\)Sincept​\(x\)=𝔼x0∼data​\[pt​\(x∣x0\)\]p\_\{t\}\(x\)=\\mathbb\{E\}\_\{x\_\{0\}\\sim\\mathrm\{data\}\}\[p\_\{t\}\(x\\mid x\_\{0\}\)\], differentiation and Bayes’ rule give

∇x​log​pt​\(x\)\\displaystyle\\nabla\_\{x\}\\log p\_\{t\}\(x\)=𝔼x0∼data​\[pt​\(x∣x0\)​∇x​log⁡pt​\(x∣x0\)\]pt​\(x\)\\displaystyle=\\frac\{\\mathbb\{E\}\_\{x\_\{0\}\\sim\\mathrm\{data\}\}\\left\[p\_\{t\}\(x\\mid x\_\{0\}\)\\nabla\_\{x\}\\log p\_\{t\}\(x\\mid x\_\{0\}\)\\right\]\}\{p\_\{t\}\(x\)\}\(80\)=𝔼\[∇xlogpt\(x∣x0\)\|xt=x\]\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\left\.\\nabla\_\{x\}\\log p\_\{t\}\(x\\mid x\_\{0\}\)\\right\|x\_\{t\}=x\\right\]=−Tt​\[x−\(1−tT\)​𝔼​\[x0∣xt=x\]\]\.\\displaystyle=\-\\frac\{T\}\{t\}\\left\[x\-\\left\(1\-\\frac\{t\}\{T\}\\right\)\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}=x\]\\right\]\.By the definition of the exact marginal mean velocity,

v∗​\(x,t\)\\displaystyle v^\{\*\}\(x,t\)=𝔼\[x0−xt\|xt=x\]=𝔼⁡\[x0∣xt=x\]−xt,\\displaystyle=\\mathbb\{E\}\\\!\\left\[\\left\.\\frac\{x\_\{0\}\-x\}\{t\}\\right\|x\_\{t\}=x\\right\]=\\frac\{\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}=x\]\-x\}\{t\},\(81\)𝔼⁡\[x0∣xt=x\]\\displaystyle\\mathbb\{E\}\[x\_\{0\}\\mid x\_\{t\}=x\]=x\+t​v∗​\(x,t\)\.\\displaystyle=x\+t\\,v^\{\*\}\(x,t\)\.Substituting this relation into Equation equation[80](https://arxiv.org/html/2609.25444#A1.E80)yields

∇x​log​pt​\(x\)\\displaystyle\\nabla\_\{x\}\\log p\_\{t\}\(x\)=−Tt​\[x−\(1−tT\)​\(x\+t​v∗​\(x,t\)\)\]\\displaystyle=\-\\frac\{T\}\{t\}\\left\[x\-\\left\(1\-\\frac\{t\}\{T\}\\right\)\\left\(x\+t\\,v^\{\*\}\(x,t\)\\right\)\\right\]\(82\)=−Tt​\[tT​x−t⁡\(T−t\)T​v∗​\(x,t\)\]\\displaystyle=\-\\frac\{T\}\{t\}\\left\[\\frac\{t\}\{T\}x\-\\frac\{t\(T\-t\)\}\{T\}v^\{\*\}\(x,t\)\\right\]=\(T−t\)​v∗​\(x,t\)−x\.\\displaystyle=\(T\-t\)v^\{\*\}\(x,t\)\-x\.
Finally, evaluating this identity atx=xtx=x\_\{t\}and substituting it into Equation equation[78](https://arxiv.org/html/2609.25444#A1.E78)gives

d​x\\displaystyle dx=\[v∗​\(xt,t\)−\(T−t\)​v∗​\(xt,t\)−xt2​T\]​d​τ\\displaystyle=\\left\[v^\{\*\}\(x\_\{t\},t\)\-\\frac\{\(T\-t\)v^\{\*\}\(x\_\{t\},t\)\-x\_\{t\}\}\{2T\}\\right\]d\\tau\(83\)=\[T\+t2​T​v∗​\(xt,t\)\+xt2​T\]​d​τ\.\\displaystyle=\\left\[\\frac\{T\+t\}\{2T\}v^\{\*\}\(x\_\{t\},t\)\+\\frac\{x\_\{t\}\}\{2T\}\\right\]d\\tau\.Replacingv∗​\(xt,t\)v^\{\*\}\(x\_\{t\},t\)by the learned approximationuθ​\(xt,t\)/tu\_\{\\theta\}\(x\_\{t\},t\)/\\sqrt\{t\}gives the practical ODE

d​x=\[T\+t2​T​t​uθ​\(xt,t\)\+xt2​T\]​d​τ\.dx=\\left\[\\frac\{T\+t\}\{2T\\sqrt\{t\}\}u\_\{\\theta\}\(x\_\{t\},t\)\+\\frac\{x\_\{t\}\}\{2T\}\\right\]d\\tau\.\(84\)This completes the derivation\.□\\square

![Refer to caption](https://arxiv.org/html/2609.25444v1/dt32.png)Figure 3:Uncurated class\-conditional samples generated by MVM on ImageNet32×3232\\times 32\. The epoch\-200 EMA checkpoint is sampled using the stochastic SA\-Solver with 65 NFE\.![Refer to caption](https://arxiv.org/html/2609.25444v1/img256.png)Figure 4:Uncurated class\-conditional samples generated by MVM on ImageNet256×256256\\times 256\. The epoch\-80 EMA checkpoint is sampled using the stochastic SA\-Solver with 90 NFE\.