MaRS: A Fast Sampler for Mean Reverting Diffusion based on ODE and SDE Solvers

Ao Li; Ge Yang; Hongbo Zhao; Le Lu; Minfeng Xu; Wei Fang

REVIEW 2 major objections 5 minor 1 cited by

MaRS: A Fast Sampler for Mean Reverting Diffusion based on ODE and SDE Solvers

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Mean-Reverting Diffusion sampling drops from hundreds of steps to 5-10 with a training-free semi-analytical sampler.

desk verdict MaRS is a genuinely useful fast sampler for Mean Reverting Diffusion with sound algebra and strong empirical evidence, but the headline 5-10 NFE claim rests on a checkpoint-specific smoothness assumption that deserves scrutiny. read the letter →

arxiv 2502.07856 v4 pith:OM4KCGGC submitted 2025-02-11 cs.CV cs.AIcs.LG

Ao Li , Wei Fang , Hongbo Zhao , Le Lu , Ge Yang , Minfeng Xu This is my paper · ORCID

classification cs.CVcs.AIcs.LG

keywords mean-revertingdiffusionfastsamplingexponentialintegratorreverse-timeSDEprobabilityflowODEimagerestorationdatapredictionnoise

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Mean-Reverting (MR) Diffusion generates a restored image by running a stochastic differential equation whose drift pulls a noisy state toward a condition image; it gives natural control for image restoration, but sampling previously required hundreds of network evaluations because fast solvers for ordinary diffusion do not apply. The paper claims that the reverse-time SDE and the equivalent probability-flow ODE of MR Diffusion can be solved semi-analytically, leaving only an integral of the network output to be approximated numerically. That integral is estimated with an exponential-integrator Taylor expansion in log-SNR, and the rest of the update is computed in closed form. The resulting sampler needs 5-10 evaluations instead of hundreds, a 10-20x speedup across ten restoration tasks, with quality comparable to posterior sampling at 100 steps, and it requires no retraining. The paper further claims that among common parameterizations, data prediction is numerically stable at small step counts while noise prediction collapses.

What carries the argument

The central object is the semi-analytical solution of the mean-reverting reverse-time SDE and PF-ODE (Propositions 1-4). An exponential integrator is a discretization that evaluates the exponential of the linear part of an equation exactly and approximates only the nonlinear forcing; here the linear part is the mean-reversion drift toward $\mu$, and the forcing is the neural-network score term. Writing the solutions in terms of the half log-SNR $\lambda_t=\log(\alpha_t/\sigma_t)$ turns the remaining integral into the form that the paper estimates with a $k$-th order Taylor expansion in $\lambda$, with derivatives computed by backward differences. This machinery is what converts a sampler that previously committed Euler-Maruyama error in both the analytical and network parts into one that is exact except for the network integral.

What would settle it

Run MR Sampler-SDE-d-2 at NFE=5 on a data-prediction MR Diffusion model that was not among the checkpoints tested here, and compute, at each step, the fraction of output components whose Taylor-series convergence radius from Eq. (30) exceeds the step size $|\lambda_t-\lambda_s|$. If that fraction is far below the near-100% the paper reports, the sampler should produce visibly degraded or collapsed samples (LPIPS/FID near posterior-sampling levels at the same NFE) instead of the stable metrics reported.

Watch

Extended reading notes

Core claim

At the paper's center are four propositions. Propositions 1 and 2 solve the noise-prediction reverse-time SDE and PF-ODE for MR Diffusion; Propositions 3 and 4 solve the same pair for data prediction. In each solution the state at time $t$ is written as a closed-form affine function of the state at time $s$, the condition $\mu$, the decay $\alpha_t=e^{-\int_0^t f(\tau)d\tau}$, and the noise scale $\sigma_t=\sigma_\infty\sqrt{1-\alpha_t^2}$, plus one integral of the network output over log-SNR $\lambda_t=\log(\alpha_t/\sigma_t)$. MaRS approximates only that integral, using the exponential-integrator Taylor expansion of Eqs. (14) and (17), so the mean-reversion part of the dynamics is computed exactly. Because backward-difference derivative estimates are unstable when the network output oscillates in $\lambda$, the paper argues data prediction should be used: its output is bounded, its Taylor series satisfies the convergence-radius test in Section 5.3, and velocity prediction can be converted to either form by Eqs. (28)-(29).

Load-bearing premise

The load-bearing premise is that the trained model's output is smooth enough as the noise level changes that a low-order Taylor expansion over each step converges; the paper shows this empirically for data prediction on the tested checkpoints but it is not a guaranteed property of every MR Diffusion model.

Editorial extensions

If this is right

At 5-10 NFEs, MR Sampler reaches LPIPS/FID comparable to posterior sampling at 100 NFEs, giving a 10-20x wall-clock speedup on the ten tested restoration tasks.
MR Sampler works without retraining on existing MR Diffusion checkpoints and supports noise, data, and velocity prediction parameterizations through explicit transformation formulas.
Data prediction should be preferred at low NFEs: noise prediction collapses below roughly 20 NFEs because its network-output derivatives fail the Taylor convergence condition.
The semi-analytical update removes the Euler-Maruyama discretization error in the mean-reversion part; posterior sampling is shown to be a special Euler-Maruyama scheme, so MR Sampler's advantage grows as NFE shrinks.
Neither ODE- nor SDE-based variant dominates: the SDE solver is usually better at larger NFEs and the ODE solver at 5 NFEs, so the choice can be tuned per task.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A practical selection rule the paper does not propose: compute the Eq. (30) convergence-radius fraction per checkpoint and NFE; when it drops well below the reported near-100%, use data prediction or more steps instead of trusting the default setting.
The derivation only assumes the affine-drift mean-reversion structure and $g^2(t)/f(t)=2\sigma_\infty^2$, so the same semi-analytical solutions should apply to other mean-reverting diffusion settings, such as latent-space or video restoration, provided the network output stays smooth in log-SNR.
The stability asymmetry suggests a training-time regularizer that penalizes high-order log-SNR derivatives of the network output could make noise-prediction MR Diffusion samplable in few steps; whether that helps is a testable extension beyond this paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core of this paper is real and worth engaging with. For Mean Reverting Diffusion, the authors derive closed-form semi-analytical solutions for both the reverse-time SDE and the PF-ODE (Propositions 1-4), with the only remaining integral being the network-output term. That is a new, nontrivial extension of the DPM-Solver/UniPC line to an affine-drift SDE, and it directly removes the practical obstacle that MR Diffusion needed hundreds of NFEs. The derivations in Appendix A.1 are internally consistent; I checked the key identity g^2 = -2 sigma^2 dlambda/dt and the variation-of-constants steps, and they hold. The parameterization transformations (noise/data/velocity) are also clean, and the experiments across ten restoration tasks consistently show 10-20x speedups with quality matching posterior sampling at 50-100 NFEs. The wall-clock tables confirm the speedup is real, not just NFE counting. Credit where due: the paper ships code, and the empirical story is broad and convincing on the tested checkpoints.

The soft spot is exactly where the stress-test note lands: the exponential-integrator step in Eqs. (14) and (17) replaces the integral with a truncated Taylor expansion in log-SNR lambda, and the paper's justification for why this works at 5-10 steps is not a proof but a heuristic stability check (Section 5.3, Eq. 30) applied to specific checkpoints. The ratio test they use is only the n=0 term, and for noise prediction it visibly fails. For data prediction it works on the checkpoints tested, but that is an empirical property of those trained networks, not a guarantee of the framework. A differently trained MR Diffusion model could plausibly violate the smoothness assumption and degrade the sampler. The authors are honest that distillation still wins below 5 NFEs, and they note the quality ceiling is unchanged.

Two smaller issues: all metrics are single-run with no variance estimates, which matters when claiming parity between samplers, and there are a few appendix typos (e.g., Algorithm 2 line 8 buffers x_{t_i} after the loop ends, though it is harmless). These are minor next to the central derivation.

Who is this for? Anyone working on inverse problems or controllable generation with diffusion models, and anyone who wants a fast, training-free sampler for an affine-drift SDE. The main quantitative claim is strong enough to deserve serious referee scrutiny, and the unproven smoothness assumption is the right thing to push on in review. I would send it to review, ask for variance estimates and a released code snapshot, and check whether the stability criterion can be stated as a verifiable condition rather than a post-hoc diagnostic.

Referee Report

2 major / 5 minor

Summary. The paper proposes MaRS, a training-free fast sampler for Mean Reverting Diffusion (MR Diffusion). Starting from the reverse-time SDE and the probability-flow ODE associated with the MRSDE, the authors derive semi-analytical solutions (Propositions 1-4) in which the analytical drift terms are computed exactly and only an integral of the neural-network output must be approximated. The integral is handled with exponential-integrator Taylor expansions in the log-SNR variable λ, yielding first- and second-order samplers for noise, data, and velocity parameterizations. Experiments on ten image-restoration tasks report that MaRS reaches comparable quality to posterior sampling at 5-20 NFEs, giving 10-20x wall-clock speedups. The paper also shows that the commonly used posterior sampler is equivalent to an Euler-Maruyama discretization, and it offers a heuristic stability analysis explaining why data prediction is more robust than noise prediction at low NFE.

Significance. If the reported results hold, this is a useful contribution: existing fast samplers such as DPM-Solver and UniPC are not directly applicable to the MRSDE because of its affine drift, and the paper supplies the first training-free acceleration scheme with derivations that are internally consistent. The algebraic core is a genuine strength: Propositions 1-4 are proved in detail in Appendix A.1, the exponential-integrator discretizations in the algorithms match those propositions, and the authors explicitly identify the equivalence between the prior posterior-sampling algorithm and Euler-Maruyama. The public code, the breadth of the evaluation (ten tasks), and the honest limitation statements about distillation and the upper bound on sampling quality are also positive. The main risk is that the 5-10 NFE speedup is an empirical property of the specific checkpoints, not a guaranteed property of the algorithm; the paper's own stability analysis does not fully close that gap.

major comments (2)

[5.3, Eq. (30)] The numerical-stability explanation for data prediction is not established as stated. Equation (30) defines the radius of convergence through a limit of c_{n+1}/c_n, but the paper immediately approximates this with the n=0 term; for a neural-network output whose total derivatives along the sampling trajectory are neither computed nor bounded, this ratio is not a convergence test. Figure 4 and Appendix D.4 illustrate the criterion on one or a few images, so the conclusion that data prediction is stable is an empirical observation about the evaluated checkpoints, not a property of the MR Diffusion framework or of data-prediction training. I request that the text be reworded to state this explicitly, and that the ratio criterion be reported quantitatively across the ten tasks with multiple images (e.g., mean and spread of the convergence ratio per step), so a reader can judge how robust the claim is. This matters because the headline claim of 5-10 NFE quality depends on the network outputs being smooth in λ over steps as large as those used at low NFE.
[5.1 and Tables 6-15] No error bars, confidence intervals, or multiple seeds are reported for any metric. Because the SDE-based samplers in Algorithms 5 and 6 are stochastic, the reported LPIPS, FID, PSNR, and SSIM values are point estimates. In several low-NFE rows the difference between MR Sampler-1 and MR Sampler-2 is large enough that sampling noise could be material (e.g., Table 3, NFE=5 FID 27.44 vs 24.02; Table 12, NFE=5 FID 41.81 vs 33.98). I ask for at least three independent runs with means and standard deviations for the main comparisons, especially at NFE=5 and 10, to support the claim that quality is maintained at low NFE.

minor comments (5)

[Algorithms 1-4] The headers of Algorithms 1-4 state that they require a 'data prediction model xθ', but the update formulas use ϵθ and the initialization uses x_T = μ + σ₈ϵ. These are noise-prediction samplers and should be labeled accordingly.
[Algorithms 1-8] The lines 'Q buffer ← xθ(...)' appear in every algorithm but the buffer is never used. They are presumably vestigial from the DPM-Solver code base and should be removed or explained.
[Algorithm 6, line 4] After the first step, Algorithm 6 contains 'Q buffer ← xθ(xti, ti)' with an undefined index i; this is a typo and should be fixed.
[Section 4.3] The paper claims support for velocity prediction, but no experiment uses the velocity-prediction transformations from Section 4.3. A small experiment would substantiate the 'supports all mainstream parameterizations' claim.
[Figures 2 and 3 and Tables 6-15] The labels 'MR Sampler-1' and 'MR Sampler-2' are used without a definition in the main text and appendix; the reader has to infer they denote first- and second-order MaRS. Please define them at first use.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: MaRS derives exact SDE/ODE solutions and applies standard exponential-integrator approximations; the 5-10 NFE claim is empirical and not fitted or defined into the derivation.

full rationale

The paper's core derivation (Propositions 1-4, Appendices A.1-A.3) is self-contained algebra: exact solutions of the reverse-time SDE and PF-ODE are obtained by Ito's formula and variation of constants, and the only non-analytic piece is an integral over the neural-network output. This integral is approximated by the standard exponential-integrator Taylor expansion (Eqs. 14 and 17), following prior independent work by Lu et al. and Hochbruck-Ostermann; no constant in the sampler is fitted to the reported quality metrics. The claim that 5-10 NFEs suffice is an experimental finding evaluated against external posterior-sampling and Euler baselines on pre-trained checkpoints, not a consequence of the algebraic derivation. The Section 5.3 stability discussion is admittedly heuristic (Eq. 30 uses the n=0 term of a ratio test that requires a limit), but it is an after-the-fact diagnosis of noise-prediction failures and is not load-bearing for the derivation; it also does not smuggle in the result because the empirical superiority of data prediction is independently visible in Table 2. There is no self-citation chain used to justify the sampler, and the cited existence facts (Anderson 1982, Song et al. 2020) are standard external results. The paper's own limitations (Appendix E: quality collapses at NFE=2; distillation is better below 5 NFEs) are stated honestly and do not indicate circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central mathematical derivation rests on the MRSDE structural identity g^2/f = 2 sigma_8^2 and on smoothness of the network output in log-SNR. No new physical entities or fitted constants are introduced; algorithm order and solver choice are compared empirically.

assumptions (5)

domain assumption For all t in [0,T], g^2(t)/f(t) = 2 sigma_8^2, with f and g from the pre-trained MR Diffusion schedule.
Used to derive the closed-form transition in Eq.(7) and the crucial substitution g^2 = -2 sigma^2 dlambda/dt in Eq.(12). Taken from Luo et al. 2023b, not proved here.
domain assumption alpha_t is strictly decreasing and sigma_t strictly increasing, so lambda_t = log(alpha_t/sigma_t) has a unique inverse t(lambda).
Needed to reparametrize the integral over tau as an integral over lambda in Section 3.1. The paper states this is deliberately designed; no proof is given.
ad hoc to paper The network output epsilon_theta or x_theta, as a function of lambda, is smooth enough that its Taylor expansion truncated at order k with backward-difference derivatives converges over each step h.
This is the load-bearing numerical assumption behind Eq.(14). Section 5.3 shows it fails for noise prediction at low NFE, so it is empirically conditional on the checkpoint and parameterization.
standard math Anderson's reverse-time SDE and the existence of the probability flow ODE apply to the MRSDE.
Standard results from Song et al. 2020b and Anderson 1982, used to write Eq.(8) and Eq.(15).
domain assumption The learned score (equivalently noise, data, or velocity prediction) is accurate enough to substitute for the true score in Eqs. (9)-(10).
Standard in diffusion sampling; if the model is poorly trained the sampler cannot recover ground truth.

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Cite this review

Pith. "Pith review of MaRS: A Fast Sampler for Mean Reverting Diffusion based on ODE and SDE Solvers." pith.science (2026). https://pith.science/paper/OM4KCGGC

@misc{pith2026250207856,
  author       = {Pith},
  title        = {Pith review of: MaRS: A Fast Sampler for Mean Reverting Diffusion based on ODE and SDE Solvers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OM4KCGGC}},
  note         = {Machine review of arXiv:2502.07856}
}

read the original abstract

In applications of diffusion models, controllable generation is of practical significance, but is also challenging. Current methods for controllable generation primarily focus on modifying the score function of diffusion models, while Mean Reverting (MR) Diffusion directly modifies the structure of the stochastic differential equation (SDE), making the incorporation of image conditions simpler and more natural. However, current training-free fast samplers are not directly applicable to MR Diffusion. And thus MR Diffusion requires hundreds of NFEs (number of function evaluations) to obtain high-quality samples. In this paper, we propose a new algorithm named MaRS (MR Sampler) to reduce the sampling NFEs of MR Diffusion. We solve the reverse-time SDE and the probability flow ordinary differential equation (PF-ODE) associated with MR Diffusion, and derive semi-analytical solutions. The solutions consist of an analytical function and an integral parameterized by a neural network. Based on this solution, we can generate high-quality samples in fewer steps. Our approach does not require training and supports all mainstream parameterizations, including noise prediction, data prediction and velocity prediction. Extensive experiments demonstrate that MR Sampler maintains high sampling quality with a speedup of 10 to 20 times across ten different image restoration tasks. Our algorithm accelerates the sampling procedure of MR Diffusion, making it more practical in controllable generation.

Figures

Figures reproduced from arXiv: 2502.07856 by the authors.

**Figure 1.** Qualitative comparisons between MR Sampler and Posterior Sampling. All images are generated by sampling from a pre-trained MR Diffusion (Luo et al., 2024a) on the RESIDE-6k (Qin et al., 2020b) dataset and the CelebA-HQ (Karras, 2017) dataset. To improve the sampling efficiency of diffusion models, various acceleration strategies have been proposed, which can be divided into two categories. The first explores methods… view at source ↗

**Figure 2.** Perceptual evaluations on low-light and motion-blurry datasets. Motion Blurry Ground Truth Posterior Sampling (10) Euler Discretization (10) MR Sampler-1 (10) MR Sampler-2 (10) (a) Sampling results. 150 100 50 0 50 100 150 200 100 50 0 50 100 150 200 PC1 PC2 GroundTruth Posterior Sampling(100) Posterior Sampling(10) Euler Discretization(10) MR Sampler-1(10) MR Sampler-2(10) (b) Trajectory [PITH_FULL_IMAGE:figures/f… view at source ↗

**Figure 3.** Sampling trajectories. In (a), we compare our method (with order 1 and order 2) and previous sampling methods (i.e., posterior sampling and Euler discretization) on a motion blurry image. The numbers in parentheses indicate the NFE. In (b), we illustrate trajectories of each sampling method. Previous methods need to take many unnecessary paths to converge. With few NFEs, they fail to reach the ground truth (i.e., t… view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Convergence of noise prediction and data prediction. In (a), we choose a low-light image for example. The numbers in parentheses indicate the NFE. In (b), we illustrate the ratio of components of neural network output that satisfy the Taylor expansion convergence requi…

**Figure 5.** Figure 5: Convergence of noise prediction and data prediction at 5 NFEs. In (a), we choose a stained image for example. The numbers in parentheses indicate the NFE. In (b), we illustrate the ratio of components of neural network output that satisfy the Taylor expansion convergen…

**Figure 6.** Figure 6: Comparisons between posterior sampling and MR Sampler-SDE-d-2 on snowy, shadowed, rainy, raindrop, noisy and low-light datasets [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]

**Figure 7.** Figure 7: Comparisons between posterior sampling and textitMR Sampler-SDE-d-2 on JPEGcompressed, hazy, inpainting and motion-blurry datasets. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_7.png]

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e´ şt 0 fpτqdτ . Proof. For ODEs which have a semi-linear structure as follows: dx dt “ Pptqx` Qpx, tq, (34) the method of

Given an initial value xs at time s P r0, Ts, the solution xt at time t P r0, ss of Eq.(15) is xt“ αt αs xs`p 1´ αt αs qµ` αt ż t s g2pτq 2ατ στ ϵθpxτ , τqdτ, (33) where αt :“ e´ şt 0 fpτqdτ . Proof. For ODEs which have a semi-linear structure as follows: dx dt “ Pptqx` Qpx, tq, (34) the method of ”variation of constants” gives the following solution: xpt...

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Given an initial value xs at time s P r0, Ts, the solution xt at time t P r0, ss of Eq.(22) is xt“ σt σs e´pλt´λsqxs` µ ˆ 1´ αt αs e´2pλt´λsq´ αt` αte´2pλt´λsq ˙ `2αt ż λt λs e´2pλt´λqxθpxλ, λqdλ` σt a 1´ e´2pλt´λsqz, (36) where z„ Np0, Iq. Proof. According to Eq.(32), we define uptq“ g2ptq σ2 t ´ fptq and ψpx, tq“ xe şt 0 upτqdτ . We substitute fpx, tq a...

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(44) Proof

Given an initial value xs at time s P r0, Ts, the solution xt at time t P r0, ss of Eq.(24) is xt“ σt σs xs` µ ˆ 1´ σt σs ` σt σs αs´ αt ˙ ` σt ż λt λs eλxθpxλ, λqdλ. (44) Proof. Note that Eq.(24) shares the same structure as Eq.(34). Let Pptq“ g2ptq 2σ2 t ´ fptq, and Qpx, tq“ „ fptq´ g2ptq 2σ2 t p1´ αtq ȷ µ´ g2ptq 2σ2 t αtxθpxt, tq. According to Eq.(12),...

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(2017) 1111 Hazy RESIDE-6k Qin et al

Task name Dataset name Reference Number of testing images Blurry GoPro Nah et al. (2017) 1111 Hazy RESIDE-6k Qin et al. (2020a) 1000 JPEG-compressing DIV2K, Flickr2K and LIVE1 Agustsson & Timofte (2017),Timofte et al. (2017), Sheikh (2005) 29 Low-light LOL Wei et al. (2018) 15 Noisy DIV2K, Flickr2K and CBSD68 Agustsson & Timofte (2017),Timofte et al. (201...

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25 Published as a conference paper at ICLR 2025 Stained Ground Truth Noise prediction (5) Data prediction (5) (a) Sampling results

We choose one image for each task. 25 Published as a conference paper at ICLR 2025 Stained Ground Truth Noise prediction (5) Data prediction (5) (a) Sampling results. 0 1 2 3 0.0 0.2 0.4 0.6 0.8 1.0 Index of timestep Noise prediction Data prediction (b) Ratio of convergence. Figure 5: Convergence of noise prediction and data prediction at 5 NFEs. In (a), ...

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Denoising diffusion implicit models

Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020a. Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems, 32,

arXiv 2010

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Score-based generative modeling through stochastic differential equations

Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456, 2020b. Yang Song, Conor Durkan, Iain Murray, and Stefano Ermon. Maximum likelihood training of score-based diffusion models. Advances in neural inform...

arXiv 2011