CHORDS: Diffusion Sampling Accelerator with Multi-core Hierarchical ODE Solvers

Haotian Ye; James Zou; Jiaqi Han; Minkai Xu; Puheng Li; Stefano Ermon

REVIEW 4 major objections 5 minor 50 references

CHORDS: Diffusion Sampling Accelerator with Multi-core Hierarchical ODE Solvers

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Diffusion models can be sampled 2-3x faster on parallel cores with no retraining, because slow, accurate solvers hierarchically correct fast, approximate ones.

desk verdict Credible speedups and a clean unifying framework, but the exact-identity guarantee is contradicted by the paper's own scheduler. read the letter →

arxiv 2507.15260 v1 pith:JFKXJT4W submitted 2025-07-21 cs.LG

Jiaqi Han , Haotian Ye , Puheng Li , Minkai Xu , James Zou , Stefano Ermon This is my paper

classification cs.LG MSC 65L0565Y0568T07

keywords diffusionsamplingaccelerationmulti-coreparallelismtraining-freeODEsolvershierarchicalrectificationparalleltext-to-imagegenerationtext-to-video

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

CHORDS claims that the expensive iterative denoising loop of a diffusion model can be made two to three times faster simply by running several copies of the sampler in parallel on separate GPU cores, without retraining the model and without measurable loss of output quality. Its mechanism is a hierarchy of solvers ordered from slow and accurate to fast and approximate, in which each slower core periodically corrects the next faster core's latent state using a rectification update whose error-reducing property is proved for continuous ODE solutions. From this framework the authors extract a closed-form recipe for where each core should start and show that it yields 2.0-2.9x speedups across three video models and two image models with four to eight cores, beating prior parallel samplers by up to 50% while keeping VBench and CLIP quality scores essentially flat. If the claim holds, any deployment with several idle GPUs can accelerate diffusion inference without touching the model.

What carries the argument

The load-bearing object is the rectification pair: the update rule $r_\theta$ of Equations (3)-(4) together with the slow-to-fast core hierarchy it links. Whenever a slower core catches up to a faster one, the faster core's already-advanced latent is corrected by $r_\theta(x^k_{t'}, x^{k+1}_{t'}, t', \delta t)$, and because each corrected core in turn rectifies the next faster core, accurate information propagates down the chain without idle bubbles in the pipeline. The second essential piece is the parameter-free initialization recipe of Theorem 2.5, derived from the linear case $f_\theta(x,t) = x$: the fastest core starts at $t^{(K)} = (s-1)/s$ for target speedup $s$, and earlier cores are placed by the local rule $t^{(k)} = t^{(k+1)}/2$ or $t^{(k)} = 2t^{(k+1)} - t^{(k+2)}$, producing concrete schedules such as $[0, 8, 16, 32]$ for four cores on 50 steps. The theory the recipe rests on is Proposition 2.1 (rectification shrinks the error of an inaccurate start) and the reward axioms of Definition 2.4 (monotonicity and trade-off) that justify using the solvable linear surrogate in place of the true, intractable error.

What would settle it

Fix a model, prompt, and seed, run the sequential 50-step solver and then Algorithm 1 with, say, eight cores, and compare the slowest core's final latent to the sequential latent element by element: the remark's guarantee predicts a latent RMSE of exactly zero, whereas any nonzero RMSE would show that the discrete Scheduler's early large jumps (for example, from step 0 to step 8) divert the 'slow' core from the sequential trajectory and void the guarantee. A second check is the small-step regime: if the fast core's VBench or CLIP score drops measurably when the total step count $N$ falls from 50 to 20, then the no-quality-degradation claim holds only on fine grids.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that multi-core diffusion acceleration can be organized as a single rectification pipeline governed by one design choice: the initialization sequence $I = [t^{(1)}, \ldots, t^{(K)}]$ that fixes how much faster each successive core is. The core operation is the update $r_\theta(x_t, \tilde{x}_t, t, \delta t) = \delta t\,(f_\theta(x_t, t) - f_\theta(\tilde{x}_t, t)) + x_t - \tilde{x}_t$, applied when a slower core reaches a faster core's time; Proposition 2.1 shows this reduces the fast core's error to $o(\|\tilde{x}_{t'} - x_{t'}\|_2)$. For the linear surrogate $f_\theta(x,t) = x$ with reward $R(I) = \sum_d \ln x^K_{1,d}$, Theorem 2.5 gives optimal start times—for a target speedup $s$ with three cores, $t^{(3)} = (s-1)/s$ and $t^{(2)} = t^{(3)}/2$ if $s \le 3$, otherwise $t^{(2)} = 2t^{(3)} - 1$—and a recursion extends the recipe to any number of cores. Discrete instantiations of the scheme report 2.0-2.9x speedups on HunyuanVideo, Wan2.1, CogVideoX1.5, SD3.5-Large, and Flux with four to eight cores, with quality scores essentially unchanged and latent RMSE well below the strongest baseline, and the paper asserts in a remark on Algorithm 1 that the pipeline's last output is identical to the sequential sampler's output.

Load-bearing premise

The load-bearing premise is that the error-reduction analysis, done for continuous ODEs with infinitesimal steps, transfers intact to the finite and uneven step sizes the implemented algorithm actually uses—in particular, that the slowest core really follows the full sequential path so its final output equals the sequential sampler's output exactly, as the remark on Algorithm 1 guarantees.

Editorial extensions

If this is right

Existing diffusion models can be sampled 2.0-2.9x faster by adding four to eight parallel cores, with VBench and CLIP quality scores essentially unchanged across image and video models.
The acceleration is training-free and model-agnostic, so it composes with distillation and with splitting model parameters across cores, as the paper explicitly notes.
Because outputs arrive progressively from fast to slow cores and the slowest core's result is claimed to equal the sequential output, deployments can stop early for latency or wait for the exact sequential result as a fallback.
Larger step budgets favor the method: at 75 and 100 total steps the measured eight-core speedup on HunyuanVideo rises to 3.4x and 3.6x, so the gain grows exactly where fidelity demands are highest.
The framework treats the two prior parallel samplers, SRDS and ParaDIGMS, as special instantiations and improves on their speed by up to 50% at eight cores.

Reading between the lines

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

Beyond the paper: the closed-form initialization recipe is the most consequential piece—if it transfers to other solver families such as high-order exponential integrators, the method becomes a drop-in inference flag rather than a per-model tuning exercise, which the paper only demonstrates for DDIM and Euler.
Beyond the paper: deployed wall-clock gains will likely trail the headline ratios, because the headline speedup counts sequential network forward calls while real runs pay synchronization and communication costs; the paper's own 'time per sample' columns are the better planning number.
Beyond the paper: a natural stress test is the small-step regime ($N = 10$-$20$), where the rectification's $o(\delta t)$ error reduction must survive large step sizes; if the fast core's quality collapses there, the method's safe operating range is bounded by $N$.
Beyond the paper: because the slowest core is an exact sequential fallback, CHORDS could double as a quality monitor—compare early fast-core outputs against the pending slow-core output to decide when to stop—a use the paper's streaming discussion motivates but does not develop.

Share X Bluesky LinkedIn Reddit HN

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 paper is worth a careful read. It unifies two existing parallel diffusion samplers (SRDS and ParaDIGMS) under a single rectification framework, and the empirical results are substantial: 2–2.9x training-free speedups on HunyuanVideo, Wan2.1, CogVideoX, SD3.5, and Flux, with VBench and CLIP scores essentially flat and low latent RMSE. If those numbers hold up, this is a practically useful inference-acceleration tool. The linear-ODE reward analysis in Theorems 2.5 and the initialization recursion give a principled way to choose core start times—something the prior methods did not offer. I also appreciate that the paper frames the slower-core rectification clearly and mentions that the framework is a generalization of prior work; the citation pattern looks fair.

The soft spots are real, but they are concentrated and fixable. The biggest one is the Remark on Algorithm 1, which claims the last output is guaranteed to be identical to the sequential run. As printed, Equation (7) does not deliver that. The scheduler sends every core—including the slowest—through the same coarse early jumps (0->8->16->32 for the four-core setting), and then the indexing becomes internally inconsistent after step K. So the slowest core is not executing the sequential 50-step trajectory, and the identity claim is false as written. That matters because this remark is the only formal quality-safety argument; the empirical quality measurements are what actually support the 'without quality degradation' claim. The stress-test note is right: the intended condition may have been 'step < k', but the paper must say so and prove the identity. This is a load-bearing bug, not a typo.

Other concerns are more moderate. No error bars are reported on any of the quality metrics or latencies, and the differences between methods are small enough that noise could matter. The continuous theory (Prop 2.1, Thm 2.5) covers exact solvers, while the implementation is discrete; the transfer is plausible but not rigorously justified. Also, the SRDS baseline is modified (pipelined) rather than used out-of-the-box, which is a reasonable choice but should be made more prominent.

Who is this for? Anyone working on diffusion inference acceleration, especially for deployment on multi-GPU setups. The empirical contribution is strong enough to deserve a serious referee, and the flaws are correctable. The authors should fix the scheduler, prove the identity condition or retract it, add error bars, and tighten the discrete–continuous discussion. I'd engage with this paper and would bring it to our reading group.

Referee Report

4 major / 5 minor

Summary. The paper proposes CHORDS, a training-free multi-core acceleration method for diffusion model sampling. The core idea is to run several ODE solvers of different speeds on different cores, initialized at increasing times, and to use the slower, more accurate solver to rectify the faster solver through a communication rule (Eq. 3). The authors give a continuous-time framework, a greedy initialization-sequence selection based on a scalar reward function, and a discrete instantiation in Algorithm 1. They evaluate CHORDS on three video diffusion models and two image diffusion models with 4, 6, and 8 cores, reporting speedups up to 2.9x with eight cores and small latent RMSE relative to the sequential solver. The paper also claims in the Remark on Algorithm 1 that the last output is guaranteed to be identical to the sequential output.

Significance. If the technical issues are resolved, this is a potentially valuable contribution: it offers a training-free, model-agnostic way to reduce diffusion sampling latency, with a streaming output structure and a unified view that subsumes prior parallel sampling methods. The experimental scope is broad, covering five modern large-scale models and both image and video generation, and the authors provide code and demos. The paper also reports latent RMSE against the sequential oracle, which is a useful fidelity check. However, the central formal guarantee is currently not supported by the printed algorithm, and the theory is stated in continuous time while the implementation is discrete. These issues need to be fixed before the paper's main claims can be accepted.

major comments (4)

[§3, Algorithm 1, Eq. (7), and Remark on Algorithm 1] The Remark asserts that "the last output is guaranteed to be identical to the output when there is no multi-core acceleration," but this does not follow from the printed Scheduler. In Eq. (7), for step < K every core, including the slowest core k=1, receives the pair (i_step, i_step+1). With the reported initialization I_hat=[0,8,16,32] and N=50, core 1 takes the jumps 0->8->16->32 during steps 1-3 before switching to the branch i_k + step - k at step 4, which is indexing-inconsistent because the trajectory has already reached time 32 but is then asked to continue from time 3. Thus the slowest core's trajectory is not the sequential 50-step discretization, and the exact-identity guarantee is false as written. The authors should correct the Scheduler so that core 1 follows the sequential schedule and prove the identity, for example by distinguishing the behavior for step < k and adding an explicit invariant, or they should remove or substantially weaken the guarantee.
[§2.1, Proposition 2.1, versus §3, Algorithm 1] The theoretical justification of rectification is stated for continuous ODE solvers with indefinitely small time steps, while Algorithm 1 uses a finite number of discrete steps with non-uniform intervals. Proposition 2.1 derives an o(||tilde_x_t' - x_t'||) error reduction from Taylor expansions around a single point; it does not automatically carry over to the discrete update in Eq. (3) when the steps are finite and the vector field is a learned network evaluated at discrete latents. No discrete local-error analysis is given. Since the "without measurable quality degradation" claim rests on this transfer, the paper needs a discrete error bound or an explicit argument that the rectification reduces one-step error for the implemented schedulers.
[§4, Tables 1 and 2, and the claim "without measurable quality degradation"] The central empirical claim of no quality degradation is supported only by single-point estimates. VBench quality scores, CLIP scores, and latent RMSE values are reported without error bars, significance tests, or per-prompt variability. For example, HunyuanVideo at K=8 drops from 84.4% to 84.1% in VBench while the latent RMSE is 0.068; without confidence intervals it is impossible to assess whether this is measurable degradation. I ask the authors to report means and standard deviations over multiple seeds or prompt subsets, and to state the number of samples used for each metric.
[§2.3, Definition 2.4, Theorem 2.5, and §4.3] The optimal-initialization result is proven only for the scalar linear model f_theta(x,t)=x with x_0=1 and the specific reward R(I)=sum_d ln x^K_{1,d}. For general nonlinear f_theta, the reward is a heuristic surrogate, and the greedy fast-to-slow recursion for K>3 is not shown to be optimal. The paper acknowledges that R is a surrogate, but Section 4.3 then describes the resulting sequence as "theoretically derived" and "optimal" in the text around Eq. (5). Please state clearly which parts are proven and which are heuristic, and consider an ablation with additional non-uniform sequences beyond the uniform baseline to support the claim that the specific choice is essential.

minor comments (5)

[§3, Algorithm 1 line 6] There is a typo: "prev, ← Scheduler" should be "prev ← Scheduler(...)", and the variable name should be consistent with the use on line 7.
[§2.2, Framework 2.2 item 4] The notation in the communication rule is hard to parse: the subscript of x^k_{t^{(k)}+nδ^{(k)}} and the arguments of r_θ are not fully aligned, and the index n is defined only for positive integers. Please define the indices more precisely.
[Appendix A.1] The proof heading says "Proof of Lemma 2.1" but the main text refers to Proposition 2.1. Please align the numbering.
[§3, Initialization paragraph] The speedup expression 1 / (1 - t(i_k) + (k-1)/N) mixes continuous time and discrete step counts; since t(i_k) is in [0,1] and (k-1)/N is a number of steps, the formula is not dimensionally consistent as written. Please clarify the definition.
[Figure 2] The caption says "the number in marks represents the wall-clock time of that solver (multiplied by 10)", which is unclear; I assume "marks" refers to the labels in the figure. Please rephrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: CHORDS' theory is self-contained and its speedups are direct measurements, not outputs of a fitted model.

full rationale

The paper's derivation chain is not circular. The rectification rule (Eqs. 3-4) is stated as a definition and then proven in Proposition 2.1 with a Taylor-expansion argument; the proof does not assume the conclusion. The initialization selection is derived from an explicitly defined surrogate reward R(I)=sum ln x^K_{1,d} for the toy vector field f_theta(x,t)=x, with Theorem 2.5 proved in the appendix under that model; although this is a proxy that may not transfer to discrete neural samplers, it is not fitted to the reported speedups or quality metrics. The headline speedups (2.1x and 2.9x) are measured wall-clock times in Tables 1-2 against sequential and baseline samplers, so they are empirical observations rather than predictions forced by a fitted parameter. Existing methods [29,30] are described as special cases of the framework, and while those citations share authors with this paper, the framework does not depend on them for its validity; their use as benchmark baselines is not load-bearing. The Remark's guarantee that the last output equals the sequential output is a potential correctness gap, since the discrete Scheduler in Eq. 7 appears inconsistent with that claim, but this is a bug or omitted proof, not a circular reduction: no equation in the paper defines the claimed output in terms of itself. Accordingly, no circularity is found.

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

The central claims rest on a smoothness assumption, a linear surrogate for choosing initialization, a heuristic recursion for many cores, and an unproved identity between the slowest core and the sequential sampler. No free parameters are fitted; all speedups are directly measured.

assumptions (5)

domain assumption fθ is sufficiently smooth for Taylor expansion in Proposition 2.1.
Proposition 2.1 requires a second-order Taylor expansion of fθ; neural networks are not guaranteed smooth, though in practice their outputs are locally Lipschitz.
domain assumption The surrogate ODE fθ(x,t)=x with x0=1 captures the essential trade-offs of real diffusion sampling.
Section 2.3 defines the reward R(I) using this linear equation; the authors claim it mirrors practical reality but provide no formal transfer argument.
ad hoc to paper The greedy fast-to-slow recursion for K>3 gives near-optimal initialization.
Theorem 2.5 only proves optimality for K=3; the recursion in Section 2.3 is a heuristic decomposition, stated without proof for general K.
ad hoc to paper The slowest core in the discrete Scheduler follows the exact sequential discretization.
The Remark on Algorithm 1 asserts the last output is identical to the sequential output, but the Scheduler (Equation 7) implies large non-uniform jumps for early steps, so this identity is not demonstrated.
domain assumption Continuous-time solver analysis transfers to discrete finite-step solvers.
The framework is derived in the continuous limit with indefinitely small steps (Section 2), but the implementation uses DDIM or Euler with finite N=50; no discrete error bound is provided.

how reviews work

0 comments

Cite this review

Pith. "Pith review of CHORDS: Diffusion Sampling Accelerator with Multi-core Hierarchical ODE Solvers." pith.science (2026). https://pith.science/paper/JFKXJT4W

@misc{pith2026250715260,
  author       = {Pith},
  title        = {Pith review of: CHORDS: Diffusion Sampling Accelerator with Multi-core Hierarchical ODE Solvers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JFKXJT4W}},
  note         = {Machine review of arXiv:2507.15260}
}

read the original abstract

Diffusion-based generative models have become dominant generators of high-fidelity images and videos but remain limited by their computationally expensive inference procedures. Existing acceleration techniques either require extensive model retraining or compromise significantly on sample quality. This paper explores a general, training-free, and model-agnostic acceleration strategy via multi-core parallelism. Our framework views multi-core diffusion sampling as an ODE solver pipeline, where slower yet accurate solvers progressively rectify faster solvers through a theoretically justified inter-core communication mechanism. This motivates our multi-core training-free diffusion sampling accelerator, CHORDS, which is compatible with various diffusion samplers, model architectures, and modalities. Through extensive experiments, CHORDS significantly accelerates sampling across diverse large-scale image and video diffusion models, yielding up to 2.1x speedup with four cores, improving by 50% over baselines, and 2.9x speedup with eight cores, all without quality degradation. This advancement enables CHORDS to establish a solid foundation for real-time, high-fidelity diffusion generation.

Figures

Figures reproduced from arXiv: 2507.15260 by the authors.

**Figure 1.** We apply CHORDS, our proposed multi-core diffusion sampling accelerator on state-of-the-art video generation model (HunyuanVideo) and image generation model (Flux). It achieves a significant 2.1× ∼ 2.9× speedup with four to eight computation cores. Abstract Diffusion-based generative models have become dominant generators of high-fidelity images and videos but remain limited by their computationally expensive infer… view at source ↗

**Figure 2.** An illustration of Framework 2.2 with K = 4 cores. According to Section 2.3, four cores are initialized at time I = [0, 0.2, 0.4, 0.7]. Then, each core solves forward simultaneously, where the number in marks represents the wall-clock time of that solver (multiplied by 10). Solid lines denote the continuous solve within a core, and dash lines across cores denote the rectification technique proposed in Section 2.1. T… view at source ↗

**Figure 3.** Qualitative comparisons with baselines on CogVideoX1.5. Our approach achieves the best quality and the lowest latency. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Scaling of CHORDS towards different numbers of cores. puting provides another promising approach to accelerate diffusion by distributing sampling computations across multiple cores. Existing parallel sampling methods include Picard iteration [30], Anderson accelerati…

**Figure 5.** Figure 5: Convergence curve on HunyuanVideo and SD3.5-Large. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: More results on image diffusion model Flux. [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

**Figure 7.** Figure 7: More results on video diffusion model HunyuanVideo. [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

50 extracted references · 28 canonical work pages

[1]

Multi-level adaptive solutions to boundary- value problems

Achi Brandt. Multi-level adaptive solutions to boundary- value problems. Mathematics of computation, 31(138):333– 390, 1977. 3

work page 1977
[2]

Scaling recti- fied flow transformers for high-resolution image synthesis

Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas M ¨uller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, et al. Scaling recti- fied flow transformers for high-resolution image synthesis. In Forty-first international conference on machine learning,
[3]

xdit: an inference engine for diffusion trans- formers (dits) with massive parallelism

Jiarui Fang, Jinzhe Pan, Xibo Sun, Aoyu Li, and Jian- nan Wang. xdit: an inference engine for diffusion trans- formers (dits) with massive parallelism. arXiv preprint arXiv:2411.01738, 2024. 2, 8

arXiv 2024
[4]

A multigrid method for multiprocessors

A Greenbaum. A multigrid method for multiprocessors. Ap- plied mathematics and computation, 19(1-4):75–88, 1986. 3

work page 1986
[5]

Geometric trajectory diffusion models

Jiaqi Han, Minkai Xu, Aaron Lou, Haotian Ye, and Stefano Ermon. Geometric trajectory diffusion models. Advances in Neural Information Processing Systems, 37:25628–25662,

work page
[6]

CLIPScore: a reference-free evaluation met- ric for image captioning

Jack Hessel, Ari Holtzman, Maxwell Forbes, Ronan Le Bras, and Yejin Choi. CLIPScore: a reference-free evaluation met- ric for image captioning. In EMNLP, 2021. 6

work page 2021
[7]

Denoising dif- fusion probabilistic models

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising dif- fusion probabilistic models. Advances in neural information processing systems, 33:6840–6851, 2020. 2, 3

2020
[8]

Video dif- fusion models

Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J Fleet. Video dif- fusion models. Advances in Neural Information Processing Systems, 35:8633–8646, 2022. 2

2022

Show all 50 references

[9]

VBench: Com- prehensive benchmark suite for video generative models

Ziqi Huang, Yinan He, Jiashuo Yu, Fan Zhang, Chenyang Si, Yuming Jiang, Yuanhan Zhang, Tianxing Wu, Qingyang Jin, Nattapol Chanpaisit, Yaohui Wang, Xinyuan Chen, Limin Wang, Dahua Lin, Yu Qiao, and Ziwei Liu. VBench: Com- prehensive benchmark suite for video generative models....

2024
[10]

Open- clip, 2021

Gabriel Ilharco, Mitchell Wortsman, Ross Wightman, Cade Gordon, Nicholas Carlini, Rohan Taori, Achal Dave, Vaishaal Shankar, Hongseok Namkoong, John Miller, Han- naneh Hajishirzi, Ali Farhadi, and Ludwig Schmidt. Open- clip, 2021. If you use this software, please cite it as below. 6

2021
[11]

Picard iteration, chebyshev polynomials and chebyshev-picard methods: Application in astrodynam- ics

John L Junkins, Ahmad Bani Younes, Robyn M Woollands, and Xiaoli Bai. Picard iteration, chebyshev polynomials and chebyshev-picard methods: Application in astrodynam- ics. The Journal of the Astronautical Sciences, 60:623–653,
[12]

Elucidating the design space of diffusion-based generative models

Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion-based generative models. Advances in Neural Information Processing Sys- tems, 35:26565–26577, 2022. 2

2022
[13]

Consistency trajectory mod- els: Learning probability flow ode trajectory of diffusion

Dongjun Kim, Chieh-Hsin Lai, Wei-Hsiang Liao, Naoki Mu- rata, Yuhta Takida, Toshimitsu Uesaka, Yutong He, Yuki Mitsufuji, and Stefano Ermon. Consistency trajectory mod- els: Learning probability flow ode trajectory of diffusion. arXiv preprint arXiv:2310.02279, 2023. 2, 8

arXiv
[14]

Hunyuanvideo: A systematic framework for large video generative models

Weijie Kong, Qi Tian, Zijian Zhang, Rox Min, Zuozhuo Dai, Jin Zhou, Jiangfeng Xiong, Xin Li, Bo Wu, Jianwei Zhang, et al. Hunyuanvideo: A systematic framework for large video generative models. arXiv preprint arXiv:2412.03603, 2024. 2, 6

2024 arXiv
[15]

Diffwave: A versatile diffusion model for audio synthesis

Zhifeng Kong, Wei Ping, Jiaji Huang, Kexin Zhao, and Bryan Catanzaro. Diffwave: A versatile diffusion model for audio synthesis. In International Conference on Learning Representations, 2021. 2

2021
[16]

Black Forest Labs. Flux. https://github.com/ black-forest-labs/flux, 2024. 6

2024
[17]

Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maxim- ilian Nickel, and Matthew Le. Flow matching for genera- tive modeling. In The Eleventh International Conference on Learning Representations, 2023. 3

2023
[18]

Audi- oldm: Text-to-audio generation with latent diffusion models

Haohe Liu, Zehua Chen, Yi Yuan, Xinhao Mei, Xubo Liu, Danilo Mandic, Wenwu Wang, and Mark D Plumbley. Audi- oldm: Text-to-audio generation with latent diffusion models. arXiv preprint arXiv:2301.12503, 2023. 2

2023 arXiv
[19]

Flow straight and fast: Learning to generate and transfer data with rectified flow

Xingchao Liu, Chengyue Gong, and qiang liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. In The Eleventh International Conference on Learning Representations, 2023. 3

2023
[20]

Simplifying, stabilizing and scaling continuous-time consistency models

Cheng Lu and Yang Song. Simplifying, stabilizing and scaling continuous-time consistency models. In The Thir- teenth International Conference on Learning Representa- tions, 2025. 8

2025
[21]

Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps

Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongxuan Li, and Jun Zhu. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps. Advances in Neural Information Processing Systems , 35:5775–5787,
[22]

Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models

Cheng Lu, Yuhao Zhou, Fan Bao, Jianfei Chen, Chongx- uan Li, and Jun Zhu. Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models. arXiv preprint arXiv:2211.01095, 2022. 2, 8

2022 arXiv
[23]

Regiondrag: Fast region-based image editing with diffusion models

Jingyi Lu, Xinghui Li, and Kai Han. Regiondrag: Fast region-based image editing with diffusion models. In Eu- ropean Conference on Computer Vision , pages 231–246. Springer, 2024. 2

2024
[24]

Sdedit: Guided image synthesis and editing with stochastic differential equa- tions

Chenlin Meng, Yutong He, Yang Song, Jiaming Song, Jia- jun Wu, Jun-Yan Zhu, and Stefano Ermon. Sdedit: Guided image synthesis and editing with stochastic differential equa- tions. arXiv preprint arXiv:2108.01073, 2021. 2

2021 arXiv
[25]

Learning transferable visual models from natural language supervi- sion

Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervi- sion. In International conference on machine learning, ...

2021
[26]

High-resolution image synthesis with latent diffusion models

Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bj ¨orn Ommer. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10684–10695, 2022. 2

2022
[27]

Mm-diffusion: Learning multi-modal diffusion mod- els for joint audio and video generation

Ludan Ruan, Yiyang Ma, Huan Yang, Huiguo He, Bei Liu, Jianlong Fu, Nicholas Jing Yuan, Qin Jin, and Baining Guo. Mm-diffusion: Learning multi-modal diffusion mod- els for joint audio and video generation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern...

2023
[28]

Progressive distillation for fast sampling of diffusion models

Tim Salimans and Jonathan Ho. Progressive distillation for fast sampling of diffusion models. arXiv preprint arXiv:2202.00512, 2022. 2, 8

2022 arXiv
[29]

Self-refining diffusion samplers: Enabling parallelization via parareal iterations

Nikil Roashan Selvam, Amil Merchant, and Stefano Ermon. Self-refining diffusion samplers: Enabling parallelization via parareal iterations. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. 2, 3, 4, 6, 7, 8

2024
[30]

Parallel sampling of diffusion models

Andy Shih, Suneel Belkhale, Stefano Ermon, Dorsa Sadigh, and Nima Anari. Parallel sampling of diffusion models. Ad- vances in Neural Information Processing Systems, 36, 2024. 2, 3, 4, 6, 7, 8

2024
[31]

Deep unsupervised learning using nonequilibrium thermodynamics

Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In International confer- ence on machine learning, pages 2256–2265. PMLR, 2015. 2

2015
[32]

Denoising diffusion implicit models

Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. arXiv preprint arXiv:2010.02502, 2020. 2, 3, 6

2010 arXiv
[33]

Improved techniques for training consistency models

Yang Song and Prafulla Dhariwal. Improved techniques for training consistency models. In The Twelfth International Conference on Learning Representations, 2024. 8

2024
[34]

Generative modeling by esti- mating gradients of the data distribution

Yang Song and Stefano Ermon. Generative modeling by esti- mating gradients of the data distribution. Advances in neural information processing systems, 32, 2019. 3

2019
[35]

Consistency models

Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models. In International Conference on Machine Learning, 2023. 2, 8

2023
[36]

Yang et al. Song. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456, 2020. 2, 3

2011 arXiv
[37]

Accelerating parallel sampling of dif- fusion models

Zhiwei Tang, Jiasheng Tang, Hao Luo, Fan Wang, and Tsung-Hui Chang. Accelerating parallel sampling of dif- fusion models. In Forty-first International Conference on Machine Learning, 2024. 4, 8

2024
[38]

Genmo Team. Mochi 1. https: // github. com/ genmoai/models, 2024. 2

2024
[39]

Wan: Open and advanced large-scale video gen- erative models

Wan Team. Wan: Open and advanced large-scale video gen- erative models. 2025. 6

2025
[40]

EM distillation for one-step diffusion models

Sirui Xie, Zhisheng Xiao, Diederik P Kingma, Tingbo Hou, Ying Nian Wu, Kevin Patrick Murphy, Tim Salimans, Ben Poole, and Ruiqi Gao. EM distillation for one-step diffusion models. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. 8

2024
[41]

Diffusion models: A comprehensive survey of methods and applications

Ling Yang, Zhilong Zhang, Yang Song, Shenda Hong, Run- sheng Xu, Yue Zhao, Wentao Zhang, Bin Cui, and Ming- Hsuan Yang. Diffusion models: A comprehensive survey of methods and applications. ACM computing surveys, 56(4): 1–39, 2023. 2

2023
[42]

Cogvideox: Text-to-video diffusion models with an expert transformer

Zhuoyi Yang, Jiayan Teng, Wendi Zheng, Ming Ding, Shiyu Huang, Jiazheng Xu, Yuanming Yang, Wenyi Hong, Xiao- han Zhang, Guanyu Feng, et al. Cogvideox: Text-to-video diffusion models with an expert transformer. arXiv preprint arXiv:2408.06072, 2024. 6

2024 arXiv
[43]

Tianwei Yin, Micha ¨el Gharbi, Taesung Park, Richard Zhang, Eli Shechtman, Fredo Durand, and William T. Freeman. Im- proved distribution matching distillation for fast image syn- thesis. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. 8

2024
[44]

One-step diffusion with distribution matching distillation

Tianwei Yin, Micha ¨el Gharbi, Richard Zhang, Eli Shecht- man, Fredo Durand, William T Freeman, and Taesung Park. One-step diffusion with distribution matching distillation. In Proceedings of the IEEE/CVF conference on computer vi- sion and pattern recognition, pages 6613–6623...

2024
[45]

From slow bidirectional to fast causal video generators

Tianwei Yin, Qiang Zhang, Richard Zhang, William T Free- man, Fredo Durand, Eli Shechtman, and Xun Huang. From slow bidirectional to fast causal video generators. arXiv preprint arXiv:2412.07772, 2024. 2

2024
[46]

Fast sampling of dif- fusion models with exponential integrator

Qinsheng Zhang and Yongxin Chen. Fast sampling of dif- fusion models with exponential integrator. arXiv preprint arXiv:2204.13902, 2022. 2, 3, 8

2022 arXiv
[47]

gDDIM: Generalized denoising diffusion implicit models

Qinsheng Zhang, Molei Tao, and Yongxin Chen. gDDIM: Generalized denoising diffusion implicit models. In The Eleventh International Conference on Learning Representa- tions, 2023. 8

2023
[48]

Dreampropeller: Supercharge text-to-3d generation with parallel sampling

Linqi Zhou, Andy Shih, Chenlin Meng, and Stefano Er- mon. Dreampropeller: Supercharge text-to-3d generation with parallel sampling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition , pages 4610–4619, 2024. 8 A. Proofs A.1. Proof of Lemma 2.1...

2024
[49]

However, based on the part of monotonicity, we can findI ′ 1 that is a prefix of I′ 2 and S(I′

≥ R(I2). However, based on the part of monotonicity, we can findI ′ 1 that is a prefix of I′ 2 and S(I′
[50]

A rotating set of crystalline spheres, each sphere releasing radiant chord progressions that illuminate a cosmic backdrop

> R(I′ 2), which is a contradiction. By contradiction, we completed the proof. A.3. Proof of Theorem 2.5 Proof. Denote xi t as the solution of the i-th core at time t, i = 1, 2, 3. Notice that the update solver solving xi t → xj t′ follows the following rules: (1) Fine solver:...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Multi-level adaptive solutions to boundary- value problems

Achi Brandt. Multi-level adaptive solutions to boundary- value problems. Mathematics of computation, 31(138):333– 390, 1977. 3

work page 1977

[2] [2]

Scaling recti- fied flow transformers for high-resolution image synthesis

Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas M ¨uller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, et al. Scaling recti- fied flow transformers for high-resolution image synthesis. In Forty-first international conference on machine learning,

[3] [3]

xdit: an inference engine for diffusion trans- formers (dits) with massive parallelism

Jiarui Fang, Jinzhe Pan, Xibo Sun, Aoyu Li, and Jian- nan Wang. xdit: an inference engine for diffusion trans- formers (dits) with massive parallelism. arXiv preprint arXiv:2411.01738, 2024. 2, 8

arXiv 2024

[4] [4]

A multigrid method for multiprocessors

A Greenbaum. A multigrid method for multiprocessors. Ap- plied mathematics and computation, 19(1-4):75–88, 1986. 3

work page 1986

[5] [5]

Geometric trajectory diffusion models

Jiaqi Han, Minkai Xu, Aaron Lou, Haotian Ye, and Stefano Ermon. Geometric trajectory diffusion models. Advances in Neural Information Processing Systems, 37:25628–25662,

work page

[6] [6]

CLIPScore: a reference-free evaluation met- ric for image captioning

Jack Hessel, Ari Holtzman, Maxwell Forbes, Ronan Le Bras, and Yejin Choi. CLIPScore: a reference-free evaluation met- ric for image captioning. In EMNLP, 2021. 6

work page 2021

[7] [7]

Denoising dif- fusion probabilistic models

Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising dif- fusion probabilistic models. Advances in neural information processing systems, 33:6840–6851, 2020. 2, 3

2020

[8] [8]

Video dif- fusion models

Jonathan Ho, Tim Salimans, Alexey Gritsenko, William Chan, Mohammad Norouzi, and David J Fleet. Video dif- fusion models. Advances in Neural Information Processing Systems, 35:8633–8646, 2022. 2

2022