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REVIEW 5 major objections 5 minor 63 references

FLEX: A Backbone for Diffusion-Based Modeling of Spatio-temporal Physical Systems

T0 review · 5 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read FLEX is a diffusion backbone that learns residual fields, achieving accurate super-resolution and forecasting of 2D turbulence in as few as two reverse diffusion steps.

desk verdict Strong empirical backbone for diffusion-based turbulence modeling, undermined by a mismatch between the stated theory (L2) and the actual training loss (L1). read the letter →

arxiv 2505.17351 v1 pith:KXIMIUJO submitted 2025-05-23 cs.LG cs.AI

N. Benjamin Erichson , Vinicius Mikuni , Dongwei Lyu , Yang Gao , Omri Azencot , Soon Hoe Lim , Michael W. Mahoney This is my paper
classification cs.LGcs.AI
keywords diffusionmodelsspatio-temporalphysicalsystemsresidual-spacelearningvelocityparameterization2Dturbulencesuper-resolutionforecastingzero-shotgeneralization
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

The paper introduces FLEX, a diffusion-model backbone aimed at spatio-temporal physical systems, and claims it can emulate 2D turbulence accurately and cheaply. The central idea is to learn residual corrections rather than full fields, together with a velocity-based parameterization that the paper argues reduces the variance of the diffusion target and stabilizes training. The architecture combines convolutional U-Net layers with a latent Transformer and uses a two-level conditioning scheme to balance reconstruction fidelity and sample diversity. On high-resolution turbulence data, FLEX outperforms U-Net and ViT baselines for super-resolution and forecasting, works with as few as two reverse diffusion steps, and produces calibrated uncertainty estimates from ensembles. It also reports zero-shot generalization to unseen Reynolds numbers, velocity fields, and boundary conditions.

What carries the argument

The central objects are the velocity-parameterized diffusion process in residual space and the hybrid U-Net/Transformer backbone. In the forward process, $Z_t = \alpha(t) R + \sigma(t)\epsilon$, and the network learns $v(t,R)=\alpha(t)\epsilon - \sigma(t) R$; the paper shows $v^*$ is a scaled score combination and that its expected squared norm is a scaled Fisher divergence, giving a variance-reduction argument for using residuals. Architecturally, the latent Transformer sits in the U-Net middle block at patch size 1 so each spatial token sees the whole field through self-attention with MLP channel mixing, while a task-specific encoder injects weak conditioning via shallow skips and strong conditioning via deep skips and bottleneck features. This mechanism carries the argument because it is what the paper claims makes FLEX fast, stable, and transferable.

What would settle it

Train FLEX twice on the same turbulence data, once with the L2 velocity loss used in Proposition 1 and once with the L1 residual loss used in the experiments; if the residual-space advantage over raw-data training shrinks under L1, the variance-reduction argument is not what explains the reported gains.

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Extended reading notes

Core claim

On its own terms, the paper claims that a diffusion model for spatio-temporal physics should be trained on residual fields rather than raw snapshots, using a velocity parameterization. The optimal velocity field is $$$v^{{*}}$(t,Z)=-\frac{\$\sigma$(t)}{\$\alpha$(t)}\bigl(Z+\nabla_Z \log p_t(Z)\bigr),$$ and Proposition 1 gives $$\mathbb{E}_{p_t}\bigl[\|$v^{{*}}$(t,Z)\|^2\bigr]=\left(\frac{\$\sigma$(t)}{\$\alpha$(t)}\right)^2 \mathrm{DF}\bigl(p_t\,\|\,\mathcal{N}(0,1)\bigr),$$ so the closer the noised data distribution is to Gaussian, the smaller the velocity variance. The paper measures this Fisher divergence on 40,000 turbulence patches and finds that both super-resolution and forecasting residuals are closer to Gaussian than raw vorticity, especially at small diffusion times. FLEX then combines a U-Net with a patch-size-1 latent Transformer and a hierarchical conditioning scheme, and the paper reports that it outperforms U-Net and ViT diffusion baselines on 2048x2048 2D turbulence, using as few as two DDIM steps and providing calibrated ensemble uncertainties.

Load-bearing premise

The theoretical argument for residual-space training assumes an L2 velocity loss, while the actual training uses an L1 residual loss, leaving a gap between the theory and the trained model.

Editorial extensions

If this is right

  • With as few as two reverse diffusion steps, FLEX produces accurate super-resolution and forecasting, so diffusion-based emulation can run at near-regression speed.
  • Training in residual space with a velocity parameterization lowers the variance of the target velocity field, which the paper argues stabilizes training and improves sample quality.
  • The hybrid U-Net plus latent Transformer backbone captures both local spatial detail and long-range dependencies, outperforming pure U-Net and pure ViT baselines.
  • The weak/strong hierarchical conditioning scheme balances reconstruction fidelity and sample diversity, yielding calibrated uncertainty estimates from sampling.
  • FLEX zero-shot generalizes to unseen Reynolds numbers, velocity observables, and Dirichlet boundary conditions, and on the PDEBench data it forecasts better than on in-distribution test cases.

Reading between the lines

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

  • Beyond the paper, the two-step reverse-diffusion results suggest that FLEX could be distilled or amortized to a single-step predictor, but the paper does not attempt that.
  • Beyond the paper, the weak/strong conditioning recipe, withholding fine details from early layers and injecting them at the decoder, is a transferable design principle for other conditional generative tasks, though the paper only demonstrates it on turbulence and temperature fields.
  • Beyond the paper, the zero-shot transfer from vorticity to velocity and to Dirichlet boundaries hints that the latent representation captures physical invariants of the Navier-Stokes equations; a direct test would be whether FLEX transfers to forced or rotating turbulence without retraining.
  • Beyond the paper, because all main results are single-seed runs with no reported standard deviations, a multi-seed comparison would establish whether the reported margins over baselines are stable.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The paper introduces FLEX, a hybrid U-Net/Transformer backbone for conditional diffusion models applied to super-resolution and forecasting of 2D turbulent flows. The main design choices are (i) modeling residuals instead of raw fields, (ii) using a velocity parameterization, (iii) inserting a latent Vision Transformer with patch size 1 into the U-Net bottleneck, and (iv) a hierarchical conditioning scheme with weak and strong task-specific encoder injections. The authors provide theoretical propositions in Appendix B intended to show that residual-space training reduces the variance of the target velocity field, and they evaluate FLEX on SuperBench NSKT data at 2048x2048 resolution, including super-resolution at several Reynolds numbers, recursive forecasting over 50 steps, and zero-shot transfer to velocity fields and to PDEBench Dirichlet-boundary data. The paper reports state-of-the-art RFNE and correlation results, calibrated uncertainty via ensembles, and success with as few as two DDIM steps. A weather-data super-resolution comparison is also included in Appendix F.

Significance. If the empirical claims hold, FLEX would be a practically valuable backbone for fast generative emulation of turbulent flows, with the two-step sampling result and the zero-shot transfer to unseen observables and boundary conditions being the most notable contributions. The paper has genuine strengths: a demanding high-resolution benchmark, a clean architectural ablation, spectral diagnostics, and explicit uncertainty-calibration analysis. However, the central theoretical motivation is undermined by a direct mismatch between the loss used in the theory and the loss stated for the deployed model, and the empirical support is weakened by the absence of error bars and by several protocol ambiguities. The claims are defensible in principle but require substantial clarification and additional evidence before publication.

major comments (5)
  1. [Sec. 3.1 (Eq. 6), Algorithm 1, App. D.1] There is a direct contradiction between the training objective used in the theory and the objective stated for the experiments. Eq. (6), Algorithm 1 line 8, and App. C.2 specify an L2 velocity loss, but App. D.1 states: "we use an ℓ1 loss between predicted and target residuals." Propositions 1 and 2 characterize the L2-optimal velocity field v*, which is the conditional mean. Under an L1 loss the learned estimator is instead a conditional median, so the identity E[||v*||^2] = (σ/α)^2 DF(pt || N(0,I)) and the residual-variance-reduction argument do not apply to the model that was actually trained. This is load-bearing because residual-space modeling is advertised as a theoretical contribution. The authors must either state that the experiments use the L2 loss of Eq. (6), or provide an analogous theory for the L1-trained estimator, or explicitly withdraw the theoretical motivation for the deployed model.
  2. [Sec. 3.2 (Eq. 7) and Algorithm 2] The DDIM update in Eq. (7) is inconsistent with the velocity parameterization and with Algorithm 2. With the cosine schedule of App. C, α(1)=0, so the term (Z_ti - σ(t_i)vθ)/α(t_i) is undefined at the initial sampling time. Moreover, substituting the velocity relation ε = α_t v + σ_t Z_t into the standard DDIM update yields Z_{t_{i-1}} = (α_{t_{i-1}}α_{t_i}+σ_{t_{i-1}}σ_{t_i})Z_{t_i} + (σ_{t_{i-1}}α_{t_i}-α_{t_{i-1}}σ_{t_i})vθ, not Eq. (7). Algorithm 2 appears to use a different and more plausible update, but the indexing convention (t0=1, tN=0) also conflicts with the statement that Eq. (7) maps Z_ti to Z_ti-1. Because the two-step sampling result is central to the paper, this inconsistency must be resolved.
  3. [App. B.2, Figure 9] The empirical support for the residual-variance-reduction claim is not reliable. Figure 9 estimates the Fisher divergence DF(pt||N(0,I)) using a Gaussian KDE over 40,000 patches of size 256x256, i.e., in roughly 65,000 dimensions. Score estimates from KDE in that dimensionality are dominated by concentration-of-measure effects and by the choice of bandwidth, and no error bars or bandwidth-selection details are given. The right panel's conclusion that residual samples reduce E[||v*||^2] therefore does not provide the stated support for Proposition 1's relevance to the trained model. The authors should either use analytically tractable lower-dimensional diagnostics, provide estimates with uncertainties that are robust in high dimensions, or present another form of evidence for the variance reduction.
  4. [Sec. 4, Tables 1-2, Fig. 8, Limitations] The headline empirical comparisons lack error bars and do not specify the DDIM sampling budgets for all baselines. The Limitations section explicitly acknowledges that results were not computed over multiple random seeds and that standard deviations are not reported. As a result, gaps such as FLEX-MT-L's 5.9 average RFNE versus DM-M U-Net's 6.7 in Table 1 cannot be assessed for significance. Additionally, Sec. 4.1 states "we use only 2 diffusion steps" but does not say whether this budget applies to the DM-S/M U-Net and U-ViT baselines or only to FLEX; if the baselines are evaluated with a different number of steps, the comparison is not apples-to-apples. Please report seed variability and the exact sampling schedule used for every diffusion baseline.
  5. [Sec. 4.1 vs. App. E.1; App. D.1] Two protocol issues affect the interpretation of the generalization results. First, Sec. 4 and Table 1 state that training Reynolds numbers are {2k,4k,8k,16k,32k}, but App. E.1 says training uses {2k,4k,8k,16k,23k}; this changes whether 32k is a seen or unseen Reynolds number and affects the meaning of the "unseen Re" claims. Second, App. D.1 states "We fine-tune standard deviation for zero-shot tasks," which implies that target-domain summary statistics are used in the reported velocity and PDEBench zero-shot experiments. The paper should clarify which results use the tuned normalization, report the values used, and discuss how much of the zero-shot performance depends on this adjustment.
minor comments (5)
  1. [App. E.1 heading] The heading "Naviar Stokes Kraichnan Turbulence" contains a typo and should read "Navier-Stokes."
  2. [Sec. 4.1, Fig. 5] The text reports that the pull distribution has σ≈1.78, while the caption to Fig. 5(a) reports a normal fit with σ=1.62; these numbers should be reconciled.
  3. [Fig. 4, Fig. 6, Fig. 7, Fig. 8, Fig. 10-13] Several figure captions and axis labels use "Engergy Spectrum" instead of "Energy Spectrum," and figure text contains minor spacing inconsistencies; a final proofreading pass is needed.
  4. [App. B.2] In the sentence before Proposition 2, "suprising" should be "surprising," and the proof of Proposition 2 contains an unbalanced parenthesis in the line following Eq. (17).
  5. [App. F.1, Table 5] The weather-data comparison reports only mean values without error bars or a statement about the number of test snapshots and whether the differences are stable across years; please add this information.

Circularity Check

1 steps flagged · score 4.0 of 10

Mild circularity: zero-shot evaluations fine-tune a normalization parameter on the target domain, and the L1 training loss falls outside the L2-based theory used to justify residual modeling.

  1. fitted input called prediction [Appendix D.1, Data Normalization; zero-shot claims in Sections 4.1 and 4.2]
    "Each input field is normalized by subtracting a mean of 0 and dividing by a standard deviation of 5.457 calculated from the average residuals across all Reynold Numbers used in this work. ... This normalization is applied consistently across tasks and Reynolds numbers. We fine-tune standard deviation for zero-shot tasks."

    The paper advertises zero-shot generalization across physical observables and boundary conditions and reports zero-shot results on velocity fields and PDEBench Dirichlet data. However, the normalization standard deviation is explicitly fine-tuned for these zero-shot tasks, meaning a target-domain statistic is fitted before the zero-shot evaluation. The model's output scale is thus adjusted to the unseen domain rather than predicted from training data alone. The zero-shot claim is therefore partially a fitted-input claim rather than a parameter-free generalization.

full rationale

The main architectural and algorithmic contributions are not circular: FLEX's residual-space, velocity-parameterized diffusion model is trained on held-out simulation data, and the theoretical results in Appendix B are standard identities (Tweedie's formula, Fisher-divergence identity, Hessian bounds) whose conclusions are empirical estimates of the data distribution, not restatements of the model's fitted parameters. The in-distribution super-resolution and forecasting comparisons against SwinIR, HAT, U-Net, U-ViT, FourCastNet, DYffusion, and the external PDEBench and ERA5 evaluations provide checkpoints independent of the fitted model. The score is elevated to 4 because the zero-shot generalization results are obtained after fine-tuning the normalization standard deviation on the target domain (Appendix D.1), and because the theoretical motivation is derived for the L2 velocity loss in Equation (6) while Appendix D.1 states that the deployed training uses an L1 loss. With an L1 objective, the learned estimator is not the conditional mean v* used in Propositions 1 and 2, so the variance-reduction argument does not strictly justify the trained model. These are overclaim and support-gap issues rather than full reductions by construction; no predicted field is equal to a fitted parameter or to a self-cited theorem. The paper's own Limitations section further acknowledges that no standard deviations are reported, which compounds the lack of error bars in the Figure 9 estimates.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities, particles, forces, or conserved quantities. The architecture components are computational and fall outside the invented-entity category.

free parameters (3)
  • Normalization standard deviation = 5.457 (training), fine-tuned per zero-shot task
    App. D.1 divides each input by a standard deviation of 5.457 computed from training residuals, and then states that the standard deviation is fine-tuned for zero-shot tasks, making it a target-dependent tuning knob.
  • Weak conditioning depth L_weak = Not reported
    Sec. 2 defines weak conditioning for resolution levels up to L_weak, but the numerical value is never given, leaving a hand-chosen architectural hyperparameter unspecified.
  • Number of DDIM sampling steps = 2
    Inference hyperparameter used for all main results; the paper claims accuracy with only 2 steps, so the central 'few-step' claim depends on this choice.
assumptions (5)
  • standard math Tweedie's formula is used to relate the score function to the conditional mean in the derivation of the optimal velocity field (Eq. 12).
    The paper cites Efron [10] for this standard result and does not prove it.
  • standard math The Hessian identity for the score function (Lemma 1) is taken from Gottwald et al. [16], App. A.1, without reproducing the proof.
    The result is used in Proposition 2 and is assumed valid from the cited source.
  • domain assumption The Fisher divergence estimates in Figure 9, computed with a Gaussian kernel density estimator over 40,000 patches, accurately approximate the true divergences.
    The Monte Carlo estimate's error is not quantified, and the KDE bandwidth is not reported.
  • domain assumption The velocity parametrization v(t,R) = alpha(t)*epsilon - sigma(t)*R is an appropriate learning target, and the chosen cosine noise schedule is suitable for all tasks.
    Sec. C.1 fixes the schedule; the effectiveness is assumed and tested empirically rather than proven.
  • domain assumption Training on 256x256 patches and stitching to 2048x2048 preserves the global statistics of the full turbulence snapshots.
    Sec. 4 trains on patches for memory reasons; the stitching is used at inference, assuming boundary artifacts are small (Figure 4 shows small artifacts).

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Cite this review

Pith. "Pith review of FLEX: A Backbone for Diffusion-Based Modeling of Spatio-temporal Physical Systems." pith.science (2026). https://pith.science/paper/KXIMIUJO

@misc{pith2026250517351,
  author       = {Pith},
  title        = {Pith review of: FLEX: A Backbone for Diffusion-Based Modeling of Spatio-temporal Physical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXIMIUJO}},
  note         = {Machine review of arXiv:2505.17351}
}
read the original abstract

We introduce FLEX (FLow EXpert), a backbone architecture for generative modeling of spatio-temporal physical systems using diffusion models. FLEX operates in the residual space rather than on raw data, a modeling choice that we motivate theoretically, showing that it reduces the variance of the velocity field in the diffusion model, which helps stabilize training. FLEX integrates a latent Transformer into a U-Net with standard convolutional ResNet layers and incorporates a redesigned skip connection scheme. This hybrid design enables the model to capture both local spatial detail and long-range dependencies in latent space. To improve spatio-temporal conditioning, FLEX uses a task-specific encoder that processes auxiliary inputs such as coarse or past snapshots. Weak conditioning is applied to the shared encoder via skip connections to promote generalization, while strong conditioning is applied to the decoder through both skip and bottleneck features to ensure reconstruction fidelity. FLEX achieves accurate predictions for super-resolution and forecasting tasks using as few as two reverse diffusion steps. It also produces calibrated uncertainty estimates through sampling. Evaluations on high-resolution 2D turbulence data show that FLEX outperforms strong baselines and generalizes to out-of-distribution settings, including unseen Reynolds numbers, physical observables (e.g., fluid flow velocity fields), and boundary conditions.

Figures

Figures reproduced from arXiv: 2505.17351 by the authors.

Figure 1
Figure 1. FLEX is a backbone for modeling spatio-temporal physical systems using diffusion models. It learns residual corrections conditioned on task-specific inputs (e.g., low-resolution or past states) and physical parameters. The architecture integrates a task-specific encoder, a common encoder, a Transformer operating in latent space, and a decoder within a U-Net-style framework. The task-specific encoder weakly condition… view at source ↗
Figure 2
Figure 2. Illustration of the multi-task FLEX backbone instantiated for super-resolution and forecasting. The backbone includes two task-specific encoders and shares a common encoder, a latent Transformer, and a decoder. During inference, only the encoder corresponding to the target task is used. • The latent Transformer T replaces the middle block of the U-Net to provide global spatial context and full channel mixing. The bo… view at source ↗
Figure 3
Figure 3. Illustration of the score-based diffusion process. During training, residual samples are progressively [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Example snapshot demonstrating FLEX’s performance on vorticity field super-resolution at [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: (a) Pull distribution analysis shows that the model provides unbiased uncertainty estimates, while [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FLEX’s zero-shot performance on velocity field super-resolution at [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FLEX’s zero-shot performance on inhomogeneous NS data from PDEbench with Dirichlet boundary [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: Forecasting accuracy over 50 time steps. (a–b) Average Pearson correlation between predicted and [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: On the left, we investigate how "Gaussian" the raw data samples and residual samples are at different [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: Forecast of vorticity at Re = 12,000. FLEX maintains coherent vortex structures and preserves fine-scale features up to 40 steps ahead, despite the highly chaotic nature of the flow. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: Velocity field forecast at Re = 12,000, a variable not used during training. FLEX generalizes well across observables, producing physically consistent structures even under cross-variable prediction. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]
Figure 12
Figure 12. Figure 12: Forecasted vorticity from PDEBench’s inhomogeneous Navier–Stokes data with Dirichlet boundaries. [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]
Figure 13
Figure 13. Figure 13: Comparison of reconstructed temperature fields from the ERA5 dataset. (top) Ground truth high [PITH_FULL_IMAGE:figures/full_fig_p027_13.png]

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Reference graph

Works this paper leans on

63 extracted references · 17 canonical work pages

  1. [1]

    Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797, 2023

    Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797, 2023

    arXiv 2023

  2. [2]

    Computational design for long-term numerical integration of the equations of fluid motion: Two- dimensional incompressible flow

    Akio Arakawa. Computational design for long-term numerical integration of the equations of fluid motion: Two- dimensional incompressible flow. part i.Journal of Computational Physics, 135(2):103–114, 1997

    work page 1997

  3. [3]

    All are worth words: A ViT backbone for diffusion models

    Fan Bao, Shen Nie, Kaiwen Xue, Yue Cao, Chongxuan Li, Hang Su, and Jun Zhu. All are worth words: A ViT backbone for diffusion models. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 22669–22679, 2023

    2023

  4. [4]

    Stable video diffusion: Scaling latent video diffusion models to large datasets.arXiv preprint arXiv:2311.15127, 2023

    Andreas Blattmann, Tim Dockhorn, Sumith Kulal, Daniel Mendelevitch, Maciej Kilian, Dominik Lorenz, Yam Levi, Zion English, Vikram V oleti, Adam Letts, et al. Stable video diffusion: Scaling latent video diffusion models to large datasets.arXiv preprint arXiv:2311.15127, 2023

    arXiv 2023

  5. [5]

    Align your latents: High-resolution video synthesis with latent diffusion models

    Andreas Blattmann, Robin Rombach, Huan Ling, Tim Dockhorn, Seung Wook Kim, Sanja Fidler, and Karsten Kreis. Align your latents: High-resolution video synthesis with latent diffusion models. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 22563–22575, 2023

    2023

  6. [6]

    Yu Cao, Jingrun Chen, Yixin Luo, and Xiang Zhou. Exploring the optimal choice for generative processes in diffusion models: Ordinary vs stochastic differential equations.Advances in Neural Information Processing Systems, 36:33420–33468, 2023

    work page 2023

  7. [7]

    Activating more pixels in image super- resolution transformer

    Xiangyu Chen, Xintao Wang, Jiantao Zhou, Yu Qiao, and Chao Dong. Activating more pixels in image super- resolution transformer. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 22367–22377, 2023

    work page 2023

  8. [8]

    Maddix, Abdul Fatir Ansari, Andrew Stuart, Michael W

    Chaoran Cheng, Boran Han, Danielle C. Maddix, Abdul Fatir Ansari, Andrew Stuart, Michael W. Mahoney, and Bernie Wang. Gradient-free generation for hard-constrained systems. InThe Thirteenth International Conference on Learning Representations, 2025

    2025

Show all 63 references
  1. [9]

    An image is worth 16x16 words: Transformers for image recognition at scale

    Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszkoreit, and Neil Houlsby. An image is worth 16x16 words: Transformers for image recognition at...

    2021

  2. [10]

    Tweedie’s formula and selection bias.Journal of the American Statistical Association, 106(496):1602– 1614, 2011

    Bradley Efron. Tweedie’s formula and selection bias.Journal of the American Statistical Association, 106(496):1602– 1614, 2011

    2011

  3. [11]

    U-shape mamba: State space model for faster diffusion.arXiv preprint arXiv:2504.13499, 2025

    Alex Ergasti, Filippo Botti, Tomaso Fontanini, Claudio Ferrari, Massimo Bertozzi, and Andrea Prati. U-shape mamba: State space model for faster diffusion.arXiv preprint arXiv:2504.13499, 2025

    2025

  4. [12]

    Shallow neural networks for fluid flow reconstruction with limited sensors.Proceedings of the Royal Society A, 476(2238):20200097, 2020

    N Benjamin Erichson, Lionel Mathelin, Zhewei Yao, Steven L Brunton, Michael W Mahoney, and J Nathan Kutz. Shallow neural networks for fluid flow reconstruction with limited sensors.Proceedings of the Royal Society A, 476(2238):20200097, 2020

    2020

  5. [13]

    Scaling rectified flow transformers for high-resolution image synthesis

    Patrick Esser, Sumith Kulal, Andreas Blattmann, Rahim Entezari, Jonas Müller, Harry Saini, Yam Levi, Dominik Lorenz, Axel Sauer, Frederic Boesel, et al. Scaling rectified flow transformers for high-resolution image synthesis. In Forty-first International Conference on Machine ...

    2024

  6. [14]

    Scalable diffusion models with state space backbone

    Zhengcong Fei, Mingyuan Fan, Changqian Yu, and Junshi Huang. Scalable diffusion models with state space backbone. arXiv preprint arXiv:2402.05608, 2024

    2024 arXiv

  7. [15]

    Dimba: Transformer- mamba diffusion models.arXiv preprint arXiv:2406.01159, 2024

    Zhengcong Fei, Mingyuan Fan, Changqian Yu, Debang Li, Youqiang Zhang, and Junshi Huang. Dimba: Transformer- mamba diffusion models.arXiv preprint arXiv:2406.01159, 2024

    2024 arXiv

  8. [16]

    Localized diffusion models for high dimensional distributions generation.arXiv preprint arXiv:2505.04417, 2025

    Georg A Gottwald, Shuigen Liu, Youssef Marzouk, Sebastian Reich, and Xin T Tong. Localized diffusion models for high dimensional distributions generation.arXiv preprint arXiv:2505.04417, 2025

    2025

  9. [17]

    Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, 33:6840–6851, 2020

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models.Advances in Neural Information Processing Systems, 33:6840–6851, 2020

    2020

  10. [18]

    Video diffusion models.Advances in Neural Information Processing Systems, 35:8633–8646, 2022

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

    2022

  11. [19]

    simple diffusion: End-to-end diffusion for high resolution images

    Emiel Hoogeboom, Jonathan Heek, and Tim Salimans. simple diffusion: End-to-end diffusion for high resolution images. InInternational Conference on Machine Learning, pages 13213–13232. PMLR, 2023

    2023

  12. [20]

    Simpler diffusion (sid2): 1.5 fid on imagenet512 with pixel-space diffusion.arXiv preprint arXiv:2410.19324, 2024

    Emiel Hoogeboom, Thomas Mensink, Jonathan Heek, Kay Lamerigts, Ruiqi Gao, and Tim Salimans. Simpler diffusion (sid2): 1.5 fid on imagenet512 with pixel-space diffusion.arXiv preprint arXiv:2410.19324, 2024

    2024 arXiv

  13. [21]

    Elucidating the design space of diffusion-based generative models.Advances in Neural Information Processing Systems, 35:26565–26577, 2022

    Tero Karras, Miika Aittala, Timo Aila, and Samuli Laine. Elucidating the design space of diffusion-based generative models.Advances in Neural Information Processing Systems, 35:26565–26577, 2022. 11

    2022

  14. [22]

    Analyzing and improving the training dynamics of diffusion models.arXiv preprint arXiv:2312.02696, 2024

    Tero Karras, Miika Aittala, and Jaakkoet al.Lehtinen. Analyzing and improving the training dynamics of diffusion models.arXiv preprint arXiv:2312.02696, 2024

    2024 arXiv

  15. [23]

    Variational diffusion models.Advances in Neural Information Processing Systems, 34:21696–21707, 2021

    Diederik Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. Variational diffusion models.Advances in Neural Information Processing Systems, 34:21696–21707, 2021

    2021

  16. [24]

    Benchmarking autoregressive conditional diffusion models for turbulent flow simulation.arXiv preprint arXiv:2309.01745, 2023

    Georg Kohl, Li-Wei Chen, and Nils Thuerey. Benchmarking autoregressive conditional diffusion models for turbulent flow simulation.arXiv preprint arXiv:2309.01745, 2023

    2023 arXiv

  17. [25]

    Srdiff: Single image super-resolution with diffusion probabilistic models.Neurocomputing, 479:47–59, 2022

    Haoying Li, Yifan Yang, Meng Chang, Shiqi Chen, Huajun Feng, Zhihai Xu, Qi Li, and Yueting Chen. Srdiff: Single image super-resolution with diffusion probabilistic models.Neurocomputing, 479:47–59, 2022

    2022

  18. [26]

    Swinir: Image restoration using swin transformer

    Jingyun Liang, Jiezhang Cao, Guolei Sun, Kai Zhang, Luc Van Gool, and Radu Timofte. Swinir: Image restoration using swin transformer. InProceedings of the IEEE/CVF international conference on computer vision, pages 1833–1844, 2021

    2021

  19. [27]

    Elucidating the design choice of probability paths in flow matching for forecasting.arXiv preprint arXiv:2410.03229, 2024

    Soon Hoe Lim, Yijin Wang, Annan Yu, Emma Hart, Michael W Mahoney, Xiaoye S Li, and N Benjamin Erichson. Elucidating the design choice of probability paths in flow matching for forecasting.arXiv preprint arXiv:2410.03229, 2024

    arXiv

  20. [28]

    Flow matching for generative modeling.arXiv preprint arXiv:2210.02747, 2022

    Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling.arXiv preprint arXiv:2210.02747, 2022

    2022 arXiv

  21. [29]

    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. InAdvances in Neural Information Processing Systems, 2022

    2022

  22. [30]

    Fit: Flexible vision transformer for diffusion model.arXiv preprint arXiv:2402.12376, 2024

    Zeyu Lu, Zidong Wang, Di Huang, Chengyue Wu, Xihui Liu, Wanli Ouyang, and Lei Bai. Fit: Flexible vision transformer for diffusion model.arXiv preprint arXiv:2402.12376, 2024

    2024 arXiv

  23. [31]

    Difffluid: Plain diffusion models are effective predictors of flow dynamics.arXiv preprint arXiv:2409.13665, 2024

    Dongyu Luo, Jianyu Wu, Jing Wang, Hairun Xie, Xiangyu Yue, and Shixiang Tang. Difffluid: Plain diffusion models are effective predictors of flow dynamics.arXiv preprint arXiv:2409.13665, 2024

    2024 arXiv

  24. [32]

    Latte: Latent diffusion transformer for video generation.arXiv preprint arXiv:2401.03048, 2024

    Xin Ma, Yaohui Wang, Gengyun Jia, Xinyuan Chen, Ziwei Liu, Yuan-Fang Li, Cunjian Chen, and Yu Qiao. Latte: Latent diffusion transformer for video generation.arXiv preprint arXiv:2401.03048, 2024

    2024 arXiv

  25. [33]

    Generative ai for fast and accurate statistical computation of fluids

    Roberto Molinaro, Samuel Lanthaler, Bogdan Raoni ´c, Tobias Rohner, Victor Armegioiu, Zhong Yi Wan, Fei Sha, Siddhartha Mishra, and Leonardo Zepeda-Núñez. Generative ai for fast and accurate statistical computation of fluids. arXiv e-prints, pages arXiv–2409, 2024

    2024

  26. [34]

    Improved denoising diffusion probabilistic models

    Alexander Quinn Nichol and Prafulla Dhariwal. Improved denoising diffusion probabilistic models. InInternational Conference on Machine Learning, pages 8162–8171. PMLR, 2021

    2021

  27. [35]

    Integrating neural operators with diffusion models improves spectral representation in turbulence modeling.arXiv preprint arXiv:2409.08477, 2024

    Vivek Oommen, Aniruddha Bora, Zhen Zhang, and George Em Karniadakis. Integrating neural operators with diffusion models improves spectral representation in turbulence modeling.arXiv preprint arXiv:2409.08477, 2024

    2024 arXiv

  28. [36]

    Kilometer-scale convection allowing model emulation using generative diffusion modeling.arXiv preprint arXiv:2408.10958, 2024

    Jaideep Pathak, Yair Cohen, Piyush Garg, Peter Harrington, Noah Brenowitz, Dale Durran, Morteza Mardani, Arash Vahdat, Shaoming Xu, Karthik Kashinath, et al. Kilometer-scale convection allowing model emulation using generative diffusion modeling.arXiv preprint arXiv:2408.10958, 2024

    2024 arXiv

  29. [37]

    Fourcastnet: A global data-driven high- resolution weather model using adaptive fourier neural operators.arXiv preprint arXiv:2202.11214, 2022

    Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chattopadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzadenesheli, et al. Fourcastnet: A global data-driven high- resolution weather model using adaptive fourier neural ope...

    2022 arXiv

  30. [38]

    Frame invariant neural network closures for kraichnan turbulence.Physica A: Statistical Mechanics and its Applications, 609:128327, 2023

    Suraj Pawar, Omer San, Adil Rasheed, and Prakash Vedula. Frame invariant neural network closures for kraichnan turbulence.Physica A: Statistical Mechanics and its Applications, 609:128327, 2023

    2023

  31. [39]

    Scalable diffusion models with transformers

    William Peebles and Saining Xie. Scalable diffusion models with transformers. InProceedings of the IEEE/CVF international conference on computer vision, pages 4195–4205, 2023

    2023

  32. [40]

    Gencast: Diffusion-based ensemble forecasting for medium-range weather.arXiv preprint arXiv:2312.15796, 2023

    Ilan Price, Alvaro Sanchez-Gonzalez, Ferran Alet, Tom R Andersson, Andrew El-Kadi, Dominic Masters, Timo Ewalds, Jacklynn Stott, Shakir Mohamed, Peter Battaglia, et al. Gencast: Diffusion-based ensemble forecasting for medium-range weather.arXiv preprint arXiv:2312.15796, 2023

    2023 arXiv

  33. [41]

    Learning transferable visual models from natural language supervision

    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 supervision. In International conference on machine learning, pa...

    2021

  34. [42]

    Benjamin Erichson, Junyi Guo, Shashank Subramanian, Omer San, Zarija Lukic, and Michael W

    Pu Ren, N. Benjamin Erichson, Junyi Guo, Shashank Subramanian, Omer San, Zarija Lukic, and Michael W. Mahoney. Superbench: A super-resolution benchmark dataset for scientific machine learning.Journal of Data-centric Machine Learning Research, 2025. 12

    2025

  35. [43]

    High-resolution image synthesis with latent diffusion models

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

    2022

  36. [44]

    U-net: Convolutional networks for biomedical image segmenta- tion

    Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmenta- tion. InMedical image computing and computer-assisted intervention–MICCAI 2015: 18th international conference, Munich, Germany, October 5-9, 2015, proceedings, part...

    2015

  37. [45]

    Dyffusion: A dynamics-informed diffusion model for spatiotemporal forecasting.Advances in neural information processing systems, 36:45259–45287, 2023

    Salva Rühling Cachay, Bo Zhao, Hailey Joren, and Rose Yu. Dyffusion: A dynamics-informed diffusion model for spatiotemporal forecasting.Advances in neural information processing systems, 36:45259–45287, 2023

    2023

  38. [46]

    Image super- resolution via iterative refinement.IEEE transactions on pattern analysis and machine intelligence, 45(4):4713–4726, 2022

    Chitwan Saharia, Jonathan Ho, William Chan, Tim Salimans, David J Fleet, and Mohammad Norouzi. Image super- resolution via iterative refinement.IEEE transactions on pattern analysis and machine intelligence, 45(4):4713–4726, 2022

    2022

  39. [47]

    Progressive distillation for fast sampling of diffusion models.arXiv preprint arXiv:2202.00512, 2022

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

    2022 arXiv

  40. [48]

    Resdiff: Combining cnn and diffusion model for image super-resolution

    Shuyao Shang, Zhengyang Shan, Guangxing Liu, LunQian Wang, XingHua Wang, Zekai Zhang, and Jinglin Zhang. Resdiff: Combining cnn and diffusion model for image super-resolution. InProceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 8975–8983, 2024

    2024

  41. [49]

    Deep unsupervised learning using nonequilibrium thermodynamics

    Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. InInternational conference on machine learning, pages 2256–2265. PMLR, 2015

    2015

  42. [50]

    Denoising diffusion implicit models

    Jiaming Song, Chenlin Meng, and Stefano Ermon. Denoising diffusion implicit models. InInternational Conference on Learning Representations, 2021

    2021

  43. [51]

    Consistency models.arXiv preprint arXiv:2303.01469, 2023

    Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models.arXiv preprint arXiv:2303.01469, 2023

    2023 arXiv

  44. [52]

    Sliced score matching: A scalable approach to density and score estimation

    Yang Song, Sahaj Garg, Jiaxin Shi, and Stefano Ermon. Sliced score matching: A scalable approach to density and score estimation. InProc. UAI, 2019

    2019

  45. [53]

    Score-based generative modeling through stochastic differential equations.arXiv preprint arXiv:2011.13456, 2020

    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, 2020

    2011 arXiv

  46. [54]

    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. InInternational Conference on Learning Representations, 2021

    2021

  47. [55]

    Pdebench: An extensive benchmark for scientific machine learning.Advances in Neural Information Processing Systems, 35:1596–1611, 2022

    Makoto Takamoto, Timothy Praditia, Raphael Leiteritz, Daniel MacKinlay, Francesco Alesiani, Dirk Pflüger, and Mathias Niepert. Pdebench: An extensive benchmark for scientific machine learning.Advances in Neural Information Processing Systems, 35:1596–1611, 2022

    2022

  48. [56]

    Statistical downscaling via high-dimensional distribution matching with generative models.arXiv preprint arXiv:2412.08079, 2024

    Zhong Yi Wan, Ignacio Lopez-Gomez, Robert Carver, Tapio Schneider, John Anderson, Fei Sha, and Leonardo Zepeda-Núñez. Statistical downscaling via high-dimensional distribution matching with generative models.arXiv preprint arXiv:2412.08079, 2024

    2024 arXiv

  49. [57]

    Sinsr: diffusion-based image super-resolution in a single step

    Yufei Wang, Wenhan Yang, Xinyuan Chen, Yaohui Wang, Lanqing Guo, Lap-Pui Chau, Ziwei Liu, Yu Qiao, Alex C Kot, and Bihan Wen. Sinsr: diffusion-based image super-resolution in a single step. InProceedings of the IEEE/CVF conference on computer vision and pattern recognition, pa...

    2024

  50. [58]

    Fitv2: Scalable and improved flexible vision transformer for diffusion model.arXiv preprint arXiv:2410.13925, 2024

    ZiDong Wang, Zeyu Lu, Di Huang, Cai Zhou, Wanli Ouyang, et al. Fitv2: Scalable and improved flexible vision transformer for diffusion model.arXiv preprint arXiv:2410.13925, 2024

    2024 arXiv

  51. [59]

    Generative diffusion-based downscaling for climate.arXiv preprint arXiv:2404.17752, 2024

    Robbie A Watt and Laura A Mansfield. Generative diffusion-based downscaling for climate.arXiv preprint arXiv:2404.17752, 2024

    2024 arXiv

  52. [60]

    Cogvideox: Text-to-video diffusion models with an expert transformer.arXiv preprint arXiv:2408.06072, 2024

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

    2024 arXiv

  53. [61]

    Improved techniques for maximum likelihood estimation for diffusion ODEs

    Kaiwen Zheng, Cheng Lu, Jianfei Chen, and Jun Zhu. Improved techniques for maximum likelihood estimation for diffusion ODEs. InInternational Conference on Machine Learning, pages 42363–42389. PMLR, 2023

    2023

  54. [62]

    Gaussian

    Daquan Zhou, Weimin Wang, Hanshu Yan, Weiwei Lv, Yizhe Zhu, and Jiashi Feng. Magicvideo: Efficient video generation with latent diffusion models.arXiv preprint arXiv:2211.11018, 2022. 13 Appendix A Related Work Diffusion models were first introduced as a thermodynamically insp...

    2022 arXiv

  55. [2017]

    Table 5 reports performance metrics, including RFNE, structural similarity index (SSIM), and peak signal-to-noise ratio (PSNR)

    Figure 13 shows an example reconstruction from the year 2016. Table 5 reports performance metrics, including RFNE, structural similarity index (SSIM), and peak signal-to-noise ratio (PSNR). FLEX outperforms HAT in all metrics, highlighting its ability to recover fine-scale fea...

    2016

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.