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REVIEW 4 major objections 6 minor 50 references

Modeling Continuous Spatial-temporal Dynamics of Turbulent Flow with Test-time Refinement

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Test-time refinement reconstructs DNS fields from LES with top accuracy

desk verdict A competent incremental paper with a real test-time refinement idea, but the kinetic-energy validation is partly circular and needs an ablation to show the constraint's actual contribution. read the letter →

arxiv 2412.19927 v1 pith:V4JQQQPC submitted 2024-12-27 physics.flu-dyn cs.AI

Shengyu Chen , Peyman Givi , Can Zheng , Xiaowei Jia This is my paper · ORCID
classification physics.flu-dyncs.AI
keywords turbulentflowsuper-resolutiontest-timerefinementphysics-guidedneuralnetworklargeeddysimulationdirectnumericalimplicitrepresentationNavier-Stokes
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 proposes Super-Resolution through Test-time Refinement (SR-TR), a neural method that reconstructs high-resolution direct numerical simulation (DNS) velocity fields from coarser large eddy simulation (LES) data. The central claim is that during testing, repeatedly degrading the model's own output back to the LES resolution and penalizing deviations, together with two physical constraints (equal mean velocity and prescribed kinetic energy), keeps long-horizon sequential predictions from drifting. The paper also claims the continuous spatial transition unit (CSTU) lets one model produce DNS at multiple resolutions, including resolutions never seen in training. On forced isotropic turbulence and Taylor-Green vortex data, SR-TR reports the highest SSIM and the lowest dissipation difference among the compared super-resolution and downscaling baselines over twenty forecast steps. A sympathetic reader would take away that test-time LES data can act as a cheap physical anchor for learned turbulence reconstruction.

What carries the argument

The load-bearing machinery is the Continuous Spatial Transition Unit (CSTU) plus a test-time refinement loop. CSTU combines the physics-guided recurrent unit (PRU), which advances the velocity field with a Runge-Kutta scheme whose temporal derivatives come from the incompressible Navier-Stokes equation, with an implicit neural representation (INR) that resamples the field onto arbitrary query points. The refinement loop uses a degradation operator that downsamples the reconstructed DNS to the LES grid and then adjusts the network's last layer at every test step to reduce the loss $L_{\text{refine}} = \alpha_0 L_{\text{deg}} + \alpha_1 L_{\text{mean}} + \alpha_2 L_{\text{kinetic}}$, where $L_{\text{deg}}$ is the mismatch with LES, $L_{\text{mean}}$ enforces equal mean velocity, and $L_{\text{kinetic}}$ pushes total kinetic energy toward a value extrapolated from the training DNS.

What would settle it

Run the trained model on a turbulent flow whose forcing varies in time or that undergoes transition, so that total kinetic energy rises or oscillates rather than staying constant or decaying linearly; if the test-time refinement then pulls reconstructions toward the wrong energy level and the SSIM or dissipation advantage over unrefined CSTU disappears, the central claim that LES-based refinement is the source of the gain is contradicted.

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Extended reading notes

Core claim

The discovery the paper argues for is that low-resolution LES data, used only at test time, are enough to stabilize learned DNS reconstruction: instead of training the network to map LES to DNS, one trains a sequential predictor on DNS alone and then, during forecasting, downsamples each reconstructed DNS field with a fixed degradation operator, compares it to the observed LES field, and updates the last layer of the model to minimize a weighted combination of the degradation loss, a mean-field loss, and a kinetic-energy loss. The reported result is that this refinement raises SSIM from about 0.936 to 0.949 on forced isotropic turbulence and from about 0.705 to 0.918 on Taylor-Green vortex data over the first ten test steps, while also lowering dissipation difference, and that it prevents the late-time error growth seen in unrefined CSTU baselines.

Load-bearing premise

The refinement assumes the total kinetic energy of the test flow follows the same simple time pattern (constant or linearly decaying) as the training period, because that pattern is extrapolated and enforced as a hard penalty rather than derived from the governing equations.

Editorial extensions

If this is right

  • Reconstructing DNS from LES no longer requires paired high-resolution data at test time, only the cheap LES stream and a short training-period DNS history.
  • The refinement loss can be attached to other sequential predictors, since it only adjusts the last layer and needs no architectural change.
  • One trained model can be applied zero-shot at target resolutions above the training resolution, with optional fine-tuning on a few high-resolution DNS samples improving it further.
  • Long-horizon forecasting of turbulent fields becomes more stable, since the degradation loop prevents cumulative error growth in the velocity and kinetic-energy statistics.
  • Using LES as an additional input brings little benefit once the test-time refinement is in place, so the cheap LES stream is best spent as a constraint rather than a feature.

Reading between the lines

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

  • Editorial inference: the same degradation-based refinement idea could work for other under-resolved simulations paired with coarse observations, such as ocean or climate models, wherever a coarse, frequently available field can serve as an anchor.
  • Editorial inference: the paper's framing suggests a testable extension where the kinetic-energy prior is replaced by an online estimator built from the LES stream itself, which would remove the main assumption of stationary or linearly decaying energy.
  • Editorial inference: because refinement only tunes the last layer, it could be run in an online or streaming setting, adapting the reconstruction as new LES frames arrive without retraining.
  • Editorial inference: the zero-shot multi-resolution results imply the INR learns a quasi-continuous spatial representation of the flow, so probing the model at arbitrary off-grid points, not just coarse and fine grids, is a direct way to test how continuous that representation really is.
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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

4 major / 6 minor

Summary. The paper proposes SR-TR (Super-Resolution through Test-time Refinement), a physics-guided neural network for reconstructing high-resolution DNS fields from lower-resolution LES data in unsteady incompressible turbulent flows. The method combines a physics-guided recurrent unit (PRU) based on a Runge-Kutta discretization of the Navier-Stokes equations with a continuous spatial transition unit (CSTU) based on implicit neural representations, enabling prediction at arbitrary resolutions. During testing, the reconstructed DNS is refined by minimizing a loss that includes a degradation term against available LES data, an equal-mean constraint, and a kinetic-energy constraint. The authors evaluate SR-TR on forced isotropic turbulence and Taylor-Green vortex flows, reporting higher SSIM and lower dissipation difference than several super-resolution, downscaling, and neural-operator baselines, as well as improved long-term behavior over 20 forecast steps. The implementation is stated to be publicly available.

Significance. The central idea of using LES data at test time through a degradation-based refinement, rather than as a training input, is valuable and potentially of broad interest for turbulence reconstruction and for other PDE-constrained super-resolution tasks. The CSTU's combination of implicit neural representation with a physics-guided recurrent unit is a sensible way to address resolution generalization, and the quantitative comparisons cover a reasonable set of baselines. The paper also ships a public implementation, which is a concrete strength. The significance of the results is currently tempered by three issues: the kinetic-energy validation is partly circular because Lkinetic directly enforces the metric used for validation; the CSTU is described as operating on 2D slices despite the 3D Navier-Stokes constraint; and the empirical tables lack error bars and contain at least one internal inconsistency. With these points addressed, the contribution would be a useful step toward physically consistent, test-time-adaptable super-resolution for turbulent flows.

major comments (4)
  1. [Sec. 3.1, Eq. (3.4); Sec. 4.2.3, Figs. 7 and 15] The kinetic-energy term Lkinetic is a fitted prior, and the validation in Sec. 4.2.3 uses the same quantity as a success metric. Since Eq. (3.4) directly minimizes |K(Q_d_hat) - K_tilde|, the agreement between reconstructed and DNS kinetic energy in Figs. 7 and 15 is achieved partly by construction rather than by the model's predictive ability. Both test flows are in the favorable regime for this prior: forced isotropic turbulence has stationary kinetic energy and the Taylor-Green vortex decays monotonically. The paper reports no ablation with Lkinetic removed, so it is unknown whether the claimed long-term stability is due to the physical constraint or merely to the enforced energy level. Please add such an ablation, report SSIM and dissipation difference for that variant, and test at least one out-of-favorable-regime case (e.g., time-varying forcing or transition), or explicitly limit the claim to flows with stationary or linearly decaying energy.
  2. [Sec. 3.2, Eqs. (3.6)-(3.7); Sec. 2.1, Eq. (2.1)] The CSTU is described as operating on "each flow data slice" in a continuous 2D feature space, but the physics constraint is the 3D Navier-Stokes equation in Eq. (2.1). If the PRU estimates spatial derivatives only in the x-y plane, the z-gradient terms are omitted and the model does not enforce the full incompressible Navier-Stokes dynamics on which the physical-consistency claim rests. Please state explicitly how the z-direction is handled, whether adjacent slices are coupled in the gradient computation, and, if gradients are only two-dimensional, provide results with full 3D gradient computation or explain why the omission does not affect the central conclusions.
  3. [Table 1] The reported SSIM for SRGAN in the w channel is 0.215, while the u and v channels are 0.891 and 0.881, and all other entries in the table are above 0.6. This is almost certainly a typographical error, but it makes the table internally inconsistent and undermines the quantitative comparison. Please correct the entry and verify all reported numbers in Tables 1 and 2.
  4. [Sec. 4.1.3, Tables 1 and 2] All quantitative results are reported as averages over the first 10 time steps, without error bars, confidence intervals, or the number of random seeds. Some of the reported differences between SR-TR and the strongest baselines are small (e.g., SSIM 0.949 vs 0.936 for SR-TR FDM vs CSTU FDM in Table 1), so the observed ranking could change with statistical variability. Please provide variance estimates or per-seed results so the reader can assess the significance of the improvements.
minor comments (6)
  1. [Appendix A.2, Sec. 4.1.2] The method is named SR-TR, but the appendix and parts of the text refer to "ST-TR FDM" and "ST-TR CNN"; please unify the notation.
  2. [Sec. 3.2] The text contains the misspelling "Runga-Kutta" and should read "Runge-Kutta."
  3. [Sec. 4.1.3, Ref. [48]] The dissipation metric is attributed to a Wikipedia article on the Laplace operator; please cite a standard fluid-dynamics reference for dissipation or pseudo-dissipation instead.
  4. [Sec. 3.1, Eq. (3.3)] The equal-mean loss is written as Lmean = |Q_l - Q_d_hat|, but comparing the full LES field with the reconstructed DNS field is ambiguous because the fields have different resolutions; clarify whether the mean is taken over spatial dimensions and whether Q_l is upsampled or Q_d_hat is downsampled before the difference is computed.
  5. [Abstract, Sec. 3.1, Sec. 3.2] The abstract states that LES data are used "only in the testing phase," but Sec. 3.2 introduces a variant that uses LES data as additional input during prediction and presumably during training; please state more precisely that LES data are used only at test time in the core SR-TR variant, while an optional variant incorporates LES as input.
  6. [Sec. 4.2.4, Fig. 8] The notation "SR-TR_f FDM" is introduced in the text but the figure caption is compressed and unclear; please spell out the subscript in the figure and caption.

Circularity Check

1 steps flagged · score 6.0 of 10

Kinetic-energy validation in Sec. 4.2.3 is partly circular because Eq. (3.4) directly optimizes the same quantity used as the validation metric; SSIM and dissipation results remain independent.

  1. fitted input called prediction [Eq. (3.4) and Sec. 4.2.3 (Validation via physical metrics)]
    "However, the value of kinetic energy for incompressible flows often follows simple patterns, e.g., constant or linearly decayed, and thus can be approximated from the DNS data in the training period, denoted by ˜K. Therefore, the loss function Lkinetic is defined as follows: (3.4) Lkinetic = |K( ˆQ d ) − ˜K|. ... The performance is also evaluated through the long-term prediction of turbulent kinetic energy. Figure 7 illustrates the energies corresponding to the target DNS, along with the reconstructed flow data from both the baselines and SR-TR-based methods for the FIT dataset."

    The test-time refinement minimizes Lkinetic at every refinement step, explicitly driving K(Qhat_d) toward Ktilde, a value extrapolated from training-period DNS. For FIT and TGV, Ktilde is deliberately chosen to match the test-period DNS energy because the energy is stationary or monotonically decaying. Therefore, the Sec. 4.2.3 demonstration that SR-TR tracks DNS kinetic energy is not an independent check: it is the optimization objective itself, provided Ktilde is a good extrapolation. The SSIM and dissipation-difference results are independent, but the kinetic-energy evidence for 'preserving physical attributes' is forced by construction.

full rationale

The paper's main quantitative evaluation against held-out DNS is not circular: SSIM and dissipation difference are computed on testing-period data and are not part of any training or refinement loss. The PRU self-citation [41] is also not load-bearing evidence; PRU is used as an architectural component, not as a justification for the paper's claims. The circularity is localized to the kinetic-energy validation. Eq. (3.4) defines a refinement loss minimizing |K(Qhat_d) - Ktilde|, and Sec. 4.2.3/Fig. 7 then uses closeness of K(Qhat_d) to DNS kinetic energy as evidence of success. Since Ktilde is extrapolated from DNS training data and the tested flows are exactly those where the constant/linear-decay energy assumption holds, the SR-TR kinetic-energy curves are close by construction. This does not invalidate the independent SSIM and dissipation results, but it does make the physical-attribute claim based on kinetic energy partly circular. Score is set to 6 rather than higher because the central reconstruction claim still rests on held-out, non-loss metrics; the circularity is partial and confined to one validation channel.

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

The central method rests on five assumptions beyond standard neural-network training: incompressible Newtonian flow, LES-as-downsampled-DNS consistency, a kinetic-energy prior from training data, periodic boundaries, and 2D slice processing. The kinetic-energy prior is the most fragile because it is a fitted curve rather than a governing law, and the two test flows were selected so that it holds.

free parameters (3)
  • PRU Runge-Kutta weights {w_n} = not listed
    Trainable weights combining the RK intermediate temporal derivatives (Section 2.1); fitted to DNS training data.
  • Refinement loss weights alpha0, alpha1, alpha2 = 1000, 1, 1
    Chosen by hand in Section A.3; control balance of LES degradation, mean velocity, and kinetic energy losses.
  • Kinetic energy target K~ = constant or linearly decaying function
    Approximated from training-period DNS (Section 3.1); used as a test-time constraint.
assumptions (5)
  • domain assumption Incompressible Newtonian flow with constant density
    Section 2 states the problem formulation; the Navier-Stokes form in Eq. (2.1) relies on it.
  • domain assumption LES data are consistent with a downsampled DNS field
    Eq. (3.2) minimizes MSE between downsampled reconstruction and LES; if the LES filter or subgrid model makes LES differ from coarse DNS, the refinement target is biased.
  • ad hoc to paper Kinetic energy of the test flow follows the same simple pattern as training DNS
    Eq. (3.4) uses K~ from training DNS; the paper justifies it as often following simple patterns but provides no physical guarantee for general flows.
  • domain assumption Boundary conditions are periodic
    Periodic data augmentation is used in Section 4.1.3; valid for FIT and TGV but not for general walls or inlets.
  • ad hoc to paper The CSTU operates on 2D slices and omits z-gradients in the Navier-Stokes constraint
    Section 3.2 describes a continuous 2D feature space and estimates spatial derivatives on slices, while the paper claims 3D reconstruction.

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Pith. "Pith review of Modeling Continuous Spatial-temporal Dynamics of Turbulent Flow with Test-time Refinement." pith.science (2026). https://pith.science/paper/V4JQQQPC

@misc{pith2026241219927,
  author       = {Pith},
  title        = {Pith review of: Modeling Continuous Spatial-temporal Dynamics of Turbulent Flow with Test-time Refinement},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V4JQQQPC}},
  note         = {Machine review of arXiv:2412.19927}
}
read the original abstract

The precise simulation of turbulent flows holds immense significance across various scientific and engineering domains, including climate science, freshwater science, and energy-efficient manufacturing. Within the realm of simulating turbulent flows, large eddy simulation (LES) has emerged as a prevalent alternative to direct numerical simulation (DNS), offering computational efficiency. However, LES cannot accurately capture the full spectrum of turbulent transport scales and is present only at a lower spatial resolution. Reconstructing high-fidelity DNS data from the lower-resolution LES data is essential for numerous applications, but it poses significant challenges to existing super-resolution techniques, primarily due to the complex spatio-temporal nature of turbulent flows. This paper proposes a novel flow reconstruction approach that leverages physical knowledge to model flow dynamics. Different from traditional super-resolution techniques, the proposed approach uses LES data only in the testing phase through a degradation-based refinement approach to enforce physical constraints and mitigate cumulative reconstruction errors over time. Furthermore, a feature sampling strategy is developed to enable flow data reconstruction across different resolutions. The results on two distinct sets of turbulent flow data indicate the effectiveness of the proposed method in reconstructing high-resolution DNS data, preserving the inherent physical attributes of flow transport, and achieving DNS reconstruction at different resolutions.

Figures

Figures reproduced from arXiv: 2412.19927 by the authors.

Figure 1
Figure 1. The data distribution of LES, DNS, and HR [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The PRU, based on the Navier-Stokes equa [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The overall structure of SR-TR, which comprises the degradation-based refinement and the CSTU. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Change of dissipation difference by different models from 1st (5.6s) to 20th (6s) time step in FIT dataset. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Reconstructed w channel by each method on a sample testing slice along the z dimension in the FIT dataset. The reconstruction results are shown at 20th (6s) in (a)-(f). (a) LES Upscaling. (b) DCS/MS. (c) CTN. (d) SR-TRFDM. (e) SR-TRCNN. (f) Target DNS [PITH_FULL_IMAGE…
Figure 6
Figure 6. Figure 6: Reconstructed w channel by each method on a sample testing slice along the z dimension in the TGV dataset. The reconstruction results are shown at 15th (110s) in (a)-(f). GAN [16], two well-known dynamic fluid downscaling methods: DCS/MS [33] and FSR [37], and the Four…
Figure 7
Figure 7. Figure 7: Change of kinetic energy produced by the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: The quantitative performance of the models in [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 11
Figure 11. Figure 11: Several observations are highlighted: (1) With [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 10
Figure 10. Figure 10: Change of SSIM produced by different models from 1st (5.6s) to 20th (6s) time step in FIT dataset. [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: Change of dissipation difference by different models from 1st (5.6s) to 20th (6s) time step in FIT [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: Change of SSIM produced by different models from 1st (80s) to 20th (120s) time step in TGV dataset. [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 13
Figure 13. Figure 13: Change of dissipation difference by different models from 1st (80s) to 20th (120s) time step in TGV [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: Reconstructed w channel by each method on a sample testing slice along the z dimension in the FIT dataset. The reconstruction results are shown at 1st (5.6s), 10th (5.8s) and 20th (6s) in (a)-(f), (g)-(l), and (m)-(r), respectively [PITH_FULL_IMAGE:figures/full_fig_p…
Figure 15
Figure 15. Figure 15: Change of kinetic energy produced by the reference DNS and different models in the TGV datasets. derivative within the CSTU can obtain similar perfor￾mance. The same conclusion can be drawn from the temporal analysis of the TGV data, shown in Figs. 12 and 13. B.1.1 Vi…
Figure 16
Figure 16. Figure 16: Reconstructed w channel by each method on a sample testing slice along the z dimension in the TGV dataset. The reconstruction results are shown at 1st (80s), 10th (100s), and 15th (110s) in (a)-(f), (g)-(l), and (m)-(r), respectively. analysis of TGV data in supplemen…

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Reference graph

Works this paper leans on

50 extracted references · 43 canonical work pages

  1. [1]

    Gulf stream shifts following enso events

    Arnold H Taylor, Michael B Jordan, and John A Stephens. Gulf stream shifts following enso events. Nature, 393(6686):638–638, 1998

    work page 1998

  2. [2]

    el ni˜ no

    J Javier D ´ ıez. About “el ni˜ no” and other concomitant phenomena. Journal of Coastal Research , 21(6):xiii– xviii, 2005

    work page 2005

  3. [3]

    Integration of fully 3d fluid dynamics and geophysical fluid dynamics models for multiphysics coastal ocean flows: Simulation of lo- cal complex free-surface phenomena

    K Qu, HS Tang, and A Agrawal. Integration of fully 3d fluid dynamics and geophysical fluid dynamics models for multiphysics coastal ocean flows: Simulation of lo- cal complex free-surface phenomena. Ocean Modelling, 135:14–30, 2019

    work page 2019

  4. [4]

    Deep learning to represent subgrid processes in climate models

    Stephan Rasp, Michael S Pritchard, and Pierre Gen- tine. Deep learning to represent subgrid processes in climate models. Proceedings of the National Academy of Sciences, 2018

    work page 2018

  5. [5]

    P. Givi. Spectral and random vortex methods in turbulent reacting flows. In P. A. Libby and F. A. Williams, editors, Turbulent Reacting Flows, chapter 8, pages 475–572. Academic Press, London, England, 1994

    work page 1994

  6. [6]

    Large eddy simulation for incompress- ible flows: an introduction

    Pierre Sagaut. Large eddy simulation for incompress- ible flows: an introduction . Springer Science & Busi- ness Media, 2005

    work page 2005

  7. [7]

    Self-contained filtered density function

    Arash G Nouri, Mehdi B Nik, Pope Givi, Daniel Livescu, and Stephen B Pope. Self-contained filtered density function. Physical Review Fluids , 2(9):094603, 2017

    work page 2017

  8. [8]

    Super-resolution image reconstruction: a technical overview

    Sung Cheol Park, Min Kyu Park, and Moon Gi Kang. Super-resolution image reconstruction: a technical overview. IEEE signal processing magazine , 20(3):21– 36, 2003

    work page 2003

Show all 50 references
  1. [9]

    Understanding of a convolutional neural net- work

    Saad Albawi, Tareq Abed Mohammed, and Saad Al- Zawi. Understanding of a convolutional neural net- work. In 2017 international conference on engineering and technology (ICET) , pages 1–6. IEEE, 2017

    2017

  2. [10]

    Learning a deep convolutional network for image super-resolution

    Chao Dong, Chen Change Loy, Kaiming He, and Xi- aoou Tang. Learning a deep convolutional network for image super-resolution. In Computer Vision–ECCV 2014: 13th European Conference, Zurich, Switzerland, September 6-12, 2014, Proceedings, Part IV 13 , pages 184–199. Springer, 2014

    2014

  3. [11]

    Residual dense network for image super- resolution

    Yulun Zhang, Yapeng Tian, Yu Kong, Bineng Zhong, and Yun Fu. Residual dense network for image super- resolution. arXiv preprint arXiv:1802.08797 , 2018

    2018 arXiv

  4. [12]

    Fast, accurate, and lightweight super-resolution with cascading residual network

    Namhyuk Ahn, Byungkon Kang, and Kyung-Ah Sohn. Fast, accurate, and lightweight super-resolution with cascading residual network. In Proceedings of the Eu- ropean conference on computer vision (ECCV) , pages 252–268, 2018

    2018

  5. [13]

    Second-order attention network for sin- gle image super-resolution

    Tao Dai, Jianrui Cai, Yongbing Zhang, Shu-Tao Xia, and Lei Zhang. Second-order attention network for sin- gle image super-resolution. In 2019 IEEE/CVF Con- ference on Computer Vision and Pattern Recognition (CVPR), pages 11057–11066, 2019

    2019

  6. [14]

    A fast and efficient super- resolution network using hierarchical dense residual learning

    Vinh Van Duong, Thuc Nguyen Huu, Jonghoon Yim, and Byeungwoo Jeon. A fast and efficient super- resolution network using hierarchical dense residual learning. In 2021 IEEE International Conference on Image Processing (ICIP) , pages 1809–1813. IEEE, 2021

    2021

  7. [15]

    Image super-resolution using very deep residual channel attention networks

    Yulun Zhang, Kunpeng Li, Kai Li, Lichen Wang, Bineng Zhong, and Yun Fu. Image super-resolution using very deep residual channel attention networks. In Proceedings of the European conference on computer vision (ECCV) , pages 286–301, 2018

    2018

  8. [16]

    Photo-realistic single image super- resolution using a generative adversarial network

    Christian Ledig, Lucas Theis, Ferenc Husz´ ar, Jose Caballero, Andrew Cunningham, Alejandro Acosta, Andrew Aitken, Alykhan Tejani, Johannes Totz, Ze- han Wang, et al. Photo-realistic single image super- resolution using a generative adversarial network. In Proceedings of the I...

    2017

  9. [17]

    Recovering realistic texture in image super- resolution by deep spatial feature transform

    Xintao Wang, Ke Yu, Chao Dong, and Chen Change Loy. Recovering realistic texture in image super- resolution by deep spatial feature transform. In Pro- ceedings of the IEEE conference on computer vision and pattern recognition, pages 606–615, 2018

    2018

  10. [18]

    Esr- gan: Enhanced super-resolution generative adversarial networks

    Xintao Wang, Ke Yu, Shixiang Wu, Jinjin Gu, Yihao Liu, Chao Dong, Yu Qiao, and Chen Change Loy. Esr- gan: Enhanced super-resolution generative adversarial networks. In ECCV Workshops , 2018

    2018

  11. [19]

    Progressive growing of gans for improved quality, stability, and variation

    Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. arXiv preprint arXiv:1710.10196, 2017

    2017 arXiv

  12. [20]

    Robust super-resolution gan, with manifold-based and percep- tion loss

    Uddeshya Upadhyay and Suyash P Awate. Robust super-resolution gan, with manifold-based and percep- tion loss. In 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019) . IEEE, 2019

    2019

  13. [21]

    Mfagan: A compression framework for memory-efficient on-device super-resolution gan

    Wenlong Cheng, Mingbo Zhao, Zhiling Ye, and Shuhang Gu. Mfagan: A compression framework for memory-efficient on-device super-resolution gan. arXiv preprint arXiv:2107.12679, 2021

    2021 arXiv

  14. [22]

    Ranksrgan: Generative adversarial networks with ranker for image super-resolution

    Zhang Wenlong, Liu Yihao, Chao Dong, and Yu Qiao. Ranksrgan: Generative adversarial networks with ranker for image super-resolution. IEEE Transac- tions on Pattern Analysis and Machine Intelligence , 44(10):1–1, 2021

    2021

  15. [23]

    Adversarial generation of continuous im- ages

    Ivan Skorokhodov, Savva Ignatyev, and Mohamed El- hoseiny. Adversarial generation of continuous im- ages. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages 10753– 10764, 2021

    2021

  16. [24]

    Learning Copyright © 2025 by SIAM Unauthorized reproduction of this article is prohibited continuous image representation with local implicit image function

    Yinbo Chen, Sifei Liu, and Xiaolong Wang. Learning Copyright © 2025 by SIAM Unauthorized reproduction of this article is prohibited continuous image representation with local implicit image function. In Proceedings of the IEEE/CVF conference on computer vision and pattern reco...

    2025

  17. [25]

    Magnet: Mesh agnostic neural pde solver

    Oussama Boussif, Yoshua Bengio, Loubna Benabbou, and Dan Assouline. Magnet: Mesh agnostic neural pde solver. Advances in Neural Information Processing Systems, 35:31972–31985, 2022

    2022

  18. [26]

    Videoinr: Learning video implicit neural representation for continuous space-time super- resolution

    Zeyuan Chen, Yinbo Chen, Jingwen Liu, Xingqian Xu, Vidit Goel, Zhangyang Wang, Humphrey Shi, and Xiaolong Wang. Videoinr: Learning video implicit neural representation for continuous space-time super- resolution. In Proceedings of the IEEE/CVF Con- ference on Computer Vision a...

    2022

  19. [27]

    Cross- modality high-frequency transformer for mr image super-resolution

    Chaowei Fang, Dingwen Zhang, Liang Wang, Yu- lun Zhang, Lechao Cheng, and Junwei Han. Cross- modality high-frequency transformer for mr image super-resolution. In Proceedings of the 30th ACM In- ternational Conference on Multimedia , pages 1584– 1592, 2022

    2022

  20. [28]

    Transformer for single image super-resolution

    Zhisheng Lu, Juncheng Li, Hong Liu, Chaoyan Huang, Linlin Zhang, and Tieyong Zeng. Transformer for single image super-resolution. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 457–466, 2022

    2022

  21. [29]

    A hybrid network of cnn and transformer for lightweight image super-resolution

    Jinsheng Fang, Hanjiang Lin, Xinyu Chen, and Kun Zeng. A hybrid network of cnn and transformer for lightweight image super-resolution. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 1103–1112, 2022

    2022

  22. [30]

    Detail-preserving transformer for light field image super-resolution

    Shunzhou Wang, Tianfei Zhou, Yao Lu, and Huijun Di. Detail-preserving transformer for light field image super-resolution. In Proceedings of the AAAI Confer- ence on Artificial Intelligence , volume 36, pages 2522– 2530, 2022

    2022

  23. [31]

    Self-calibrated efficient trans- former for lightweight super-resolution

    Wenbin Zou, Tian Ye, Weixin Zheng, Yunchen Zhang, Liang Chen, and Yi Wu. Self-calibrated efficient trans- former for lightweight super-resolution. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 930–939, 2022

    2022

  24. [32]

    Light field image super-resolution with transformers

    Zhengyu Liang, Yingqian Wang, Longguang Wang, Jungang Yang, and Shilin Zhou. Light field image super-resolution with transformers. IEEE Signal Pro- cessing Letters, 29:563–567, 2022

    2022

  25. [33]

    Super-resolution reconstruction of turbulent flows with machine learning

    Kai Fukami, Koji Fukagata, and Kunihiko Taira. Super-resolution reconstruction of turbulent flows with machine learning. Journal of Fluid Mechanics , 870:106–120, 2019

    2019

  26. [34]

    Machine-learning-based spatio-temporal super resolu- tion reconstruction of turbulent flows

    Kai Fukami, Koji Fukagata, and Kunihiko Taira. Machine-learning-based spatio-temporal super resolu- tion reconstruction of turbulent flows. Journal of Fluid Mechanics, 2021

    2021

  27. [35]

    Deep learning methods for super-resolution re- construction of turbulent flows

    Bo Liu, Jiupeng Tang, Haibo Huang, and Xi-Yun Lu. Deep learning methods for super-resolution re- construction of turbulent flows. Physics of Fluids , 32(2):025105, 2020

    2020

  28. [36]

    Super-resolution reconstruction of turbulent velocity fields using a generative adver- sarial network-based artificial intelligence framework

    Zhiwen Deng, Chuangxin He, Yingzheng Liu, and Kyung Chun Kim. Super-resolution reconstruction of turbulent velocity fields using a generative adver- sarial network-based artificial intelligence framework. Physics of Fluids , 31(12):125111, 2019

    2019

  29. [37]

    Super-resolution reconstruction for the three-dimensional turbulence flows with a back- projection network

    Zhen Yang et al. Super-resolution reconstruction for the three-dimensional turbulence flows with a back- projection network. Physics of Fluids , 35(5), 2023

    2023

  30. [38]

    Super-resolution reconstruction of turbulent flows with a transformer-based deep learning framework

    Qin Xu et al. Super-resolution reconstruction of turbulent flows with a transformer-based deep learning framework. Physics of Fluids , 2023

    2023

  31. [39]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895, 2020

    2010 arXiv

  32. [40]

    Multi-scale message passing neural pde solvers

    L´ eonard Equer, T Konstantin Rusch, and Siddhartha Mishra. Multi-scale message passing neural pde solvers. arXiv preprint arXiv:2302.03580 , 2023

    2023 arXiv

  33. [41]

    Physics guided neural networks for spatio-temporal super- resolution of turbulent flows

    Tianshu Bao, Shengyu Chen, Taylor T Johnson, Pey- man Givi, Shervin Sammak, and Xiaowei Jia. Physics guided neural networks for spatio-temporal super- resolution of turbulent flows. In The 38th Conference on Uncertainty in Artificial Intelligence , 2022

    2022

  34. [42]

    Forced isotropic turbulence data set (extended)

    Wan Minping, Shiyi Chen, Gregory Eyink, Charles Meneveau, Perry Johnson, Eric Perlman, Randal Burns, Yi Li, Alex Szalay, and Stephen Hamilton. Forced isotropic turbulence data set (extended). 2012

    2012

  35. [43]

    The taylor-green vortex and fully developed turbulence

    Marc E Brachet, D Meiron, S Orszag, B Nickel, R Morf, and Uriel Frisch. The taylor-green vortex and fully developed turbulence. Journal of Statistical Physics , 34(5):1049–1063, 1984

    1984

  36. [44]

    Runge-kutta methods

    John Butcher. Runge-kutta methods. Scholarpedia, 2(9):3147, 2007

    2007

  37. [45]

    Numerical partial differen- tial equations: finite difference methods , volume 22

    James William Thomas. Numerical partial differen- tial equations: finite difference methods , volume 22. Springer Science & Business Media, 2013

    2013

  38. [46]

    Long short- term memory

    Sepp Hochreiter and J¨ urgen Schmidhuber. Long short- term memory. Neural computation , 9(8):1735–1780, 1997

    1997

  39. [47]

    Image quality assessment: from error visibility to structural similarity

    Zhou Wang, Alan C Bovik, Hamid R Sheikh, and Eero P Simoncelli. Image quality assessment: from error visibility to structural similarity. IEEE transac- tions on image processing , 13(4):600–612, 2004

    2004

  40. [48]

    Laplace operator — Wikipedia, the free encyclopedia, 2022

    Wikipedia contributors. Laplace operator — Wikipedia, the free encyclopedia, 2022. [Online; ac- cessed 23-January-2023]

    2022

  41. [49]

    zero- shot

    Assaf Shocher, Nadav Cohen, and Michal Irani. “zero- shot” super-resolution using deep internal learning. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3118–3126, 2018

    2018

  42. [50]

    Adam: A method for stochastic optimization

    Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. Copyright © 2025 by SIAM Unauthorized reproduction of this article is prohibited A Experimental Settings A.1 Dataset To evaluate the performance of the proposed me...

    2014 arXiv

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