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arxiv: 2605.17888 · v1 · pith:57HUYUU2new · submitted 2026-05-18 · ⚛️ physics.flu-dyn · cs.LG

Long-horizon prediction of three-dimensional wall-bounded turbulence with CTA-Swin-UNet and resolvent analysis

Bo Chen , Yitong Fan , Jie Yao , Weipeng Li This is my paper

Pith reviewed 2026-05-20 01:24 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.LG
keywords wall-bounded turbulenceautoregressive predictionmachine learningresolvent analysisSwin-UNetspectral linear stochastic estimation3D flow reconstruction
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The pith

Hybrid neural model predicts 3D wall turbulence stably for 300 steps then reconstructs full fields via resolvent analysis

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

The paper presents a hybrid framework that first uses a channel-time-attention Swin-UNet to forecast velocity components inside a single wall-parallel plane. A multi-time-scale fusion correction strategy is added to suppress error growth during repeated autoregressive steps, allowing stable forecasts out to 300 time intervals where standard networks break down after 20-50 steps. From each accurate planar snapshot the method applies resolvent-based spectral linear stochastic estimation to recover the complete three-dimensional velocity field together with its energy spectra. A sympathetic reader would care because direct numerical simulation of wall turbulence at high Reynolds number remains prohibitively expensive for long times, and a method that learns only in two dimensions while still delivering usable three-dimensional statistics could make extended forecasts practical.

Core claim

The CTA-Swin-UNet outperforms LSTM, FNO and conventional Swin-UNet on both single-step accuracy and autoregressive rollout stability, remaining usable for roughly 150 steps; the MTFC correction extends this horizon to 300 steps. Resolvent-based SLSE reconstruction applied to the predicted planar fields recovers the dominant three-dimensional flow structures and the correct energy spectral distributions, demonstrating that the combined pipeline supplies an effective and computationally efficient route to long-horizon prediction of 3D wall-bounded turbulence.

What carries the argument

Channel-time-attention Swin-UNet (CTA-Swin-UNet) augmented by multi-time-scale fusion correction (MTFC) for planar forecasting, followed by resolvent-based spectral linear stochastic estimation (SLSE) that lifts the two-dimensional predictions to full three-dimensional fields.

If this is right

  • The CTA module improves rollout stability relative to LSTM, FNO and plain Swin-UNet architectures.
  • The MTFC correction extends stable prediction from roughly 150 to 300 steps at fixed temporal spacing.
  • Resolvent SLSE recovers coherent three-dimensional structures and spectral distributions from planar inputs alone.
  • The overall pipeline reduces the computational cost of generating long sequences of 3D turbulent data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Focusing learning on planar slices could lower the volume of training data required compared with full 3D supervision.
  • The same planar-plus-reconstruction pattern might transfer to other chaotic flow problems where full-volume data are expensive.
  • Checking performance at higher Reynolds numbers would test whether the resolvent reconstruction remains faithful when small-scale turbulence intensifies.
  • Coupling the method to existing CFD solvers could allow inexpensive extension of existing simulation trajectories.

Load-bearing premise

The planar flow predictions must stay accurate enough over hundreds of autoregressive steps that the resolvent operator reconstructs the three-dimensional velocity field and spectra without large structural or energetic errors.

What would settle it

Generate 300-step rollouts with the trained model, reconstruct the 3D fields with SLSE, and compare instantaneous structures plus energy spectra against a reference direct numerical simulation at the same Reynolds number; large discrepancies would show the claim does not hold.

Figures

Figures reproduced from arXiv: 2605.17888 by Bo Chen, Jie Yao, Weipeng Li, Yitong Fan.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematic of the proposed 3D wall-bounded turbulence reconstruction framework. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The proposed CTA-Swin-UNet architecture. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Schematic of the MTFC procedure with fusion interval [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Training and validation loss curves for all four models. [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Single-step velocity predictions at a random test timestep. Each row shows one [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Error propagation during autoregressive rollout. (a) MSE evolution for all four [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Streamwise velocity time series at two randomly selected probe points. Left [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Streamwise one-dimensional energy spectra [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Spanwise one-dimensional energy spectra [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Training and validation loss of the L-SM. [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Pearson correlation coefficient [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Effect of initial fusion point ( [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Pearson correlation coefficient [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14: Temporal evolution of the three velocity components at a probe location from [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15: Streamwise energy spectra [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16: Spanwise energy spectra [PITH_FULL_IMAGE:figures/full_fig_p028_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17: Instantaneous streamwise velocity on wall-parallel (XZ) planes at [PITH_FULL_IMAGE:figures/full_fig_p029_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18: Streamwise velocity in the [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
read the original abstract

Long-horizon prediction of three-dimensional (3D) wall-bounded turbulence with machine-learning methods remains a challenging task, due to the rapid accumulation of autoregressive errors and the substantially computational cost. To address these challenges, we present a hybrid machine-learning framework, in which a channel-time-attention Swin-UNet (CTA-Swin-UNet) and a multi-time-scale fusion correction (MTFC) strategy are developed to predict the turbulent flow fields in a wall-parallel plane, with affordable computational cost. Then, 3D flow fields are reconstructed via a resolvent-based spectral linear stochastic estimation (SLSE), rooting from the predicted planar flow. Results show that the CTA-Swin-UNet outperforms the baseline models (LSTM, FNO and traditional Swin-UNet) in both single-step prediction and autoregressive rollouts, indicating the effectiveness of introducing the CTA module into the Swin-UNet architecture. At the same temporal interval, the CTA-Swin-UNet remains stable for approximately 150 rollout steps, while the baseline models fail within 20 to 50 rollout steps. After introducing the MTFC strategy, a longer horizon upto 300 steps is achieved. Using the resolvent-based SLSE reconstruction further recovers the 3D flow structures and energy spectral distributions from the predicted planar inputs, which demonstrates that the proposed framework provides an effective and computationally efficient approach for long-horizon autoregressive prediction of 3D wall-bounded turbulence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a hybrid framework for long-horizon 3D wall-bounded turbulence prediction. A channel-time-attention Swin-UNet (CTA-Swin-UNet) combined with a multi-time-scale fusion correction (MTFC) strategy is used to generate autoregressive predictions of wall-parallel planar flow fields. These planar fields then serve as input to a resolvent-based spectral linear stochastic estimation (SLSE) procedure that reconstructs the full 3D velocity field and energy spectra. The authors report that CTA-Swin-UNet outperforms LSTM, FNO, and standard Swin-UNet baselines in both single-step accuracy and rollout stability, achieving stable predictions to ~150 steps without MTFC and ~300 steps with MTFC, while the SLSE step recovers 3D structures.

Significance. If the quantitative validation holds, the work demonstrates a computationally efficient route to extended autoregressive forecasting of turbulence by restricting the expensive ML component to planar slices and delegating the wall-normal reconstruction to a physics-based linear estimator. The reported extension of stable rollout horizon from 20-50 steps (baselines) to 300 steps would be a meaningful advance for data-driven turbulence modeling if supported by error metrics that confirm the planar fields remain statistically close enough to the training distribution for SLSE to remain accurate.

major comments (2)
  1. [Abstract] Abstract: the central claim that the framework enables 'long-horizon autoregressive prediction of 3D wall-bounded turbulence' rests on the assertion that planar predictions remain accurate enough over 300 steps for resolvent SLSE to recover correct 3D structures and spectra. No quantitative measures of reconstruction error (e.g., wall-normal velocity RMS, two-point correlation error, or spectral L2 discrepancy) are reported as a function of rollout length, leaving the weakest link in the pipeline unquantified.
  2. [Results] Results section (implied by the reported rollout numbers): the statement that CTA-Swin-UNet 'remains stable for approximately 150 rollout steps' and reaches 300 steps with MTFC is presented without accompanying error curves, dataset split details, or held-out validation statistics. Because SLSE is a linear estimator whose fidelity depends on the input statistics remaining near the training distribution, the absence of these metrics makes it impossible to verify that accumulated autoregressive drift does not invalidate the 3D reconstruction.
minor comments (2)
  1. The manuscript should specify the Reynolds number, domain size, grid resolution, and training/validation split sizes for the turbulence dataset.
  2. Clarify the precise implementation of the channel-time-attention (CTA) module and how it differs from the standard Swin-UNet attention blocks; a diagram or equation would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. These points correctly identify the need for stronger quantitative support of the 3D reconstruction fidelity over long rollouts. We address each major comment below and will revise the manuscript accordingly to improve clarity and verifiability.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the framework enables 'long-horizon autoregressive prediction of 3D wall-bounded turbulence' rests on the assertion that planar predictions remain accurate enough over 300 steps for resolvent SLSE to recover correct 3D structures and spectra. No quantitative measures of reconstruction error (e.g., wall-normal velocity RMS, two-point correlation error, or spectral L2 discrepancy) are reported as a function of rollout length, leaving the weakest link in the pipeline unquantified.

    Authors: We agree that explicit quantitative reconstruction-error metrics versus rollout length are necessary to substantiate the central claim. In the revised manuscript we will add a new figure (or panel set) in the Results section that reports wall-normal velocity RMS error, two-point correlation error, and spectral L2 discrepancy of the SLSE-reconstructed fields as functions of autoregressive steps, computed on held-out test data. These curves will be shown both with and without the MTFC correction to demonstrate that the planar predictions remain sufficiently close to the training distribution for the linear estimator to remain accurate up to 300 steps. revision: yes

  2. Referee: [Results] Results section (implied by the reported rollout numbers): the statement that CTA-Swin-UNet 'remains stable for approximately 150 rollout steps' and reaches 300 steps with MTFC is presented without accompanying error curves, dataset split details, or held-out validation statistics. Because SLSE is a linear estimator whose fidelity depends on the input statistics remaining near the training distribution, the absence of these metrics makes it impossible to verify that accumulated autoregressive drift does not invalidate the 3D reconstruction.

    Authors: We acknowledge that the current Results section relies primarily on reported rollout horizons and basic planar-field metrics without full error curves or explicit dataset-split information. In the revision we will expand the Results section to include (i) time-evolving error curves (MSE and correlation coefficients) for the planar predictions on both training and held-out validation sets, (ii) a clear statement of the train/validation/test split ratios and any temporal separation used to avoid leakage, and (iii) a brief discussion of how the predicted planar statistics remain within the regime where the resolvent-based SLSE operator was calibrated. These additions will directly address the concern about accumulated drift affecting the 3D reconstruction. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical NN training plus independent physics reconstruction

full rationale

The paper describes a standard hybrid pipeline: CTA-Swin-UNet (with MTFC) is trained on turbulence snapshot data to produce planar predictions, after which an established resolvent-based SLSE method reconstructs the 3D field from those predictions. No equation or claim reduces the reported long-horizon stability or spectral recovery to a fitted parameter by construction, nor does any load-bearing step rest on a self-citation whose validity is assumed rather than independently verified. The workflow remains empirically falsifiable against held-out DNS data and external benchmarks, satisfying the criteria for a self-contained, non-circular result.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Abstract-only review; full details on model hyperparameters, training data, and exact resolvent assumptions are unavailable.

free parameters (1)
  • CTA-Swin-UNet hyperparameters and training settings
    Neural-network architectures contain numerous tunable parameters whose specific values are not reported.
axioms (1)
  • domain assumption Resolvent analysis supplies a reliable linear mapping from wall-parallel plane statistics to full 3D flow structures
    Invoked to reconstruct 3D fields from the 2D predictions.

pith-pipeline@v0.9.0 · 5808 in / 1479 out tokens · 71132 ms · 2026-05-20T01:24:06.295032+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

81 extracted references · 81 canonical work pages · 1 internal anchor

  1. [1]

    Physical Review Fluids , volume =

    Drag reduction via opposition control in turbulent channel flows at high Reynolds numbers , author =. Physical Review Fluids , volume =. 2025 , doi =

    work page 2025

  2. [2]

    and Byeon, Wonmin and Wan, Zhong Y

    Vlachas, Pantelis R. and Byeon, Wonmin and Wan, Zhong Y. and Sapsis, Themistoklis P. and Koumoutsakos, Petros , title =. Proceedings of the Royal Society A , year =

    work page

  3. [3]

    APL Machine Learning , volume =

    Autoregressive transformers for data-driven spatiotemporal learning of turbulent flows , author =. APL Machine Learning , volume =. 2023 , doi =

    work page 2023

  4. [4]

    Physical Review Fluids , volume =

    Predictions of turbulent shear flows using deep neural networks , author =. Physical Review Fluids , volume =. 2019 , doi =

    work page 2019

  5. [5]

    International Journal of Heat and Fluid Flow , volume =

    Recurrent neural networks and Koopman-based frameworks for temporal prediction in a low-order model of turbulence , author =. International Journal of Heat and Fluid Flow , volume =. 2021 , doi =

    work page 2021

  6. [6]

    Physica D: Nonlinear Phenomena , volume =

    Learning on predictions: Fusing training and autoregressive inference for long-term spatiotemporal forecasts , author =. Physica D: Nonlinear Phenomena , volume =. 2024 , doi =

    work page 2024

  7. [7]

    Annual Review of Fluid Mechanics , volume =

    Machine Learning for Fluid Mechanics , author =. Annual Review of Fluid Mechanics , volume =. 2020 , doi =

    work page 2020

  8. [8]

    Nature Communications , volume =

    Solera-Rico, Alberto and Sanmiguel Vila, Carlos and G. Nature Communications , volume =. 2024 , doi =

    work page 2024

  9. [9]

    GAMM-Mitteilungen , volume =

    A Perspective on Machine Learning Methods in Turbulence Modeling , author =. GAMM-Mitteilungen , volume =. 2021 , doi =

    work page 2021

  10. [10]

    2000 , publisher =

    Turbulent Flows , author =. 2000 , publisher =

    work page 2000

  11. [11]

    Annual Review of Fluid Mechanics , volume =

    Cascades in Wall-Bounded Turbulence , author =. Annual Review of Fluid Mechanics , volume =. 2012 , doi =

    work page 2012

  12. [12]

    Annual Review of Fluid Mechanics , volume =

    High-Reynolds Number Wall Turbulence , author =. Annual Review of Fluid Mechanics , volume =. 2011 , doi =

    work page 2011

  13. [13]

    and Hutchins, N

    Mathis, R. and Hutchins, N. and Marusic, I. , journal =. A predictive inner–outer model for streamwise turbulence statistics in wall-bounded flows , year =

    work page

  14. [14]

    , institution =

    Cess, R.D. , institution =. A survey on the literature on heat transfer in turbulent tube flow , year =

    work page

  15. [15]

    Del. J. Fluid Mech. , title =. 2006 , pages =

    work page 2006

  16. [16]

    Journal of Fluid Mechanics , volume =

    Mathis, Romain and Hutchins, Nicholas and Marusic, Ivan , title =. Journal of Fluid Mechanics , volume =. 2009 , doi =

    work page 2009

  17. [17]

    and Hutchins, Nicholas and Marusic, Ivan , title =

    Baars, Woutijn J. and Hutchins, Nicholas and Marusic, Ivan , title =. Physical Review Fluids , volume =. 2016 , doi =

    work page 2016

  18. [18]

    and Monty, Jason P

    Illingworth, Simon J. and Monty, Jason P. and Marusic, Ivan , title =. Journal of Fluid Mechanics , volume =. 2018 , doi =

    work page 2018

  19. [19]

    and McKeon, Beverley J

    Sharma, Arvind S. and McKeon, Beverley J. , title =. Journal of Fluid Mechanics , volume =. 2013 , doi =

    work page 2013

  20. [20]

    arXiv preprint , year =

    Mohan, Arvind and Daniel, Don and Chertkov, Michael and Livescu, Daniel , title =. arXiv preprint , year =

    work page

  21. [21]

    Materials Today Communications , volume =

    Koldo Portal-Porras and Unai Fernandez-Gamiz and Ekaitz Zulueta and Oscar Irigaray and Roberto Garcia-Fernandez , title =. Materials Today Communications , volume =. 2023 , doi =

    work page 2023

  22. [22]

    Journal of Fluid Mechanics , year =

    Racca, Alberto and Doan, Nguyen Anh Khoa and Magri, Luca , title =. Journal of Fluid Mechanics , year =

    work page

  23. [23]

    Duraisamy, G

    Duraisamy, K. and Iaccarino, G. and Xiao, H. , title =. Annual Review of Fluid Mechanics , year =. doi:10.1146/annurev-fluid-010518-040547 , url =

    work page doi:10.1146/annurev-fluid-010518-040547

  24. [24]

    and Fukagata, K

    Fukami, K. and Fukagata, K. and Taira, K. , title =. Journal of Fluid Mechanics , year =. doi:10.1017/jfm.2019.238 , url =

    work page doi:10.1017/jfm.2019.238 2019

  25. [25]

    and Tang, J

    Liu, Y. and Tang, J. and Huang, H. and Lu, X.-Y. , title =. Physics of Fluids , year =. doi:10.1063/1.5125706 , url =

    work page doi:10.1063/1.5125706

  26. [26]

    Yousif, M. Z. and Yu, L. and Lim, H.-C. , title =. Physics of Fluids , year =. doi:10.1063/5.0074724 , url =

    work page doi:10.1063/5.0074724

  27. [27]

    and coauthors , title =

    Pang, Z. and coauthors , title =. Computers & Fluids , year =

    work page

  28. [28]

    and Xu, Z

    Cao, Q. and Xu, Z. and Ma, C. and Yang, X. and Chen, Y. , title =. arXiv preprint , year =. 2401.15913 , url =

    work page arXiv

  29. [29]

    and coauthors , title =

    Mao, S. and coauthors , title =. Journal of Hydrology , year =

    work page

  30. [30]

    and Mohan, A

    Maulik, R. and Mohan, A. and Lusch, B. and Balaprakash, P. and Livescu, D. , title =. Physica D , year =. doi:10.1016/j.physd.2020.132368 , url =

    work page doi:10.1016/j.physd.2020.132368 2020

  31. [31]

    Srinivasan, P. A. and Guastoni, L. and Azizpour, H. and Schlatter, P. and Vinuesa, R. , title =. Physical Review Fluids , year =

    work page

  32. [32]

    Physics of Fluids , volume =

    Deng, Zhiwen and Chen, Yujia and Liu, Yingzheng and Kim, Kyung Chun , title =. Physics of Fluids , volume =. 2019 , doi =

    work page 2019

  33. [33]

    Journal of Hydrodynamics , year =

    Wang, Tong-sheng and Xi, Guang and Sun, Zhong-guo and Huang, Zhu , title =. Journal of Hydrodynamics , year =

    work page

  34. [34]

    Michael , title =

    Drikakis, Dimitris and Kokkinakis, Ioannis William and Fung, Daryl and Spottswood, S. Michael , title =. Physics of Fluids , volume =. 2024 , doi =

    work page 2024

  35. [35]

    Proceedings of the 10th International Workshop on Climate Informatics , pages =

    Chattopadhyay, Ashesh and Mustafa, Mustafa and Hassanzadeh, Pedram and Kashinath, Karthik , title =. Proceedings of the 10th International Workshop on Climate Informatics , pages =

    work page

  36. [36]

    and Zhang, Meng and Yu, Linqi and Vinuesa, Ricardo and Lim, HeeChang , title =

    Yousif, Mustafa Z. and Zhang, Meng and Yu, Linqi and Vinuesa, Ricardo and Lim, HeeChang , title =. Journal of Fluid Mechanics , volume =. 2023 , doi =

    work page 2023

  37. [37]

    Science China Physics, Mechanics & Astronomy , volume =

    Yang, Huiyu and Wang, Yunpeng and Wang, Jianchun , title =. Science China Physics, Mechanics & Astronomy , volume =. 2026 , doi =

    work page 2026

  38. [38]

    and Zhang, Y

    Liu, X. and Zhang, Y. and Guo, T. and others , title =. Nonlinear Dynamics , volume =. 2025 , doi =

    work page 2025

  39. [39]

    and Xin, J

    Peng, K. and Xin, J. and Zhu, X. and Cao, X. and Wang, Z. and Ma, Y. and others , title =. Geophysical Research Letters , volume =. 2025 , doi =

    work page 2025

  40. [40]

    and Inanc, E

    Sarma, R. and Inanc, E. and Aach, M. and Lintermann, A. , title =. Mathematics , volume =. 2024 , doi =

    work page 2024

  41. [41]

    Physics of Fluids , volume =

    Han, Rui and Wang, Yifan and Zhang, Yifan and others , title =. Physics of Fluids , volume =

    work page

  42. [42]

    Physics of Fluids , volume =

    Nakamura, Taichi and Fukami, Kai and Hasegawa, Kazuto and Nabae, Yusuke and Fukagata, Koji , title =. Physics of Fluids , volume =

    work page

  43. [43]

    and Fukami, K

    Hasegawa, K. and Fukami, K. and Murata, T. and Fukagata, K. , title =. Theoretical and Computational Fluid Dynamics , year =. doi:10.1007/s00162-020-00528-w , url =

    work page doi:10.1007/s00162-020-00528-w

  44. [44]

    and coauthors , title =

    Jeong, S. and coauthors , title =. Mathematical Problems in Engineering , year =. doi:10.1155/2021/5575722 , url =

    work page doi:10.1155/2021/5575722 2021

  45. [45]

    and coauthors , title =

    Author, A. and coauthors , title =. Flow, Turbulence and Combustion , year =

    work page

  46. [46]

    and Fukagata, K

    Fukami, K. and Fukagata, K. and Taira, K. , title =. Physics of Fluids , year =. doi:10.1063/5.0038655 , url =

    work page doi:10.1063/5.0038655

  47. [47]

    and coauthors , title =

    Ramakrishna, R. and coauthors , title =. arXiv preprint , year =. 2402.05372 , url =

    work page arXiv

  48. [48]

    and Kovachki, N

    Li, Z. and Kovachki, N. and Azizzadenesheli, K. and others , title =. Journal of Machine Learning Research , year =

    work page

  49. [49]

    and others , title =

    Li, Z. and others , title =. Engineering Applications of Computational Fluid Mechanics , year =. doi:10.1080/19942060.2022.2071488 , url =

    work page doi:10.1080/19942060.2022.2071488 2022

  50. [50]

    and coauthors , title =

    Wang, Y. and coauthors , title =. Physical Review Fluids , year =. doi:10.1103/PhysRevFluids.9.084604 , url =

    work page doi:10.1103/physrevfluids.9.084604

  51. [51]

    and coauthors , title =

    Li, Z. and coauthors , title =. Physics of Fluids , year =. doi:10.1063/5.0150682 , url =

    work page doi:10.1063/5.0150682

  52. [52]

    and coauthors , title =

    Li, Z. and coauthors , title =. arXiv preprint , year =. 2403.03051 , url =

    work page arXiv

  53. [53]

    and coauthors , title =

    Geng, S. and coauthors , title =. Engineering Applications of Artificial Intelligence , year =. doi:10.1016/j.engappai.2023.107980 , url =

    work page doi:10.1016/j.engappai.2023.107980 2023

  54. [54]

    and coauthors , title =

    Zhang, X. and coauthors , title =. Physics of Fluids , year =. doi:10.1063/5.0183187 , url =

    work page doi:10.1063/5.0183187

  55. [55]

    and coauthors , title =

    Zhang, X. and coauthors , title =. npj Computational Materials , year =

    work page

  56. [56]

    and coauthors , title =

    Liu, B. and coauthors , title =. Remote Sensing , year =. doi:10.3390/rs16234464 , url =

    work page doi:10.3390/rs16234464

  57. [57]

    and coauthors , title =

    Tang, S. and coauthors , title =. Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV) , year =

    work page

  58. [58]

    and Lozano-Dur

    Towne, A. and Lozano-Dur. Resolvent-based estimation of space--time flow statistics , journal =. 2020 , volume =. doi:10.1017/jfm.2019.796 , url =

    work page doi:10.1017/jfm.2019.796 2020

  59. [59]

    and Baddoo, P

    Herrmann, B. and Baddoo, P. J. and Semaan, R. and Brunton, S. L. and McKeon, B. J. , title =. Journal of Fluid Mechanics , year =. doi:10.1017/jfm.2021.337 , url =

    work page doi:10.1017/jfm.2021.337 2021

  60. [60]

    Ribeiro, J. H. M. and Yeh, C.-A. and Taira, K. , title =. Physical Review Fluids , year =. doi:10.1103/PhysRevFluids.5.033902 , url =

    work page doi:10.1103/physrevfluids.5.033902

  61. [61]

    and Towne, A

    Farghadan, A. and Towne, A. , title =. arXiv preprint , year =. 2309.04617 , url =

    work page arXiv

  62. [62]

    Rolandi, L. V. and coauthors , title =. Theoretical and Computational Fluid Dynamics , year =

    work page

  63. [63]

    Sharma, A. S. and McKeon, B. J. , title =. Input--Output Analysis in Fluid Mechanics , publisher =. 2025 , note =

    work page 2025

  64. [64]

    Gomez and Lukasz Kaiser and Illia Polosukhin , title =

    Ashish Vaswani and Noam Shazeer and Niki Parmar and Jakob Uszkoreit and Llion Jones and Aidan N. Gomez and Lukasz Kaiser and Illia Polosukhin , title =. Advances in Neural Information Processing Systems , volume =

    work page

  65. [65]

    An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale

    Alexey Dosovitskiy and Lucas Beyer and Alexander Kolesnikov and Dirk Weissenborn and Xiaohua Zhai and Thomas Unterthiner and Mostafa Dehghani and Matthias Minderer and Georg Heigold and Sylvain Gelly and Jakob Uszkoreit and Neil Houlsby , title =. International Conference on Learning Representations , year =. 2010.11929 , archivePrefix=

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  66. [66]

    Proceedings of the IEEE/CVF International Conference on Computer Vision , year =

    Ze Liu and Yutong Lin and Yue Cao and Han Hu and Yixuan Wei and Zheng Zhang and Stephen Lin and Baining Guo , title =. Proceedings of the IEEE/CVF International Conference on Computer Vision , year =

    work page

  67. [67]

    Theoretical and Computational Fluid Dynamics , volume =

    Kai Fukami and Koji Fukagata and Kunihiko Taira , title =. Theoretical and Computational Fluid Dynamics , volume =. 2023 , doi =

    work page 2023

  68. [68]

    Physics of Fluids , volume =

    Hamidreza Eivazi and Hadi Veisi and Mohammad Hossein Naderi and Vahid Esfahanian , title =. Physics of Fluids , volume =. 2020 , doi =

    work page 2020

  69. [69]

    Physics of Fluids , volume =

    Xiaoqiang Zhang and Zhenliang Wang and Wenqiang Zhao and Shengli Li , title =. Physics of Fluids , volume =. 2023 , doi =

    work page 2023

  70. [70]

    Physics of Fluids , volume =

    Shuai Wang and Xiang Li and Qiang Sun and Yanan Zhao , title =. Physics of Fluids , volume =. 2024 , doi =

    work page 2024

  71. [71]

    Journal of Fluids Engineering , note =

    Yue Yang and Bowen Wang and Xiang Yang and Kun Li , title =. Journal of Fluids Engineering , note =. 2024 , eprint =

    work page 2024

  72. [72]

    APL Machine Learning , volume =

    Prathamesh Patil and Mohammad Farazmand , title =. APL Machine Learning , volume =. 2023 , doi =

    work page 2023

  73. [73]

    and Perdikaris, Paris and Turner, Richard E

    Lippe, Phillip and Veeling, Bastiaan S. and Perdikaris, Paris and Turner, Richard E. and Brandstetter, Johannes , booktitle =

    work page

  74. [74]

    Physics of Fluids , volume =

    Zhiyuan Li and Yucheng Shi and Qifeng Li and Liang Chen , title =. Physics of Fluids , volume =. 2024 , doi =

    work page 2024

  75. [75]

    2025 , eprint =

    Zichao Liu and Yichi Zhang and Yuchi Ma and Jian Zhang , title =. 2025 , eprint =

    work page 2025

  76. [76]

    2024 , eprint =

    Sunwoong Yang and Ricardo Vinuesa and Namwoo Kang , title =. 2024 , eprint =

    work page 2024

  77. [77]

    Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining , pages =

    Wang, Rui and Kashinath, Karthik and Mustafa, Mustafa and Albert, Adrian and Yu, Rose , title =. Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining , pages =. 2020 , publisher =

    work page 2020

  78. [78]

    Computer Methods in Applied Mechanics and Engineering , volume =

    Ren, Pu and Rao, Chengping and Liu, Yang and Wang, Jianxun and Sun, Hao , title =. Computer Methods in Applied Mechanics and Engineering , volume =. 2022 , doi =

    work page 2022

  79. [79]

    Physics of Fluids , volume =

    Wu, Pin and Qiu, Feng and Feng, Weibing and Fang, Fangxing and Pain, Christopher , title =. Physics of Fluids , volume =. 2022 , doi =

    work page 2022

  80. [80]

    Physics of Fluids , volume =

    Hemmasian, AmirPouya and Barati Farimani, Amir , title =. Physics of Fluids , volume =. 2023 , doi =

    work page 2023

Showing first 80 references.