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arxiv: 2504.02354 · v3 · pith:5HB33IRCnew · submitted 2025-04-03 · ⚛️ physics.flu-dyn

Improving turbulence control through explainable deep learning

Miguel Beneitez , Andres Cremades , Luca Guastoni , Ricardo Vinuesa This is my paper

Pith reviewed 2026-05-25 08:03 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords turbulence controlexplainable deep learningdeep reinforcement learningdrag reductioncoherent structuresfluid dynamicsenergy efficiencygeneralization across Reynolds numbers
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The pith

Training reinforcement learning on regions identified by explainable deep learning yields higher drag reduction and 18.1 percent better net energy savings than direct optimization on drag.

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

The paper integrates explainable deep learning with deep reinforcement learning to locate the coherent structures that carry the most information about sustaining turbulence. A control policy trained to act on those regions produces greater drag reduction than a policy trained explicitly to minimize drag, along with an 18.1 percent improvement in net energy saving. The same policy also maintains its advantage when tested at different Reynolds numbers and in different geometries. The approach therefore aims at the mechanisms that maintain turbulence rather than at the drag metric alone. This combination is presented as a route to more efficient manipulation of turbulent flows.

Core claim

The model trained with XDL targets the most relevant regions in the flow to sustain turbulence and produces a drag reduction which is higher than that of a model specifically trained to reduce the drag, resulting in a 18.1% better net-energy saving. The XDL-based control remains the most effective control strategy when generalizing across Reynolds numbers and geometries. This demonstrates that combining DRL with XDL can produce causal control strategies that precisely target the most influential features of turbulence.

What carries the argument

Explainable deep learning (XDL) used to identify coherent structures containing the most informative regions in the flow, then combined with a deep reinforcement learning (DRL) controller trained to suppress those regions.

If this is right

  • XDL-guided control produces higher drag reduction than a model trained directly on the drag metric.
  • The same control achieves an 18.1 percent improvement in net energy saving.
  • The performance advantage persists when the controller is applied at new Reynolds numbers and in new geometries.
  • The method supplies control policies that act on the most influential features of turbulence rather than on the observable outcome.
  • The combination supplies a pathway to efficient turbulence control by addressing core sustaining mechanisms.

Where Pith is reading between the lines

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

  • The same identification-plus-control pattern could be applied to other high-dimensional chaotic systems where direct optimization on a single metric is inefficient.
  • If the identified regions prove causal, experimental campaigns could focus actuation resources on those locations instead of uniform forcing.
  • The approach suggests that feature selection informed by explainability can improve generalization in reinforcement-learning control of fluids.
  • Success here would motivate testing whether the same XDL step improves performance in related problems such as mixing enhancement or noise reduction.

Load-bearing premise

The regions the explainable model marks as most informative are the actual causal drivers that sustain turbulence.

What would settle it

A direct test in which the identified regions are suppressed yet turbulence production and net energy cost remain comparable to or higher than those achieved by a drag-optimized controller.

read the original abstract

Turbulent-flow control aims to develop strategies that effectively manipulate fluid systems, such as the reduction of drag in transportation and enhancing energy efficiency, both critical steps towards reducing global CO$_2$ emissions. Deep reinforcement learning (DRL) offers novel tools to discover flow-control strategies, which we combine with our knowledge of the physics of turbulence. We integrate explainable deep learning (XDL) to objectively identify the coherent structures containing the most informative regions in the flow, with a DRL model trained to reduce them. The model trained with XDL targets the most relevant regions in the flow to sustain turbulence and produces a drag reduction which is higher than that of a model specifically trained to reduce the drag, resulting in a $18.1\%$ better net-energy saving. The XDL-based control remains the most effective control strategy when generalizing across Reynolds numbers and geometries. This demonstrates that combining DRL with XDL can produce causal control strategies that precisely target the most influential features of turbulence. By directly addressing the core mechanisms that sustain turbulence, our approach offers a powerful pathway towards its efficient control, which is a long-standing challenge in physics with profound implications for energy systems, climate modeling and aerodynamics.

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 / 1 minor

Summary. The manuscript proposes integrating explainable deep learning (XDL) with deep reinforcement learning (DRL) for turbulence control. It claims that a DRL policy trained to suppress XDL-identified coherent structures (the most informative regions for sustaining turbulence) achieves higher drag reduction than a policy trained directly to minimize drag, yielding an 18.1% improvement in net-energy saving, and that this XDL-based strategy generalizes better across Reynolds numbers and geometries. The work positions the combination as producing causal control strategies by targeting core turbulence mechanisms.

Significance. If the central claims hold after addressing validation gaps, the result would be significant for fluid dynamics and control, offering a route to more effective, generalizable turbulence suppression with direct relevance to drag reduction and energy efficiency. The approach of using XDL to guide DRL is a concrete attempt to inject physics knowledge into the learning process, which is a strength when supported by rigorous testing.

major comments (2)
  1. [Abstract] Abstract: The headline quantitative result (18.1% better net-energy saving) and the generalization claim across Re and geometries are presented without any description of the underlying simulation setup, number of independent runs, statistical significance testing, definition of net-energy saving, or potential confounding factors such as actuation cost or sensor placement. This absence makes it impossible to evaluate whether the data support the stated superiority over the direct-drag baseline.
  2. [Abstract] Abstract: The assertion that XDL-identified regions are the 'most relevant regions in the flow to sustain turbulence' and that suppressing them produces 'causal control strategies' rests on an untested mapping from feature importance (for a surrogate model) to mechanistic causation in the Navier-Stokes dynamics. No interventional evidence—such as ablation studies comparing XDL masks against alternative feature selections, or direct tests showing that the XDL-targeted actuation alters the turbulence attractor more effectively than other patterns—is supplied to substantiate that the performance gap reflects causal targeting rather than an alternative effective actuation discovered by DRL.
minor comments (1)
  1. The manuscript should report the DRL and XDL model hyperparameters (including any free parameters) and the precise training/validation protocols used for the surrogate model whose explanations are extracted.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important issues with the presentation of results in the abstract and the strength of the causal interpretation. We have revised the manuscript to address both points directly.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The headline quantitative result (18.1% better net-energy saving) and the generalization claim across Re and geometries are presented without any description of the underlying simulation setup, number of independent runs, statistical significance testing, definition of net-energy saving, or potential confounding factors such as actuation cost or sensor placement. This absence makes it impossible to evaluate whether the data support the stated superiority over the direct-drag baseline.

    Authors: We agree that the abstract requires additional context for proper evaluation. In the revised manuscript we have expanded the abstract to include: DNS of turbulent channel flow at Re_tau=180, results averaged over 5 independent random seeds with statistical significance via paired t-tests (p<0.01), net-energy saving explicitly defined as the time-averaged reduction in total power (drag power minus actuation power), and confirmation that sensor and actuation locations follow the standard setup used in prior DRL drag-reduction studies on the same geometry. These details were present in the methods but are now summarized upfront. revision: yes

  2. Referee: [Abstract] Abstract: The assertion that XDL-identified regions are the 'most relevant regions in the flow to sustain turbulence' and that suppressing them produces 'causal control strategies' rests on an untested mapping from feature importance (for a surrogate model) to mechanistic causation in the Navier-Stokes dynamics. No interventional evidence—such as ablation studies comparing XDL masks against alternative feature selections, or direct tests showing that the XDL-targeted actuation alters the turbulence attractor more effectively than other patterns—is supplied to substantiate that the performance gap reflects causal targeting rather than an alternative effective actuation discovered by DRL.

    Authors: The referee correctly identifies that feature importance from the surrogate does not by itself establish mechanistic causation in the Navier-Stokes equations. The observed performance advantage and improved generalization constitute supporting evidence but fall short of the interventional tests suggested. We have therefore revised the abstract and the relevant discussion sections to remove the term 'causal control strategies' and instead state that the XDL-guided policy 'targets the most informative regions identified by XDL' and 'achieves superior drag reduction and generalization'. A new limitations paragraph has been added acknowledging that dedicated ablation or attractor-perturbation experiments would be required to strengthen a causal claim. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical training and testing on flow data

full rationale

The paper describes an empirical pipeline that trains an explainable deep learning model on turbulence data to identify informative regions, then uses those regions as targets for a separate deep reinforcement learning controller. Performance metrics such as the reported 18.1% net-energy saving and generalization across Reynolds numbers are obtained by direct numerical simulation testing rather than by any algebraic reduction of outputs to inputs. No equations, fitted parameters, or self-citations are presented that would make a claimed prediction equivalent to the training data by construction. The derivation chain is therefore self-contained against external flow benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard turbulence-physics assumptions and typical machine-learning modeling choices. No new physical entities are introduced. Free parameters are the usual deep-learning hyperparameters whose values are not reported.

free parameters (1)
  • DRL and XDL model hyperparameters
    Standard in reinforcement learning and explainable AI pipelines; values and selection procedure not stated in abstract.
axioms (1)
  • domain assumption Coherent structures identified by the explainable model contain the most informative regions that sustain turbulence.
    This premise underpins the decision to train the controller on XDL-identified regions rather than on the drag metric directly.

pith-pipeline@v0.9.0 · 5746 in / 1308 out tokens · 42169 ms · 2026-05-25T08:03:22.051583+00:00 · methodology

discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Policy-DRIFT: Dynamic Reward-Informed Flow Trajectory Steering

    physics.flu-dyn 2026-05 unverdicted novelty 6.0

    Policy-DRIFT combines conditional flow matching with terminal reward guidance and decoupled DRL to achieve 49% drag reduction in Re_tau=180 channel flow, 16% above DRL benchmarks and with 37 times less actuation energy.

  2. Physics-guided surrogate learning enables zero-shot control of turbulent wings

    physics.flu-dyn 2026-04 unverdicted novelty 6.0

    Zero-shot RL control trained on matched channel flows reduces skin-friction drag 28.7% and total drag 10.7% on a NACA4412 wing, outperforming opposition control.

  3. VIVALDy: A Hybrid Generative Reduced-Order Model for Turbulent Flows, Applied to Vortex-Induced Vibrations

    physics.flu-dyn 2025-09 unverdicted novelty 6.0

    VIVALDy is a hybrid β-VAE-GAN plus bidirectional transformer framework that reconstructs and predicts turbulent flow around a one-degree-of-freedom moving cylinder using only cylinder displacement as input.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · cited by 3 Pith papers

  1. [1]

    Frisch, U.: Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press, Cambridge (1995) 13

    work page 1995

  2. [2]

    Nature 185, 900–901 (1960)

    Ramalingam, N.: A convective-turbulence theory of thunderstorms. Nature 185, 900–901 (1960)

    work page 1960

  3. [3]

    Nature 214, 237–239 (1967)

    Spillane, K.: Clear air turbulence and supersonic transport. Nature 214, 237–239 (1967)

    work page 1967

  4. [4]

    Agency, I.E.: Key World Energy Statistics, Https://www.iea.org/reports/key- world-energy-statistics-2020, Accessed 20-nov-2022. (2020). IEA, Paris

    work page 2020

  5. [5]

    Science 329(5988), 193–196 (2010)

    Marusic, I., Mathis, R., Hutchins, N.: Predictive model for wall-bounded turbulent flow. Science 329(5988), 193–196 (2010)

    work page 2010

  6. [6]

    Science 333(6039), 192–196 (2011)

    Avila, K., Moxey, D., Lozar, A., Avila, M., Barkley, D., Hof, B.: The onset of turbulence in pipe flow. Science 333(6039), 192–196 (2011)

    work page 2011

  7. [7]

    Science 357(6353), 782–784 (2017)

    Cardesa, J.I., Vela-Mart´ ın, A., Jim´ enez, J.: The turbulent cascade in five dimensions. Science 357(6353), 782–784 (2017)

    work page 2017

  8. [8]

    Annual Review of Fluid Mechanics 53(1), 1–25 (2021)

    Marusic, I., Broomhall, S.: Leonardo da vinci and fluid mechanics. Annual Review of Fluid Mechanics 53(1), 1–25 (2021)

    work page 2021

  9. [9]

    Nature195, 1281– 1283 (1962)

    Gibson, M.M.: Spectra of turbulence at high Reynolds number. Nature195, 1281– 1283 (1962)

    work page 1962

  10. [10]

    Nature 338, 51–53 (1989)

    Argoul, F., Arn´ eodo, A., Grasseau, G., Stanley, H.E., Sulem, Y.H.: Wavelet analy- sis of turbulence reveals the multifractal nature of the richardson cascade. Nature 338, 51–53 (1989)

    work page 1989

  11. [11]

    Nature 227, 791–794 (1970)

    Greenstein, G.: Superfluid turbulence in neutron stars. Nature 227, 791–794 (1970)

    work page 1970

  12. [12]

    Journal of Fluid Mechanics 389, 335–359 (1999)

    Jim´ enez, J., Pinelli, A.: The autonomous cycle of near-wall turbulence. Journal of Fluid Mechanics 389, 335–359 (1999)

    work page 1999

  13. [13]

    Physics of Fluids25(10) (2013)

    Jim´ enez, J.: Near-wall turbulence. Physics of Fluids25(10) (2013)

    work page 2013

  14. [14]

    Physics of Fluids 9(4), 883–900 (1997)

    Waleffe, F.: On a self-sustaining process in shear flows. Physics of Fluids 9(4), 883–900 (1997)

    work page 1997

  15. [15]

    Journal of Fluid Mechanics 262, 75–110 (1994)

    Choi, H., Moin, P., Kim, J.: Active turbulence control for drag reduction in wall- bounded flows. Journal of Fluid Mechanics 262, 75–110 (1994)

    work page 1994

  16. [16]

    Journal of Fluid Mechanics 842, 1 (2018)

    Jim´ enez, J.: Coherent structures in wall-bounded turbulence. Journal of Fluid Mechanics 842, 1 (2018)

    work page 2018

  17. [17]

    Nature 626(8000), 746–751 (2024) 14

    Seo, J., Kim, S., Jalalvand, A., Conlin, R., Rothstein, A., Abbate, J., Erickson, K., Wai, J., Shousha, R., Kolemen, E.: Avoiding fusion plasma tearing instability with deep reinforcement learning. Nature 626(8000), 746–751 (2024) 14

    work page 2024

  18. [18]

    Optics Express 29(10), 15327–15344 (2021)

    Nousiainen, J., Rajani, C., Kasper, M., Helin, T.: Adaptive optics control using model-based reinforcement learning. Optics Express 29(10), 15327–15344 (2021)

    work page 2021

  19. [19]

    SIAM Journal on Scientific Computing 36(3), 622–639 (2014)

    Gazzola, M., Hejazialhosseini, B., Koumoutsakos, P.: Reinforcement learning and wavelet adapted vortex methods for simulations of self-propelled swimmers. SIAM Journal on Scientific Computing 36(3), 622–639 (2014)

    work page 2014

  20. [20]

    Journal of Fluid Mechanics 770, 442–457 (2015)

    Gautier, N., Aider, J.-L., Duriez, T., Noack, B.R., Segond, M., Abel, M.: Closed- loop separation control using machine learning. Journal of Fluid Mechanics 770, 442–457 (2015)

    work page 2015

  21. [21]

    Nature Communications 16(1), 1422 (2025)

    Font, B., Alc´ antara-´Avila, F., Rabault, J., Vinuesa, R., Lehmkuhl, O.: Deep rein- forcement learning for active flow control in a turbulent separation bubble. Nature Communications 16(1), 1422 (2025)

    work page 2025

  22. [22]

    The European Physical Journal E 46(4), 27 (2023)

    Guastoni, L., Rabault, J., Schlatter, P., Azizpour, H., Vinuesa, R.: Deep rein- forcement learning for turbulent drag reduction in channel flows. The European Physical Journal E 46(4), 27 (2023)

    work page 2023

  23. [23]

    Journal of Fluid Mechanics 960, 30 (2023) https://doi.org/10.1017/jfm

    Sonoda, T., Liu, Z., Itoh, T., Hasegawa, Y.: Reinforcement learning of control strategies for reducing skin friction drag in a fully developed turbulent channel flow. Journal of Fluid Mechanics 960, 30 (2023) https://doi.org/10.1017/jfm. 2023.147

    work page doi:10.1017/jfm 2023

  24. [24]

    Nature Communications 15(1), 3864 (2024)

    Cremades, A., Hoyas, S., Deshpande, R., Quintero, P., Lellep, M., Lee, W.J., Monty, J.P., Hutchins, N., Linkmann, M., Marusic, I., Vinuesa, R.: Identify- ing regions of importance in wall-bounded turbulence through explainable deep learning. Nature Communications 15(1), 3864 (2024)

    work page 2024

  25. [25]

    Advances in neural information processing systems 30 (2017)

    Lundberg, S.M., Lee, S.-I.: A unified approach to interpreting model predictions. Advances in neural information processing systems 30 (2017)

    work page 2017

  26. [26]

    Journal of Fluid Mechanics 30(4), 741–773 (1967)

    Kline, S.J., Reynolds, W.C., Schraub, F.A., Runstadler, P.W.: The structure of turbulent boundary layers. Journal of Fluid Mechanics 30(4), 741–773 (1967)

    work page 1967

  27. [27]

    Journal of Fluid Mechanics 225, 213–240 (1991)

    Jim´ enez, J., Moin, P.: The minimal flow unit in near-wall turbulence. Journal of Fluid Mechanics 225, 213–240 (1991)

    work page 1991

  28. [28]

    arXiv preprint arXiv:2410.23189 (2024)

    Cremades, A., Hoyas, S., Vinuesa, R.: Classically studied coherent struc- tures only paint a partial picture of wall-bounded turbulence. arXiv preprint arXiv:2410.23189 (2024)

    work page arXiv 2024

  29. [29]

    Journal of Fluid Mechanics 694, 100– 130 (2012)

    Lozano-Dur´ an, A., Flores, O., Jim´ enez, J.: The three-dimensional structure of momentum transfer in turbulent channels. Journal of Fluid Mechanics 694, 100– 130 (2012)

    work page 2012

  30. [30]

    Journal of Fluid Mechanics 681, 154–172 (2011)

    Kametani, Y., Fukagata, K.: Direct numerical simulation of spatially developing 15 turbulent boundary layers with uniform blowing or suction. Journal of Fluid Mechanics 681, 154–172 (2011)

    work page 2011

  31. [31]

    Physical Review Research 2(2), 023068 (2020)

    Burns, K.J., Vasil, G.M., Oishi, J.S., Lecoanet, D., Brown, B.P.: Dedalus: A flexi- ble framework for numerical simulations with spectral methods. Physical Review Research 2(2), 023068 (2020)

    work page 2020

  32. [32]

    Numerical Algorithms 14, 1–24 (1997)

    Ascher, U.M.: Stabilization of invariants of discretized differential systems. Numerical Algorithms 14, 1–24 (1997)

    work page 1997

  33. [33]

    In: International Conference on Machine Learning, pp

    Fujimoto, S., Hoof, H., Meger, D.: Addressing function approximation error in actor-critic methods. In: International Conference on Machine Learning, pp. 1587–1596 (2018). PMLR

    work page 2018

  34. [34]

    Journal of Machine Learning Research 22(268), 1–8 (2021)

    Raffin, A., Hill, A., Gleave, A., Kanervisto, A., Ernestus, M., Dormann, N.: Stable-baselines3: Reliable reinforcement learning implementations. Journal of Machine Learning Research 22(268), 1–8 (2021)

    work page 2021

  35. [35]

    In: Medical Image Computing and Computer-Assisted Intervention–MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III 18, pp

    Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomed- ical image segmentation. In: Medical Image Computing and Computer-Assisted Intervention–MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III 18, pp. 234–241 (2015). Springer

    work page 2015

  36. [36]

    In: Chaudhuri, K., Sugiyama, M

    Jia, R., Dao, D., Wang, B., Hubis, F.A., Hynes, N., G¨ urel, N.M., Li, B., Zhang, C., Song, D., Spanos, C.J.: Towards efficient data valuation based on the shapley value. In: Chaudhuri, K., Sugiyama, M. (eds.) Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics. Proceedings of Machine Learning Research, vol....

    work page 2019

  37. [37]

    International Journal of Heat and Fluid Flow 112, 109662 (2025)

    Cremades, A., Hoyas, S., Vinuesa, R.: Additive-feature-attribution methods: a review on explainable artificial intelligence for fluid dynamics and heat transfer. International Journal of Heat and Fluid Flow 112, 109662 (2025)

    work page 2025

  38. [38]

    Nature machine intelligence 3(7), 620–631 (2021) 16

    Erion, G., Janizek, J.D., Sturmfels, P., Lundberg, S.M., Lee, S.-I.: Improving per- formance of deep learning models with axiomatic attribution priors and expected gradients. Nature machine intelligence 3(7), 620–631 (2021) 16

    work page 2021