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

Mathis Bode; Pol Suarez; Ricardo Vinuesa; Yuning Wang

arxiv: 2604.09434 · v2 · submitted 2026-04-10 · ⚛️ physics.flu-dyn · cs.AI

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

Yuning Wang , Pol Suarez , Mathis Bode , Ricardo Vinuesa This is my paper

Pith reviewed 2026-05-10 16:54 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn cs.AI

keywords reinforcement learningflow controlturbulent boundary layerdrag reductionzero-shot transferNACA4412skin-friction dragopposition control

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The pith

Zero-shot reinforcement learning policies trained on matched channel flows reduce skin-friction drag on a wing by 28.7 percent.

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

The paper shows that reinforcement learning policies for controlling turbulent boundary layers can be trained in simpler channel flows whose statistics are matched to those on a wing, then transferred directly to the wing geometry with no additional training. This zero-shot approach delivers 28.7 percent skin-friction drag reduction and 10.7 percent total drag reduction on a NACA4412 airfoil at a chord Reynolds number of 200000, while cutting training costs by four orders of magnitude compared with direct wing training. A sympathetic reader would care because realistic aerodynamic surfaces involve spatial variations and adverse pressure gradients that make direct training prohibitively expensive, and the method still outperforms conventional opposition control. If the transfer holds, it points to a practical route for active drag reduction on engineering-scale wings.

Core claim

Policies trained in turbulent channel flows matched to wing boundary-layer statistics are deployed directly onto a NACA4412 wing at Re_c=2×10^5 without further training. This zero-shot control achieves a 28.7% reduction in skin-friction drag and a 10.7% reduction in total drag, outperforming opposition control by 40% in friction drag reduction and 5% in total drag. Training cost is reduced by four orders of magnitude relative to on-wing training.

What carries the argument

Physics-guided matching of local turbulence statistics between channel flows and wing boundary layers, enabling zero-shot transfer of reinforcement learning control policies.

Load-bearing premise

Turbulent channel flows can be matched to wing boundary-layer statistics such that the learned policy transfers zero-shot to the NACA4412 geometry under adverse pressure gradients without retraining or performance loss.

What would settle it

Direct numerical simulation or high-fidelity run of the transferred policy on the NACA4412 wing at the stated Reynolds number that yields skin-friction drag reduction substantially below 28.7 percent or fails to exceed opposition control.

read the original abstract

Turbulent boundary layers over aerodynamic surfaces are a major source of aircraft drag, yet their control remains challenging due to multiscale dynamics and spatial variability, particularly under adverse pressure gradients. Reinforcement learning has outperformed state-of-the-art strategies in canonical flows, but its application to realistic geometries is limited by computational cost and transferability. Here we show that these limitations can be overcome by exploiting local structures of wall-bounded turbulence. Policies are trained in turbulent channel flows matched to wing boundary-layer statistics and deployed directly onto a NACA4412 wing at $Re_c=2\times10^5$ without further training, being the so-called zero-shot control. This achieves a 28.7% reduction in skin-friction drag and a 10.7% reduction in total drag, outperforming the state-of-the-art opposition control by 40% in friction drag reduction and 5% in total drag. Training cost is reduced by four orders of magnitude relative to on-wing training, enabling scalable flow control.

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.

Desk Editor's Note private letter to a colleague

Zero-shot transfer from matched channel flow to NACA4412 wing drag control is the headline result, but the turbulence matching under adverse pressure gradients is the part that needs the most scrutiny.

read the letter

The main thing to know is that this paper reports a zero-shot reinforcement learning policy for drag reduction on a NACA4412 wing, trained only on turbulent channel flow data that was matched to the wing's boundary layer statistics. It claims 28.7 percent skin-friction drag cut and 10.7 percent total drag cut, better than opposition control, at a fraction of the training cost. What works here is the practical framing. Training in a simple channel and transferring directly avoids the huge cost of simulating and training on the full wing geometry. Four orders of magnitude savings is the kind of number that makes people pay attention if the performance holds. The outperformance over a standard method like opposition control adds to the case. The soft spot is the zero-shot transfer under adverse pressure gradients. Channel flow is a zero-pressure-gradient setup by design, while the wing has streamwise pressure changes that alter turbulence production and structures. The paper relies on matching low-order statistics for the transfer to work, but without checks on whether higher-order moments or pressure-gradient-sensitive quantities line up at the control locations, it's unclear if the policy is controlling the right things on the wing. The abstract gives the headline numbers but no error bars, run counts, or matching validation details, so the central claim needs the full methods to assess. This is for researchers in computational fluid dynamics and data-driven control who are trying to scale reinforcement learning beyond simple geometries. A reader working on surrogate models or transfer learning in turbulence would see value in the cost argument and the specific geometry choice. The work shows clear thinking on the computational bottleneck and engages with real aerodynamic conditions, so it deserves a serious referee. The referees can press on the matching procedure and ask for more evidence that the transfer is robust. I would send this to peer review.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that reinforcement learning policies trained in turbulent channel flows whose low-order statistics are matched to a wing boundary layer can be deployed zero-shot onto a NACA4412 airfoil at Re_c=2×10^5, yielding a 28.7% reduction in skin-friction drag and 10.7% reduction in total drag while outperforming opposition control by 40% and 5% respectively; training cost is reduced by four orders of magnitude relative to direct on-wing training.

Significance. If the zero-shot transfer is robust, the approach would provide a scalable route to RL-based flow control on realistic aerodynamic geometries by exploiting local wall-bounded turbulence structures, substantially lowering the computational barrier that currently limits such methods to canonical flows.

major comments (3)

[Methods (surrogate training and matching)] The matching procedure between channel-flow statistics and wing boundary-layer statistics (described in the methods section on surrogate training) supplies no quantitative validation that higher-order moments, structure functions, or adverse-pressure-gradient-sensitive quantities remain equivalent at the actuator locations. This directly underpins the zero-shot transfer claim, as the NACA4412 experiences streamwise APG absent from equilibrium channel flow.
[Results (drag reduction figures)] The reported drag reductions (28.7% skin-friction, 10.7% total) in the results section are presented without error bars, confidence intervals, number of independent realizations, or statistical significance tests relative to opposition control. The abstract states quantitative improvements, yet the absence of these details prevents assessment of whether the gains are load-bearing or within run-to-run variability.
[Results (zero-shot deployment)] The zero-shot deployment section does not report any diagnostic checks (e.g., comparison of near-wall streak spacing, production profiles, or coherent-structure lifetimes) confirming that the policy exploits only features preserved under the APG of the NACA4412. Without such checks, the performance advantage over opposition control cannot be attributed unambiguously to the physics-guided matching.

minor comments (2)

[Throughout] Notation for the surrogate model and policy network is introduced without a dedicated nomenclature table; several symbols (e.g., those denoting matched statistics) are reused across sections without redefinition.
[Figures] Figure captions for the flow visualizations and drag time histories do not state the number of snapshots or averaging windows used, reducing reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify and strengthen the presentation of our results. We address each major comment point by point below, indicating where revisions will be incorporated.

read point-by-point responses

Referee: [Methods (surrogate training and matching)] The matching procedure between channel-flow statistics and wing boundary-layer statistics (described in the methods section on surrogate training) supplies no quantitative validation that higher-order moments, structure functions, or adverse-pressure-gradient-sensitive quantities remain equivalent at the actuator locations. This directly underpins the zero-shot transfer claim, as the NACA4412 experiences streamwise APG absent from equilibrium channel flow.

Authors: We agree that the current matching focuses on low-order statistics and that explicit validation of higher-order quantities would better support the zero-shot claim. The successful transfer itself provides supporting evidence that the preserved local structures suffice for control, but to address the concern directly we will add quantitative comparisons of selected higher-order moments, structure functions, and APG-sensitive profiles at actuator locations in the revised methods section. revision: yes
Referee: [Results (drag reduction figures)] The reported drag reductions (28.7% skin-friction, 10.7% total) in the results section are presented without error bars, confidence intervals, number of independent realizations, or statistical significance tests relative to opposition control. The abstract states quantitative improvements, yet the absence of these details prevents assessment of whether the gains are load-bearing or within run-to-run variability.

Authors: We acknowledge that statistical details are essential for assessing robustness. The reported values derive from long-time averages, but we will revise the results section to include error bars computed from multiple independent realizations, state the number of runs performed, and add statistical significance tests against opposition control. revision: yes
Referee: [Results (zero-shot deployment)] The zero-shot deployment section does not report any diagnostic checks (e.g., comparison of near-wall streak spacing, production profiles, or coherent-structure lifetimes) confirming that the policy exploits only features preserved under the APG of the NACA4412. Without such checks, the performance advantage over opposition control cannot be attributed unambiguously to the physics-guided matching.

Authors: We appreciate the request for direct diagnostics. While the performance advantage is the central result, we will add diagnostic comparisons (near-wall streak spacing and production profiles) between the matched channel and the NACA4412 wing in the revised zero-shot deployment section to more explicitly link the gains to the preserved features. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical transfer results are measured, not derived by construction

full rationale

The paper reports measured drag reductions (28.7% skin-friction, 10.7% total) from direct deployment of a policy trained on statistically matched channel flows onto the NACA4412 wing. These outcomes are obtained from independent CFD evaluations on the target geometry, not from any closed-form derivation, fitted parameter renamed as prediction, or self-referential definition. The matching procedure supplies training data statistics but does not mathematically entail the observed control performance; the zero-shot claim is validated by the reported simulation results rather than assumed. No load-bearing self-citations, uniqueness theorems, or ansatzes reduce the central result to its inputs. The derivation chain consists of standard RL training followed by empirical testing and is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based on abstract only; the central claim rests on the unverified domain assumption that channel-flow turbulence statistics can be matched to wing boundary layers for policy transfer. No free parameters or invented entities are explicitly quantified in the abstract.

axioms (1)

domain assumption Turbulent channel flows can be statistically matched to wing boundary-layer statistics under adverse pressure gradients
This matching is invoked to justify training the policy in channels for zero-shot deployment on the wing.

pith-pipeline@v0.9.0 · 5476 in / 1289 out tokens · 70339 ms · 2026-05-10T16:54:36.266257+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

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.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · cited by 1 Pith paper · 2 internal anchors

[1]

Abbas,et al., Drag reduction via turbulent boundary layer flow control.Sci

A. Abbas,et al., Drag reduction via turbulent boundary layer flow control.Sci. China Technol. Sci.60, 1281–1290 (2017)

work page 2017
[2]

Fukagata, K

K. Fukagata, K. Iwamoto, Y. Hasegawa, Turbulent drag reduction by streamwise traveling waves of wall-normal forcing.Annu. Rev. Fluid Mech.56, 69–90 (2024)

work page 2024
[3]

F. Z. Wang, I. L. Animasaun, T. Muhammad, S. S. Okoya, Recent advancements in fluid dynamics: drag reduction, lift generation, computational fluid dynamics, turbulence modelling, and multiphase flow.Arab. J. Sci. Eng.49, 1–13 (2024)

work page 2024
[4]

J. I. Cardesa, A. Vela-Mart ´ın, J. Jim´enez, The turbulent cascade in five dimensions.Science 357(6353), 782–784 (2017), doi:10.1126/science.aan7933

work page doi:10.1126/science.aan7933 2017
[5]

Seifert, A

A. Seifert, A. Darabi, I. Wygnanski, Delay of airfoil stall by periodic excitation.J. Aircr.33(4), 691–698 (1996)

work page 1996
[6]

L. N. Cattafesta, M. Sheplak, Actuators for active flow control.Annu. Rev. Fluid Mech.43, 247–272 (2011)

work page 2011
[7]

S. S. Collis, R. D. Joslin, A. Seifert, V. Theofilis, Issues in active flow control: theory, control, simulation, and experiment.Prog. Aerosp. Sci.40(4–5), 237–289 (2004)

work page 2004
[8]

S. L. Brunton, B. R. Noack, Closed-loop turbulence control: progress and challenges.Appl. Mech. Rev.67(5), 050801 (2015)

work page 2015
[9]

S. L. Brunton, Applying machine learning to study fluid mechanics.Acta Mech. Sin.37(12), 1718–1726 (2021)

work page 2021
[10]

Rabault, M

J. Rabault, M. Kuchta, A. Jensen, U. R ´eglade, N. Cerardi, Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control.J. Fluid Mech.865, 281–302 (2019)

work page 2019
[11]

Vignon, J

C. Vignon, J. Rabault, R. Vinuesa, Recent advances in applying deep reinforcement learning for flow control: perspectives and future directions.Phys. Fluids35(3), 031301 (2023). 19

work page 2023
[12]

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

work page 1994
[13]

A. J. Smits, B. J. McKeon, I. Marusic, High-Reynolds number wall turbulence.Annu. Rev. Fluid Mech.43, 353–375 (2011)

work page 2011
[14]

Marusic, R

I. Marusic, R. Mathis, N. Hutchins, Predictive model for wall-bounded turbulent flow.Science 329(5988), 193–196 (2010), doi:10.1126/science.1188765

work page doi:10.1126/science.1188765 2010
[15]

Stroh, B

A. Stroh, B. Frohnapfel, P. Schlatter, Y. Hasegawa, A comparison of opposition control in turbulent boundary layer and turbulent channel flow.Phys. Fluids27(7), 075101 (2015)

work page 2015
[16]

Dacome, R

G. Dacome, R. M ¨orsch, M. Kotsonis, W. J. Baars, Opposition flow control for reducing skin- friction drag of a turbulent boundary layer.Phys. Rev. Fluids9(6), 064602 (2024)

work page 2024
[17]

J. Yao, E. Garc´ıa, F. Hussain, Drag reduction via opposition control in turbulent channel flows at high Reynolds numbers.Phys. Rev. Fluids10(9), 094604 (2025)

work page 2025
[18]

J. M. Hamilton, J. Kim, F. Waleffe, Regeneration mechanisms of near-wall turbulence struc- tures.J. Fluid Mech.287, 317–348 (1995)

work page 1995
[19]

Jim ´enez, S

J. Jim ´enez, S. Hoyas, M. P. Simens, Y. Mizuno, Turbulent boundary layers and channels at moderate Reynolds numbers.J. Fluid Mech.657, 335–360 (2010)

work page 2010
[20]

Raissi, A

M. Raissi, A. Yazdani, G. E. Karniadakis, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations.Science367(6481), 1026–1030 (2020), doi:10.1126/ science.aaw4741

work page 2020
[21]

Su ´arez,et al., Flow control of three-dimensional cylinders transitioning to turbulence via multi-agent reinforcement learning.Commun

P. Su ´arez,et al., Flow control of three-dimensional cylinders transitioning to turbulence via multi-agent reinforcement learning.Commun. Eng.4(1), 113 (2025)

work page 2025
[22]

Guastoni, J

L. Guastoni, J. Rabault, P. Schlatter, H. Azizpour, R. Vinuesa, Deep reinforcement learning for turbulent drag reduction in channel flows.Eur. Phys. J. E46(4), 27 (2023)

work page 2023
[23]

Sonoda, Z

T. Sonoda, Z. Liu, T. Itoh, Y. Hasegawa, Reinforcement learning of control strategies for reducing skin friction drag in a fully developed turbulent channel flow.J. Fluid Mech.960, A30 (2023). 20

work page 2023
[24]

Su ´arez,et al., Active flow control for drag reduction through multi-agent reinforcement learning on a turbulent cylinder at𝑅𝑒 𝐷 =3900.Flow Turbul

P. Su ´arez,et al., Active flow control for drag reduction through multi-agent reinforcement learning on a turbulent cylinder at𝑅𝑒 𝐷 =3900.Flow Turbul. Combust.115, 3–27 (2025)

work page 2025
[25]

Varela,et al., Deep reinforcement learning for flow control exploits different physics for increasing Reynolds number regimes.Actuators11(12), 359 (2022)

P. Varela,et al., Deep reinforcement learning for flow control exploits different physics for increasing Reynolds number regimes.Actuators11(12), 359 (2022)

work page 2022
[26]

Q. Liu, L. J. Trujillo Corona, D. Espinoza, F. Shu, A. Gross, Design and dimensional transfer of reinforcement learning-based closed-loop airfoil flow control.Theor. Comput. Fluid Dyn. 39(6), 49 (2025)

work page 2025
[27]

V. Belus,et al., Exploiting locality and translational invariance to design effective deep rein- forcement learning control of the 1-dimensional unstable falling liquid film.AIP Adv.9(12), 125141 (2019)

work page 2019
[28]

Rabault, A

J. Rabault, A. Kuhnle, Accelerating deep reinforcement learning strategies of flow control through a multi-environment approach.Phys. Fluids31(9), 094105 (2019)

work page 2019
[29]

Cremades,et al., Identifying regions of importance in wall-bounded turbulence through explainable deep learning.Nat

A. Cremades,et al., Identifying regions of importance in wall-bounded turbulence through explainable deep learning.Nat. Commun.15(1), 3864 (2024)

work page 2024
[30]

Sanchis-Agudo, A

M. Sanchis-Agudo, A. Cremades, A. Martinez-Sanchez, A. Lozano-Duran, R. Vinuesa, X- CAL: explaining latent causality in physical space for fluid mechanics.arXiv:2601.03311 (2026)

work page arXiv 2026
[31]

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

work page 2025
[32]

Diff-sport: Diffusion-based sensor placement optimization and reconstruction of turbulent ﬂows in urban environments

A. Vishwasrao,et al., Diff-SPORT: diffusion-based sensor placement optimization and recon- struction of turbulent flows in urban environments.arXiv:2506.00214(2025)

work page arXiv 2025
[33]

Weiner, J

A. Weiner, J. Reyes, Model-based deep reinforcement learning for accelerated learning from flow simulations.arXiv:2402.16543(2024)

work page arXiv 2024
[34]

Zhang,et al., Efficient active flow control strategy for confined square cylinder wake using deep learning-based surrogate model and reinforcement learning.arXiv:2408.14232(2024)

Z. Zhang,et al., Efficient active flow control strategy for confined square cylinder wake using deep learning-based surrogate model and reinforcement learning.arXiv:2408.14232(2024). 21

work page arXiv 2024
[35]

and Moxey, D

K. Avila,et al., The onset of turbulence in pipe flow.Science333(6039), 192–196 (2011), doi:10.1126/science.1203223

work page doi:10.1126/science.1203223 2011
[36]

Lagemann,et al., HydroGym: a reinforcement learning platform for fluid dynamics, arXiv:2512.17534 (2025)

C. Lagemann,et al., HydroGym: a reinforcement learning platform for fluid dynamics, arXiv:2512.17534 (2025)

work page arXiv 2025
[37]

Vinuesa, A

R. Vinuesa, A. Bobke, R. ¨Orl¨ u, P. Schlatter, On determining characteristic length scales in pressure-gradient turbulent boundary layers.Phys. Fluids28(5), 055101 (2016)

work page 2016
[38]

Harun, J

Z. Harun, J. P. Monty, R. Mathis, I. Marusic, Pressure gradient effects on the large-scale structure of turbulent boundary layers.J. Fluid Mech.715, 477–498 (2013)

work page 2013
[39]

Atzori,et al., Uniform blowing and suction applied to nonuniform adverse-pressure-gradient wing boundary layers.Phys

M. Atzori,et al., Uniform blowing and suction applied to nonuniform adverse-pressure-gradient wing boundary layers.Phys. Rev. Fluids6(11), 113904 (2021)

work page 2021
[40]

Y. Wang, M. Atzori, R. Vinuesa, Opposition control applied to turbulent wing sections.J. Fluid Mech.1010, A29 (2025)

work page 2025
[41]

Vinuesa,et al., Turbulent boundary layers around wing sections up to𝑅𝑒 𝑐 =1,000,000.Int

R. Vinuesa,et al., Turbulent boundary layers around wing sections up to𝑅𝑒 𝑐 =1,000,000.Int. J. Heat Fluid Flow72, 86–99 (2018)

work page 2018
[42]

and Vinuesa, R

M. Atzori,et al., Aerodynamic effects of uniform blowing and suction on a NACA4412 airfoil. Flow Turbul. Combust.105(3), 735–759 (2020), doi:10.1007/s10494-020-00135-z

work page doi:10.1007/s10494-020-00135-z 2020
[43]

Quadrio, P

M. Quadrio, P. Ricco, Critical assessment of turbulent drag reduction through spanwise wall oscillations.J. Fluid Mech.667, 135–157 (2011), doi:10.1017/S002211201000367X

work page doi:10.1017/s002211201000367x 2011
[44]

Quadrio, P

M. Quadrio, P. Ricco, C. Viotti, Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction.J. Fluid Mech.627, 161–178 (2009)

work page 2009
[45]

Marusic,et al., An energy-efficient pathway to turbulent drag reduction.Nat

I. Marusic,et al., An energy-efficient pathway to turbulent drag reduction.Nat. Commun.12, 5805 (2021)

work page 2021
[46]

P. F. Fischer, J. W. Lottes, S. G. Kerkemeier, Nek5000 web page (2008), http://nek5000.mcs.anl.gov. 22

work page 2008
[47]

B. B. De Moura, M. R. Machado, S. Dey, T. Mukhopadhyay, Manipulating flexural waves to enhance broadband vibration mitigation through programmed disorder on smart rainbow metamaterials.Appl. Math. Modelling125, 650–671 (2024)

work page 2024
[48]

Maday, A

Y. Maday, A. T. Patera, Spectral element methods for the incompressible Navier-Stokes equa- tions.State-of-the-Art Surveys on Computational Mechanicspp. 71–143 (1989)

work page 1989
[49]

Tanarro, R

´A. Tanarro, R. Vinuesa, P. Schlatter, Effect of adverse pressure gradients on turbulent wing boundary layers.J. Fluid Mech.883, A8 (2020)

work page 2020
[50]

P. S. Negi, R. Vinuesa, A. Hanifi, P. Schlatter, D. S. Henningson, Unsteady aerodynamic effects in small-amplitude pitch oscillations of an airfoil.Int. J. Heat Fluid Flow71, 378–391 (2018)

work page 2018
[51]

F. R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J.32(8), 1598–1605 (1994)

work page 1994
[52]

S. Dong, G. E. Karniadakis, C. Chryssostomidis, A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains.J. Comput. Phys.261, 83–105 (2014)

work page 2014
[53]

Schlatter, R

P. Schlatter, R. ¨Orl¨ u, Turbulent boundary layers at moderate Reynolds numbers: inflow length and tripping effects.J. Fluid Mech.710, 5–34 (2012)

work page 2012
[54]

Vinuesa, S

R. Vinuesa, S. M. Hosseini, A. Hanifi, D. S. Henningson, P. Schlatter, Pressure-gradient turbulent boundary layers developing around a wing section.Flow Turbul. Combust.99, 613– 641 (2017)

work page 2017
[55]

E. P. Hammond, T. R. Bewley, P. Moin, Observed mechanisms for turbulence attenuation and enhancement in opposition-controlled wall-bounded flows.Phys. Fluids10(9), 2421–2423 (1998)

work page 1998
[56]

T. P. Lillicrap,et al., Continuous control with deep reinforcement learning.arXiv:1509.02971 (2015). 23

work page internal anchor Pith review arXiv 2015
[57]

Fujimoto, H

S. Fujimoto, H. Hoof, D. Meger, Addressing function approximation error in actor-critic methods, inProceedings of the 35th International Conference on Machine Learning(PMLR) (2018), pp. 1587–1596

work page 2018
[58]

T. Lee, J. Kim, C. Lee, Turbulence control for drag reduction through deep reinforcement learning.Phys. Rev. Fluids8(2), 024604 (2023)

work page 2023
[59]

Sonoda, Z

T. Sonoda, Z. Liu, T. Itoh, Y. Hasegawa, Reinforcement learning of control strategies for reducing skin friction drag in a fully developed turbulent channel flow.J. Fluid Mech.960, A30 (2023)

work page 2023
[60]

W ¨alchli, L

D. W ¨alchli, L. Guastoni, R. Vinuesa, P. Koumoutsakos, Drag reduction in a minimal channel flow with scientific multi-agent reinforcement learning.J. Phys.: Conf. Ser.2753(1), 012024 (2024)

work page 2024
[61]

G. M. Cavallazzi, L. Guastoni, R. Vinuesa, A. Pinelli, Deep reinforcement learning for the management of the wall regeneration cycle in wall-bounded turbulent flows.Flow Turbul. Combust.115(3), 1291–1317 (2025)

work page 2025
[62]

Improving turbulence control through explainable deep learning

M. Beneitez, A. Cremades, L. Guastoni, R. Vinuesa, Improving turbulence control through explainable deep learning.arXiv:2504.02354(2025)

work page internal anchor Pith review arXiv 2025
[63]

Z. Zhou, M. Zhang, X. Zhu, Reinforcement-learning-based control of turbulent channel flows at high Reynolds numbers.J. Fluid Mech.1006, A12 (2025)

work page 2025
[64]

Raffin,et al., Stable-Baselines3: reliable reinforcement learning implementations.J

A. Raffin,et al., Stable-Baselines3: reliable reinforcement learning implementations.J. Mach. Learn. Res.22(268), 1–8 (2021)

work page 2021
[65]

J. K. Terry,et al., PettingZoo: gym for multi-agent reinforcement learning.arXiv:2009.14471 (2020). Acknowledgments Funding:R.V. acknowledges financial support from ERC grant no. ‘2021-CoG-101043998, DEEPCONTROL ’. Views and opinions expressed are those of the authors only and do not neces- 24 sarily reflect those of the European Union or the European Res...

work page arXiv 2009

[1] [1]

Abbas,et al., Drag reduction via turbulent boundary layer flow control.Sci

A. Abbas,et al., Drag reduction via turbulent boundary layer flow control.Sci. China Technol. Sci.60, 1281–1290 (2017)

work page 2017

[2] [2]

Fukagata, K

K. Fukagata, K. Iwamoto, Y. Hasegawa, Turbulent drag reduction by streamwise traveling waves of wall-normal forcing.Annu. Rev. Fluid Mech.56, 69–90 (2024)

work page 2024

[3] [3]

F. Z. Wang, I. L. Animasaun, T. Muhammad, S. S. Okoya, Recent advancements in fluid dynamics: drag reduction, lift generation, computational fluid dynamics, turbulence modelling, and multiphase flow.Arab. J. Sci. Eng.49, 1–13 (2024)

work page 2024

[4] [4]

J. I. Cardesa, A. Vela-Mart ´ın, J. Jim´enez, The turbulent cascade in five dimensions.Science 357(6353), 782–784 (2017), doi:10.1126/science.aan7933

work page doi:10.1126/science.aan7933 2017

[5] [5]

Seifert, A

A. Seifert, A. Darabi, I. Wygnanski, Delay of airfoil stall by periodic excitation.J. Aircr.33(4), 691–698 (1996)

work page 1996

[6] [6]

L. N. Cattafesta, M. Sheplak, Actuators for active flow control.Annu. Rev. Fluid Mech.43, 247–272 (2011)

work page 2011

[7] [7]

S. S. Collis, R. D. Joslin, A. Seifert, V. Theofilis, Issues in active flow control: theory, control, simulation, and experiment.Prog. Aerosp. Sci.40(4–5), 237–289 (2004)

work page 2004

[8] [8]

S. L. Brunton, B. R. Noack, Closed-loop turbulence control: progress and challenges.Appl. Mech. Rev.67(5), 050801 (2015)

work page 2015

[9] [9]

S. L. Brunton, Applying machine learning to study fluid mechanics.Acta Mech. Sin.37(12), 1718–1726 (2021)

work page 2021

[10] [10]

Rabault, M

J. Rabault, M. Kuchta, A. Jensen, U. R ´eglade, N. Cerardi, Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control.J. Fluid Mech.865, 281–302 (2019)

work page 2019

[11] [11]

Vignon, J

C. Vignon, J. Rabault, R. Vinuesa, Recent advances in applying deep reinforcement learning for flow control: perspectives and future directions.Phys. Fluids35(3), 031301 (2023). 19

work page 2023

[12] [12]

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

work page 1994

[13] [13]

A. J. Smits, B. J. McKeon, I. Marusic, High-Reynolds number wall turbulence.Annu. Rev. Fluid Mech.43, 353–375 (2011)

work page 2011

[14] [14]

Marusic, R

I. Marusic, R. Mathis, N. Hutchins, Predictive model for wall-bounded turbulent flow.Science 329(5988), 193–196 (2010), doi:10.1126/science.1188765

work page doi:10.1126/science.1188765 2010

[15] [15]

Stroh, B

A. Stroh, B. Frohnapfel, P. Schlatter, Y. Hasegawa, A comparison of opposition control in turbulent boundary layer and turbulent channel flow.Phys. Fluids27(7), 075101 (2015)

work page 2015

[16] [16]

Dacome, R

G. Dacome, R. M ¨orsch, M. Kotsonis, W. J. Baars, Opposition flow control for reducing skin- friction drag of a turbulent boundary layer.Phys. Rev. Fluids9(6), 064602 (2024)

work page 2024

[17] [17]

J. Yao, E. Garc´ıa, F. Hussain, Drag reduction via opposition control in turbulent channel flows at high Reynolds numbers.Phys. Rev. Fluids10(9), 094604 (2025)

work page 2025

[18] [18]

J. M. Hamilton, J. Kim, F. Waleffe, Regeneration mechanisms of near-wall turbulence struc- tures.J. Fluid Mech.287, 317–348 (1995)

work page 1995

[19] [19]

Jim ´enez, S

J. Jim ´enez, S. Hoyas, M. P. Simens, Y. Mizuno, Turbulent boundary layers and channels at moderate Reynolds numbers.J. Fluid Mech.657, 335–360 (2010)

work page 2010

[20] [20]

Raissi, A

M. Raissi, A. Yazdani, G. E. Karniadakis, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations.Science367(6481), 1026–1030 (2020), doi:10.1126/ science.aaw4741

work page 2020

[21] [21]

Su ´arez,et al., Flow control of three-dimensional cylinders transitioning to turbulence via multi-agent reinforcement learning.Commun

P. Su ´arez,et al., Flow control of three-dimensional cylinders transitioning to turbulence via multi-agent reinforcement learning.Commun. Eng.4(1), 113 (2025)

work page 2025

[22] [22]

Guastoni, J

L. Guastoni, J. Rabault, P. Schlatter, H. Azizpour, R. Vinuesa, Deep reinforcement learning for turbulent drag reduction in channel flows.Eur. Phys. J. E46(4), 27 (2023)

work page 2023

[23] [23]

Sonoda, Z

T. Sonoda, Z. Liu, T. Itoh, Y. Hasegawa, Reinforcement learning of control strategies for reducing skin friction drag in a fully developed turbulent channel flow.J. Fluid Mech.960, A30 (2023). 20

work page 2023

[24] [24]

Su ´arez,et al., Active flow control for drag reduction through multi-agent reinforcement learning on a turbulent cylinder at𝑅𝑒 𝐷 =3900.Flow Turbul

P. Su ´arez,et al., Active flow control for drag reduction through multi-agent reinforcement learning on a turbulent cylinder at𝑅𝑒 𝐷 =3900.Flow Turbul. Combust.115, 3–27 (2025)

work page 2025

[25] [25]

Varela,et al., Deep reinforcement learning for flow control exploits different physics for increasing Reynolds number regimes.Actuators11(12), 359 (2022)

P. Varela,et al., Deep reinforcement learning for flow control exploits different physics for increasing Reynolds number regimes.Actuators11(12), 359 (2022)

work page 2022

[26] [26]

Q. Liu, L. J. Trujillo Corona, D. Espinoza, F. Shu, A. Gross, Design and dimensional transfer of reinforcement learning-based closed-loop airfoil flow control.Theor. Comput. Fluid Dyn. 39(6), 49 (2025)

work page 2025

[27] [27]

V. Belus,et al., Exploiting locality and translational invariance to design effective deep rein- forcement learning control of the 1-dimensional unstable falling liquid film.AIP Adv.9(12), 125141 (2019)

work page 2019

[28] [28]

Rabault, A

J. Rabault, A. Kuhnle, Accelerating deep reinforcement learning strategies of flow control through a multi-environment approach.Phys. Fluids31(9), 094105 (2019)

work page 2019

[29] [29]

Cremades,et al., Identifying regions of importance in wall-bounded turbulence through explainable deep learning.Nat

A. Cremades,et al., Identifying regions of importance in wall-bounded turbulence through explainable deep learning.Nat. Commun.15(1), 3864 (2024)

work page 2024

[30] [30]

Sanchis-Agudo, A

M. Sanchis-Agudo, A. Cremades, A. Martinez-Sanchez, A. Lozano-Duran, R. Vinuesa, X- CAL: explaining latent causality in physical space for fluid mechanics.arXiv:2601.03311 (2026)

work page arXiv 2026

[31] [31]

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

work page 2025

[32] [32]

Diff-sport: Diffusion-based sensor placement optimization and reconstruction of turbulent ﬂows in urban environments

A. Vishwasrao,et al., Diff-SPORT: diffusion-based sensor placement optimization and recon- struction of turbulent flows in urban environments.arXiv:2506.00214(2025)

work page arXiv 2025

[33] [33]

Weiner, J

A. Weiner, J. Reyes, Model-based deep reinforcement learning for accelerated learning from flow simulations.arXiv:2402.16543(2024)

work page arXiv 2024

[34] [34]

Zhang,et al., Efficient active flow control strategy for confined square cylinder wake using deep learning-based surrogate model and reinforcement learning.arXiv:2408.14232(2024)

Z. Zhang,et al., Efficient active flow control strategy for confined square cylinder wake using deep learning-based surrogate model and reinforcement learning.arXiv:2408.14232(2024). 21

work page arXiv 2024

[35] [35]

and Moxey, D

K. Avila,et al., The onset of turbulence in pipe flow.Science333(6039), 192–196 (2011), doi:10.1126/science.1203223

work page doi:10.1126/science.1203223 2011

[36] [36]

Lagemann,et al., HydroGym: a reinforcement learning platform for fluid dynamics, arXiv:2512.17534 (2025)

C. Lagemann,et al., HydroGym: a reinforcement learning platform for fluid dynamics, arXiv:2512.17534 (2025)

work page arXiv 2025

[37] [37]

Vinuesa, A

R. Vinuesa, A. Bobke, R. ¨Orl¨ u, P. Schlatter, On determining characteristic length scales in pressure-gradient turbulent boundary layers.Phys. Fluids28(5), 055101 (2016)

work page 2016

[38] [38]

Harun, J

Z. Harun, J. P. Monty, R. Mathis, I. Marusic, Pressure gradient effects on the large-scale structure of turbulent boundary layers.J. Fluid Mech.715, 477–498 (2013)

work page 2013

[39] [39]

Atzori,et al., Uniform blowing and suction applied to nonuniform adverse-pressure-gradient wing boundary layers.Phys

M. Atzori,et al., Uniform blowing and suction applied to nonuniform adverse-pressure-gradient wing boundary layers.Phys. Rev. Fluids6(11), 113904 (2021)

work page 2021

[40] [40]

Y. Wang, M. Atzori, R. Vinuesa, Opposition control applied to turbulent wing sections.J. Fluid Mech.1010, A29 (2025)

work page 2025

[41] [41]

Vinuesa,et al., Turbulent boundary layers around wing sections up to𝑅𝑒 𝑐 =1,000,000.Int

R. Vinuesa,et al., Turbulent boundary layers around wing sections up to𝑅𝑒 𝑐 =1,000,000.Int. J. Heat Fluid Flow72, 86–99 (2018)

work page 2018

[42] [42]

and Vinuesa, R

M. Atzori,et al., Aerodynamic effects of uniform blowing and suction on a NACA4412 airfoil. Flow Turbul. Combust.105(3), 735–759 (2020), doi:10.1007/s10494-020-00135-z

work page doi:10.1007/s10494-020-00135-z 2020

[43] [43]

Quadrio, P

M. Quadrio, P. Ricco, Critical assessment of turbulent drag reduction through spanwise wall oscillations.J. Fluid Mech.667, 135–157 (2011), doi:10.1017/S002211201000367X

work page doi:10.1017/s002211201000367x 2011

[44] [44]

Quadrio, P

M. Quadrio, P. Ricco, C. Viotti, Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction.J. Fluid Mech.627, 161–178 (2009)

work page 2009

[45] [45]

Marusic,et al., An energy-efficient pathway to turbulent drag reduction.Nat

I. Marusic,et al., An energy-efficient pathway to turbulent drag reduction.Nat. Commun.12, 5805 (2021)

work page 2021

[46] [46]

P. F. Fischer, J. W. Lottes, S. G. Kerkemeier, Nek5000 web page (2008), http://nek5000.mcs.anl.gov. 22

work page 2008

[47] [47]

B. B. De Moura, M. R. Machado, S. Dey, T. Mukhopadhyay, Manipulating flexural waves to enhance broadband vibration mitigation through programmed disorder on smart rainbow metamaterials.Appl. Math. Modelling125, 650–671 (2024)

work page 2024

[48] [48]

Maday, A

Y. Maday, A. T. Patera, Spectral element methods for the incompressible Navier-Stokes equa- tions.State-of-the-Art Surveys on Computational Mechanicspp. 71–143 (1989)

work page 1989

[49] [49]

Tanarro, R

´A. Tanarro, R. Vinuesa, P. Schlatter, Effect of adverse pressure gradients on turbulent wing boundary layers.J. Fluid Mech.883, A8 (2020)

work page 2020

[50] [50]

P. S. Negi, R. Vinuesa, A. Hanifi, P. Schlatter, D. S. Henningson, Unsteady aerodynamic effects in small-amplitude pitch oscillations of an airfoil.Int. J. Heat Fluid Flow71, 378–391 (2018)

work page 2018

[51] [51]

F. R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J.32(8), 1598–1605 (1994)

work page 1994

[52] [52]

S. Dong, G. E. Karniadakis, C. Chryssostomidis, A robust and accurate outflow boundary condition for incompressible flow simulations on severely-truncated unbounded domains.J. Comput. Phys.261, 83–105 (2014)

work page 2014

[53] [53]

Schlatter, R

P. Schlatter, R. ¨Orl¨ u, Turbulent boundary layers at moderate Reynolds numbers: inflow length and tripping effects.J. Fluid Mech.710, 5–34 (2012)

work page 2012

[54] [54]

Vinuesa, S

R. Vinuesa, S. M. Hosseini, A. Hanifi, D. S. Henningson, P. Schlatter, Pressure-gradient turbulent boundary layers developing around a wing section.Flow Turbul. Combust.99, 613– 641 (2017)

work page 2017

[55] [55]

E. P. Hammond, T. R. Bewley, P. Moin, Observed mechanisms for turbulence attenuation and enhancement in opposition-controlled wall-bounded flows.Phys. Fluids10(9), 2421–2423 (1998)

work page 1998

[56] [56]

T. P. Lillicrap,et al., Continuous control with deep reinforcement learning.arXiv:1509.02971 (2015). 23

work page internal anchor Pith review arXiv 2015

[57] [57]

Fujimoto, H

S. Fujimoto, H. Hoof, D. Meger, Addressing function approximation error in actor-critic methods, inProceedings of the 35th International Conference on Machine Learning(PMLR) (2018), pp. 1587–1596

work page 2018

[58] [58]

T. Lee, J. Kim, C. Lee, Turbulence control for drag reduction through deep reinforcement learning.Phys. Rev. Fluids8(2), 024604 (2023)

work page 2023

[59] [59]

Sonoda, Z

T. Sonoda, Z. Liu, T. Itoh, Y. Hasegawa, Reinforcement learning of control strategies for reducing skin friction drag in a fully developed turbulent channel flow.J. Fluid Mech.960, A30 (2023)

work page 2023

[60] [60]

W ¨alchli, L

D. W ¨alchli, L. Guastoni, R. Vinuesa, P. Koumoutsakos, Drag reduction in a minimal channel flow with scientific multi-agent reinforcement learning.J. Phys.: Conf. Ser.2753(1), 012024 (2024)

work page 2024

[61] [61]

G. M. Cavallazzi, L. Guastoni, R. Vinuesa, A. Pinelli, Deep reinforcement learning for the management of the wall regeneration cycle in wall-bounded turbulent flows.Flow Turbul. Combust.115(3), 1291–1317 (2025)

work page 2025

[62] [62]

Improving turbulence control through explainable deep learning

M. Beneitez, A. Cremades, L. Guastoni, R. Vinuesa, Improving turbulence control through explainable deep learning.arXiv:2504.02354(2025)

work page internal anchor Pith review arXiv 2025

[63] [63]

Z. Zhou, M. Zhang, X. Zhu, Reinforcement-learning-based control of turbulent channel flows at high Reynolds numbers.J. Fluid Mech.1006, A12 (2025)

work page 2025

[64] [64]

Raffin,et al., Stable-Baselines3: reliable reinforcement learning implementations.J

A. Raffin,et al., Stable-Baselines3: reliable reinforcement learning implementations.J. Mach. Learn. Res.22(268), 1–8 (2021)

work page 2021

[65] [65]

J. K. Terry,et al., PettingZoo: gym for multi-agent reinforcement learning.arXiv:2009.14471 (2020). Acknowledgments Funding:R.V. acknowledges financial support from ERC grant no. ‘2021-CoG-101043998, DEEPCONTROL ’. Views and opinions expressed are those of the authors only and do not neces- 24 sarily reflect those of the European Union or the European Res...

work page arXiv 2009