Optimizing Metachronal Paddling with Reinforcement Learning at Low Reynolds Number

Alana A. Bailey; Robert D. Guy

REVIEW 3 major objections 5 minor 33 references

Optimizing Metachronal Paddling with Reinforcement Learning at Low Reynolds Number

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that for a rigid-paddle swimmer at zero Reynolds number, the back-to-front (antiplectic) wave is the most efficient learned stroke for two, three, and four paddle pairs, even though the fastest stroke changes with paddle…

desk verdict A clean RL study that recovers antiplectic metachrony as efficient, but the efficiency claim rests on a drag coefficient measured for one paddle posture and used for all strokes. read the letter →

arxiv 2507.18849 v1 pith:STB2WWS2 submitted 2025-07-24 physics.flu-dyn cs.LGstat.ML

Alana A. Bailey , Robert D. Guy This is my paper · ORCID

classification physics.flu-dyncs.LGstat.ML

keywords metachronalpaddlingreinforcementlearninglowReynoldsnumberStokesflowantiplecticwaveQ-learningswimmingefficiencyrigidpaddles

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether a low-Reynolds-number swimmer made of rigid, unjointed paddles will rediscover metachronal paddling—the wave-like, time-staggered limb rhythm used by ciliated organisms, krill, and mantis shrimp—when left to learn on its own. A tabular Q-learning agent optimizes forward displacement for swimmers with two, three, and four pairs of straight paddles at several fixed spacings. It learns two main gaits: a back-to-front (antiplectic) wave at tight spacings and a front-to-back (symplectic) stroke at wide spacings, with additional non-wave strokes appearing for three paddles at the largest spacings. The paper's central claim is that the fastest stroke depends on the number of paddles, but the most efficient stroke is the back-to-front wave-like stroke for every paddle count considered. If correct, this points to efficiency, not speed, as a likely selection pressure behind the antiplectic metachronal rhythm observed across biological swimmers.

What carries the argument

The central object is a two-dimensional 'paddler': a long body with pairs of straight, inflexible paddles, each with one angular degree of freedom discretized into 11 states and a stepwise action moving it left, right, or not at all. The learning loop is tabular Q-learning with the update rule $Q(s,a)\leftarrow (1-\alpha)Q(s,a)+\alpha[r+\gamma\max_{a'}Q(s',a')]$, and the reward for each step is the swimmer's net forward displacement over that time interval. Hydrodynamic forces and swimming speed are computed by solving the Stokes equations with the method of regularized Stokeslets, subject to zero net force on the swimmer. Stroke performance is then compared with the efficiency ratio $\eta = \zeta U^2/P$, using a drag coefficient measured by towing the swimmer with paddles held perpendicular to the body. Since the limbs cannot bend, any swimming motion must come from rhythm alone, which is what lets the paper attribute the learned gaits purely to coordination timing.

What would settle it

Take two learned strokes at the same spacing and paddle count, measure the time-averaged drag force during each stroke by integrating the computed forces over one cycle, and recompute each efficiency with the measured drag value. If the front-to-back stroke's efficiency then reaches or exceeds the back-to-front stroke's, the paper's central conclusion fails for that configuration.

Watch

Extended reading notes

Core claim

Across the strokes the agent learns, the back-to-front wave is the most efficient coordination pattern for two, three, and four paddle pairs, while the fastest stroke is not universal: with two and four paddles a front-to-back, inward-tilted stroke is faster at wide spacings, and with three paddles the back-to-front stroke is fastest. The gaits are learned by Q-learning maximizing each move's net forward displacement, so efficiency is not part of the reward; it is measured afterward using the standard ratio $\eta = \zeta U^2/P$, where $U$ is the average swimming speed, $P$ the average mechanical power, and $\zeta$ the drag coefficient. Because the paddles are rigid and each has a single angular degree of freedom, the only time asymmetry available to the agent is the ordering and timing of paddle motions. The paper takes this result as evidence that the antiplectic rhythm seen in nature can be an efficiency-optimal solution even when the swimmer is free to choose any coordination.

Load-bearing premise

The paper assumes that the water resistance measured when the paddles are held straight out is close to the resistance the swimmer actually feels during each learned stroke; if that is wrong, the efficiency ranking could change.

Editorial extensions

If this is right

In this model, an antiplectic back-to-front wave is the efficiency-optimal coordination for two, three, and four paddle pairs, so the biologically common rhythm can arise from a pure efficiency objective without any prescribed wave rule.
Because the speed-optimal and efficiency-optimal strokes differ in most configurations, a microrobot designer must choose explicitly between maximizing distance per time and maximizing distance per energy; the same gait will not do both.
The learned gait switches from back-to-front to front-to-back near a paddle spacing of about 2 body units for all paddle counts, identifying appendage spacing as a control parameter for gait selection.
The back-to-front stroke is both fastest and most efficient at tight spacings, which is consistent with closely packed cilia arrays in biology and suggests a simple design rule: pack appendages tightly for efficient antiplectic swimming.

Reading between the lines

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

The efficiency comparison could be sharpened by measuring each stroke's own time-averaged drag instead of using the single towed value; this is a direct, testable check of whether the efficiency ranking survives the paddles' actual tilted orientations.
An agent trained to maximize efficiency directly, rather than speed, might settle on a different gait; the paper notes this requires changing the action space or reward, so an actor-critic or deep-Q variant is the natural next experiment.
Because Stokes flow has no intrinsic length scale, the spacing threshold near 2 units should scale with paddle length and body geometry; varying those dimensions in the same model would test whether the threshold is a ratio rather than an absolute distance.
The fact that speed and efficiency diverge most at wide spacings suggests that organisms with widely spaced limbs may face a sharper speed-efficiency trade-off than those with tight arrays; comparing learned gaits across more paddle counts would map this trade-off.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here is my take on the Bailey–Guy paper. The headline result is worth knowing: tabular Q-learning on a rigid-paddle swimmer at zero Re rediscovers the antiplectic back-to-front metachronal wave at tight spacings, and the same algorithm finds faster front-to-back strokes at wide spacings. The speed result is robust and new. The efficiency claim—that back-to-front is always the most efficient stroke regardless of paddle number—is the part I would push on.

What the paper does well: the model is minimal but appropriate, the RL setup is standard and clearly described, and the authors actually test the learning parameters. Section 3.5 shows that their conclusions depend on the discount factor, and they report that sensitivity honestly. The stroke metrics (phase lag, amplitude, mean tilt) are sensible, and the Appendix C table gives enough detail to see what the learned strokes look like. Citations are fair.

The soft spot is the drag coefficient used in the efficiency metric. In Section 3.4 and Appendix B, ζ is computed by towing the swimmer with all paddles pinned perpendicular to the body, and that single value is used for every learned stroke at that spacing. But the strokes themselves have systematically different paddle orientations: Table C2 lists mean paddle states up to ±3.72 (about 33° tilt) for the front-to-back strokes. In Stokes flow the drag of a flat appendage depends strongly on orientation, so the perpendicular-paddle ζ is not obviously representative. If the time-averaged drag of the front-to-back strokes is larger than the perpendicular value—which seems likely, since the paddles are tilted into the flow—their efficiency would be underestimated. That could reorder the efficiency comparison, which is the paper's central claim. The speed results do not depend on this assumption, so they stand.

Secondary issues: no code or data posted, just 'available upon reasonable request,' and the main plots show scatter from five runs but no error bars. Those are minor next to the drag issue.

This paper deserves a real referee. The speed finding is solid and the efficiency question is exactly the right one to raise. A referee should ask the authors to check the drag approximation, either by computing a stroke-averaged ζ or by reporting efficiency over a range of plausible values. If the efficiency ranking survives that check, the paper is a nice contribution. I would bring it up at our group meeting, but I would not cite it until the drag issue is resolved.

Referee Report

3 major / 5 minor

Summary. The paper uses tabular Q-learning to discover optimal paddling gaits for a two-dimensional swimmer at zero Reynolds number. The swimmer consists of an elongated body with two, three, or four pairs of rigid paddles, and the paddle angles are discretized into eleven states. The reward is the net forward displacement per action, so the learning agent optimizes swimming speed. For tight paddle spacings, the agent learns a back-to-front (antiplectic) metachronal stroke, while for wide spacings it learns front-to-back strokes in which paddles tilt inward. The speed-optimal stroke depends on paddle number, but the authors report that the back-to-front stroke is the most efficient across two, three, and four paddle pairs. The hydrodynamic problem is solved with regularized Stokeslets, and efficiency is computed with a Lighthill-type metric using a drag coefficient obtained by towing the swimmer with perpendicular paddles. A parameter study in Section 3.5 examines sensitivity to training length, discount factor, exploration rate, and learning rate.

Significance. If the results are robust, the paper provides a useful demonstration that model-free reinforcement learning can recover biologically observed antiplectic metachrony in a minimal rigid-paddle model, and it identifies a clear separation between speed-optimal and efficiency-optimal coordination. The qualitative comparison with the biological literature on ciliates and other metachronal swimmers is a strength, and the parameter study in Section 3.5 is a genuine robustness check. The hydrodynamic model and Q-learning setup are standard, and the finding that efficiency, rather than speed, selects the antiplectic wave across different paddle numbers is potentially valuable for designing low-Reynolds-number robotic swimmers.

major comments (3)

[Section 3.4, Appendix B (Table B1)] The efficiency metric η = ζU²/P uses a single drag coefficient ζ per paddle number and spacing, obtained by towing the swimmer with all paddles fixed perpendicular to the body (θ=0). The learned strokes are not small perturbations of this reference: Table C2 reports mean paddle states of about ±2 (≈18°) for the back-to-front strokes and values up to ±3.72 (≈33°) for the front-to-back strokes. Because the drag of an inclined flat appendage in Stokes flow depends on orientation, the perpendicular-paddle ζ may not be representative of either stroke. The bias likely runs in the direction of overestimating the efficiency of the more tilted front-to-back strokes, which would make the main conclusion conservative, but the paper does not state this or test it. I recommend adding a sensitivity analysis that recomputes η using ζ estimated from a stroke-specific reference configuration (for example, the time-averaged or mean-tilt configuration) and reporting whether the efficiency ordering changes.
[Section 3.5.1, Figure 9] The parameter study reveals a strong sensitivity to the discount factor. At paddle spacing 2 with the longest training loop, γ=0.99 converges to the front-to-back stroke in every trial, but γ=0.98 converges to a suboptimal back-to-front stroke in every trial, and γ=0.97 produces ineffective swimming in all trials. This indicates that the learned strokes are not robustly optimal with respect to small hyperparameter changes. Since the abstract and discussion repeatedly refer to 'optimal' strokes, the paper should either provide a formal optimality check (for example, policy iteration on the converged Q-table, multiple random seeds with the final chosen parameters, or a systematic grid over γ) or consistently soften the terminology to 'learned' or 'RL-selected' strokes. As written, the claim of optimality is stronger than the evidence.
[Section 3.4, Figure 8] The headline efficiency claim rests on comparisons of point estimates from only five runs per spacing. The text states that for the two- and four-paddle swimmers the maximum front-to-back efficiency is 'comparable' to the minimum back-to-front efficiency, which suggests a small margin that could be comparable to run-to-run variation. No confidence intervals, standard deviations, or effect sizes are reported for η, and Figure 8 does not include error bars. Given that the central conclusion is an efficiency ranking, the authors should report the quantitative margins (for example, the maximum and minimum η values for each stroke type) and assess whether the ranking is robust to the observed variation across the five runs.

minor comments (5)

[Section 2.2] The phrase '968 states' should be '968 state-action pairs': the calculation is 121 paddle configurations times 8 actions, which is the number of state-action pairs, not the number of states.
[Appendix C, Table C2 caption] The word 'mantitude' should be 'magnitude' in the caption of Table C2.
[Figure 9] The caption does not clearly explain the row and column layout of the histograms or what the numbers above the bars represent beyond 'mean swimming speed'; adding a sentence that defines the training-loop combinations for each panel would improve readability.
[Section 3.5.1] The term 'suboptimal' is used without defining the comparison baseline; since the strokes are outputs of the same learning algorithm, the authors should specify that 'suboptimal' means slower than the best stroke found with other hyperparameter choices.
[Data Availability] The code for the Q-learning algorithm and the regularized Stokeslet solver is not provided, which makes it difficult to reproduce the stochastic RL results; making the code and random seeds available would strengthen reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the RL objective is displacement, efficiency is evaluated post hoc, and no load-bearing step reduces to a fitted input or self-citation.

full rationale

The paper's central claim—that a back-to-front metachronal stroke is the most efficient while the fastest stroke depends on paddle number—is not circular. The reinforcement learning reward is the net displacement of the swimmer (Eq. 3), so the learning objective is swimming speed; efficiency is computed afterward using the Lighthill metric η = ζU²/P. The drag coefficient ζ in Appendix B is a model parameter obtained by towing the same discretized swimmer with paddles fixed perpendicular to the body; it is not fitted to the efficiency results, and it is not defined in terms of the stroke being evaluated. The efficiency ranking therefore has independent content: it compares strokes that were selected for speed, not for efficiency. There are no self-citations by Bailey and Guy in the reference list, and the cited external results about antiplectic metachrony are used as context, not as premises that force the numerical outcome. The reader's concern that the fixed perpendicular-paddle drag coefficient may not represent the time-averaged drag of tilted learned strokes is a legitimate modeling-robustness question, but it is a correctness risk rather than circularity: no quantity in the derivation is equivalent by construction to the quantity it is supposed to predict. Accordingly, no circular step is identified.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central results rest on standard Stokes-flow hydrodynamics and a set of hand-chosen numerical and learning parameters. No parameters are fitted to target data, but the qualitative conclusions, especially the efficiency ranking, depend on modeling choices such as the fixed drag coefficient and the Q-learning hyperparameters.

free parameters (7)

regularization parameter epsilon = 0.05
Chosen for the regularized Stokeslet method (ref 22). Affects near-field hydrodynamic forces and hence stroke speeds.
Stokeslet discretization spacing = 0.1
Body and paddles are discretized into points equally spaced by 0.1; resolution affects force computation and swimming speed.
paddle angle discretization = 11 states, pi/20 increments over [-pi/4, pi/4]
The RL action/state space is discrete; coarser or finer discretization could change which strokes are reachable and the optimality of the learned policy.
time per action = 1 unit
Each action lasts one time unit with angular speed pi/20; sets the stroke time scale and the reward integration interval.
discount factor gamma = 0.99 (2 paddle), 0.999 (3,4 paddle)
Chosen from the parameter study in Section 3.5.1; small changes (0.98) lead to suboptimal strokes, so the central results are sensitive to this hand-tuned value.
number of episodes and learning steps = 50 episodes x 50,000 steps (2); 500 x 500,000 (3,4)
Chosen to balance exploration and convergence; the 3/4-paddle choices were not validated with large numbers of trials due to computational cost.
drag coefficient for efficiency = listed per spacing in Table B1
Computed by towing the paddler with paddles fixed perpendicular; enters the Lighthill efficiency metric. This is a modeling choice that may bias the efficiency comparison.

assumptions (6)

domain assumption The fluid is governed by the incompressible Stokes equations at zero Reynolds number, solved with regularized Stokeslets in free space.
Used throughout Section 2.3; assumes 2D free-space Stokes flow and no inertial effects.
standard math Reciprocal motions produce zero net displacement (scallop theorem), so propulsion requires time-asymmetric coordination.
Invoked in the Introduction to motivate the model and in the Discussion to interpret results.
domain assumption Top and bottom paddle pairs beat symmetrically, so the body does not rotate and motion is purely horizontal.
Stated in Section 2.1; this reduces the state space and removes rotational degrees of freedom.
domain assumption Tabular Q-learning with the chosen discretization, discount factor, and training length converges to the optimal policy for the discounted reward objective.
Relied on for all 'optimal stroke' claims; Section 3.5 shows the result is sensitive to gamma, so convergence to the true optimum is not guaranteed.
domain assumption The drag coefficient measured with paddles perpendicular to the body is representative of the drag during the learned strokes.
Used in the efficiency metric in Section 3.4 and Appendix B; paddle orientation varies during strokes, so the fixed drag coefficient is an approximation.
domain assumption The 2D model with rigid paddles captures the essential physics of metachronal paddling relevant to the conclusions.
The study restricts to a simplified 2D rigid-paddle geometry to isolate rhythm effects; real ciliates are 3D with deformable cilia.

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Cite this review

Pith. "Pith review of Optimizing Metachronal Paddling with Reinforcement Learning at Low Reynolds Number." pith.science (2026). https://pith.science/paper/STB2WWS2

@misc{pith2026250718849,
  author       = {Pith},
  title        = {Pith review of: Optimizing Metachronal Paddling with Reinforcement Learning at Low Reynolds Number},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STB2WWS2}},
  note         = {Machine review of arXiv:2507.18849}
}

read the original abstract

Metachronal paddling is a swimming strategy in which an organism oscillates sets of adjacent limbs with a constant phase lag, propagating a metachronal wave through its limbs and propelling it forward. This limb coordination strategy is utilized by swimmers across a wide range of Reynolds numbers, which suggests that this metachronal rhythm was selected for its optimality of swimming performance. In this study, we apply reinforcement learning to a swimmer at zero Reynolds number and investigate whether the learning algorithm selects this metachronal rhythm, or if other coordination patterns emerge. We design the swimmer agent with an elongated body and pairs of straight, inflexible paddles placed along the body for various fixed paddle spacings. Based on paddle spacing, the swimmer agent learns qualitatively different coordination patterns. At tight spacings, a back-to-front metachronal wave-like stroke emerges which resembles the commonly observed biological rhythm, but at wide spacings, different limb coordinations are selected. Across all resulting strokes, the fastest stroke is dependent on the number of paddles, however, the most efficient stroke is a back-to-front wave-like stroke regardless of the number of paddles.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

In: Physics and Our World: Reissue of the Proceedings of a Symposium in Honor of Vic- tor F Weisskopf, pp

Purcell, E.M.: Life at low reynolds number. In: Physics and Our World: Reissue of the Proceedings of a Symposium in Honor of Vic- tor F Weisskopf, pp. 47–67 (2014). World Scientiﬁc

work page 2014

[2] [2]

Reports on progress in physics 78(5), 056601 (2015)

Elgeti, J., Winkler, R.G., Gompper, G.: Physics of microswimmers—single particle motion and collective behavior: a review. Reports on progress in physics 78(5), 056601 (2015)

work page 2015

[3] [3]

: Metachronal motion across scales: current challenges and future directions

Byron, M.L., Murphy, D.W., Katija, K., Hoover, A.P., Daniels, J., Garayev, K., Tak- agi, D., Kanso, E., Gemmell, B.J., Ruszczyk, M., et al. : Metachronal motion across scales: current challenges and future directions. Inte- grative and comparative biology 61(5), 1674– 1688 (2021)

work page 2021

[4] [4]

Physical Review E 95(5), 052412 (2017)

Ghorbani, A., Najaﬁ, A.: Symplectic and antiplectic waves in an array of beating cilia attached to a closed body. Physical Review E 95(5), 052412 (2017)

work page 2017

[5] [5]

Journal of Cell Science 3(32), 503–521 (1954)

Knight-Jones, E.: Relations between metachronism and the direction of ciliary beat in metazoa. Journal of Cell Science 3(32), 503–521 (1954)

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Physics of Fluids 26(9) (2014)

Guo, H., Nawroth, J., Ding, Y., Kanso, E.: Cilia beating patterns are not hydrodynami- cally optimal. Physics of Fluids 26(9) (2014)

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Marine biology 158, 2541–2554 (2011)

Murphy, D., Webster, D., Kawaguchi, S., King, R., Yen, J.: Metachronal swimming in antarctic krill: gait kinematics and sys- tem design. Marine biology 158, 2541–2554 (2011)

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Integrative and Comparative Biology 61(5), 1631–1643 (2021)

Garayev, K., Murphy, D.W.: Metachronal swimming of mantis shrimp: kinematics and interpleopod vortex interactions. Integrative and Comparative Biology 61(5), 1631–1643 (2021)

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