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arxiv: 2606.22440 · v1 · pith:DYWPYYG7new · submitted 2026-06-21 · ⚛️ physics.bio-ph · cond-mat.soft· nlin.AO· physics.data-an· q-bio.CB

Data-driven geometric phase in biological locomotion

Pyae Hein Htet , Kenta Ishimoto This is my paper

Pith reviewed 2026-06-26 09:32 UTC · model grok-4.3

classification ⚛️ physics.bio-ph cond-mat.softnlin.AOphysics.data-anq-bio.CB
keywords geometric phaseKoopman autoencoderbiological locomotionspermnematodegauge theorylimit cyclesensitivity function
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The pith

A Koopman autoencoder extracts geometric phase and sensitivity from biological locomotion data without mechanical models.

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

This paper presents a data-driven approach using a theory-guided Koopman autoencoder to recover limit cycles from imperfect cyclic biological data. The method extracts shape gaits and geometric phase from examples like sperm and nematodes, even when data is noisy and weakly periodic. It also defines a geometric phase sensitivity function to assess how shape perturbations influence locomotion. This matters because it allows mechanical insights to be drawn purely from gauge-theoretic principles applied to real-world movement observations.

Core claim

We develop a theory-guided, data-driven Koopman autoencoder to recover the limit cycle embedded in imperfect cyclic data and extract shape gaits and geometric phase from sperm and nematode data. We introduce a geometric phase sensitivity function that quantifies responses to shape perturbations and reveals mechanical information using only gauge-theoretic structure, without assuming mechanical laws.

What carries the argument

Koopman autoencoder for limit cycle recovery combined with the geometric phase sensitivity function based on gauge theory.

If this is right

  • The geometric phase can be computed directly from observed shape changes in biological swimmers.
  • Responses to perturbations can be quantified to predict changes in net locomotion.
  • Mechanical details can be inferred solely from the mathematical structure of gauge theory.
  • The approach works for both sperm and nematode data sets as demonstrations.

Where Pith is reading between the lines

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

  • The sensitivity function might guide the design of artificial swimmers by identifying effective shape modifications.
  • Similar data-driven techniques could apply to other periodic biological processes beyond locomotion.
  • Comparison with traditional mechanical simulations on the same data would test the method's independence from physical assumptions.

Load-bearing premise

The assumption that the Koopman autoencoder accurately reconstructs the underlying limit cycle despite noise, sparsity, and weak periodicity in the biological data.

What would settle it

A mismatch between the predicted sensitivity to shape perturbations and the actual change in observed locomotion when shapes are altered in experiments on sperm or nematodes.

Figures

Figures reproduced from arXiv: 2606.22440 by Kenta Ishimoto, Pyae Hein Htet.

Figure 1
Figure 1. Figure 1: , the encoder z = φ(q) is designed to approxi￾mate the Koopman eigenfunction, which follows linear dynamics in latent space. The decoder ψ(z) provides a two-dimensional manifold embedded in the shape space, functioning as a denoiser for the stochastic time series. Given the shape data time series qn = q(tn), we en￾force linear dynamics zn+1 = Kzn for the latent variable zn = φ(qn) and train our deep neural… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a-c). The arrow indicates the normal direction to the shape, and the magnitude reflects the size of Z α W (α = x, y, θ). In the zebrafish sperm, the geometric PSF is approximately uniform along the flagellum, with a maximum at the middle of the flagellum. In contrast, the bull sperm has its maximum geometric PSF in the distal region, reflecting the flagellar waveform with a large dis￾tal beat amplitude an… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
read the original abstract

Geometric phase quantifies net locomotion in dissipative media via gauge theory, but linking this theoretical quantity to noisy, sparse, and weakly periodic biological shape data is challenging. We develop a theory-guided, data-driven Koopman autoencoder to recover the limit cycle embedded in imperfect cyclic data and extract shape gaits and geometric phase from sperm and nematode data. We introduce a geometric phase sensitivity function that quantifies responses to shape perturbations and reveals mechanical information using only gauge-theoretic structure, without assuming mechanical laws.

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 develops a theory-guided Koopman autoencoder to recover embedded limit cycles from noisy, sparse, and weakly periodic shape data of sperm and nematodes, extracts shape gaits and geometric phases, and introduces a geometric phase sensitivity function that quantifies responses to perturbations and reveals mechanical information solely from gauge-theoretic structure without assuming mechanical laws.

Significance. If the autoencoder recovery step is shown to be reliable, the work would provide a practical bridge between geometric phase theory and experimental biological locomotion data, enabling extraction of gauge-theoretic quantities and mechanical insights from imperfect cyclic observations in dissipative media. This could extend to other systems where direct mechanical modeling is intractable.

major comments (2)
  1. [Methods (Koopman autoencoder recovery)] The central claim that the extracted geometric phase and sensitivity function are meaningful gauge-theoretic quantities depends on accurate recovery of the underlying limit cycle. The manuscript must provide reconstruction error metrics, uniqueness arguments, or synthetic-data benchmarks (e.g., in the Methods or Results sections) demonstrating that the autoencoder does not introduce artifacts from noise or multiple possible embeddings; absent these, the quantities risk being fitting artifacts.
  2. [Results (application to biological data)] Validation against known analytic cases or independent measurements for sperm and nematode locomotion is needed to confirm that the sensitivity function reveals genuine mechanical information rather than model-dependent features. Cross-checks or error analysis should be reported explicitly.
minor comments (2)
  1. [Theory section] Notation for the geometric phase sensitivity function should be clarified with an explicit definition or equation to avoid ambiguity with standard phase definitions.
  2. [Figures] Figure captions for data reconstructions should include quantitative error measures (e.g., mean squared reconstruction error) for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which highlight important aspects of validation for the Koopman autoencoder and the extracted quantities. We agree that strengthening the evidence for reliable limit-cycle recovery and confirming the sensitivity function's validity will improve the manuscript. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Methods (Koopman autoencoder recovery)] The central claim that the extracted geometric phase and sensitivity function are meaningful gauge-theoretic quantities depends on accurate recovery of the underlying limit cycle. The manuscript must provide reconstruction error metrics, uniqueness arguments, or synthetic-data benchmarks (e.g., in the Methods or Results sections) demonstrating that the autoencoder does not introduce artifacts from noise or multiple possible embeddings; absent these, the quantities risk being fitting artifacts.

    Authors: We agree that explicit validation of the limit-cycle recovery is necessary to support the claims. In the revised manuscript we will add reconstruction error metrics (e.g., mean-squared reconstruction error on shape variables) computed on both the experimental datasets and on controlled synthetic examples. We will also include synthetic-data benchmarks in which noisy, sparsely sampled trajectories are generated from known analytic limit cycles; the autoencoder will be shown to recover the original cycle, gait, and geometric phase to within quantifiable error bounds. A brief discussion of how the theory-guided loss constrains the embedding and mitigates non-uniqueness will be added to the Methods section. revision: yes

  2. Referee: [Results (application to biological data)] Validation against known analytic cases or independent measurements for sperm and nematode locomotion is needed to confirm that the sensitivity function reveals genuine mechanical information rather than model-dependent features. Cross-checks or error analysis should be reported explicitly.

    Authors: We accept that additional validation is warranted. The revised manuscript will contain a new subsection presenting results on synthetic data drawn from analytic geometric-phase models of locomotion; these tests will confirm that the sensitivity function recovers the expected perturbation responses. For the sperm and nematode applications we will report explicit cross-checks against published gait and phase values from the literature together with quantitative error bars on the extracted sensitivity functions. Because the datasets do not include simultaneous independent mechanical measurements, we will emphasize that the mechanical inferences remain gauge-theoretic and will not claim direct experimental confirmation of forces or torques. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies gauge-theoretic geometric phase to data recovered via a theory-guided Koopman autoencoder. No quoted step shows the phase or sensitivity function reducing to a fitted quantity by construction, a self-defined input, or a load-bearing self-citation chain; the central extraction step remains independent of the data-fitting procedure and is not forced by renaming or ansatz smuggling.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; full text required to audit.

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    Stuart-Landau oscillator and oscillator PSF The synthesized data is generated by a stochastic os- cillator in Eq. 7, whose deterministic part is known as the Stuart-Landau oscillator. Its principal Koopman eigen- function is analytically obtained as ψ(r, ϕ) = 1 r exp iΘ(r, ϕ) , (B1) with eigenvalue µ = −b+iωeff . Here ωeff = ω +b presents the phase velocity...

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