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Data Driven Modeling of Nonlinear Dynamics in a Rotating Detonation Combustor via Finite Dimensional Approximations of the Koopman Operator

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A sequence of Koopman-based DMD variants separates rotating detonation luminosity into two counter-rotating traveling waves and a nonlinear remainder, allowing quantification of nonlinear amplification at wave collisions.

desk verdict Solid engineering paper: a tailored DMD pipeline that cleanly extracts counter-rotating waves from RDC luminosity data, but the claim that the residual isolates nonlinear amplification is overstated and the key percentages lack validation against held-out data or uncertainty estimates. read the letter →

arxiv 2607.22457 v1 pith:KPTDX7N6 submitted 2026-07-24 math.OC cs.CE

David Oexle , Tobias Breiten , Myles D. Bohon This is my paper · ORCID
classification math.OCcs.CE
keywords rotatingdetonationcombustorKoopmanoperatordynamicmodedecompositiontime-delayembeddingcounter-rotatingwavesnonlinearamplificationflameluminositydata-drivenmodeling
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

This paper claims that a sequence of data-driven methods built on finite-dimensional Koopman operator approximations can take high-speed video of flame luminosity in a rotating detonation combustor and separate it into two counter-rotating traveling waves plus a remainder that captures nonlinear amplification at wave collisions. By subtracting the scaled traveling-wave reconstructions from the full signal, the authors obtain a residual R that peaks sharply when the waves collide and decays between collisions. Tracking this remainder at the wave crests across three operating modes shows that it accounts for roughly half to two-thirds of the luminosity at collision (54.47%, 61.57%, 55.79%) and that its decay profile differs between primary and secondary waves. If the decomposition is trustworthy, it gives combustor researchers a purely data-driven way to quantify nonlinear interaction strength and to survey how different operating modes amplify and sustain counter-rotating waves.

What carries the argument

The load-bearing object is a finite-dimensional approximation of the Koopman operator constructed from an extended data matrix built by time-delay embedding the state (stacking s shifted snapshots), computed with an Exact DMD and refined with an optimized DMD. The pivotal step is the change of frame of reference: circularly shifting each snapshot so that the CW or CCW wave appears stationary, which makes that wave's modes low-rank and cleanly recoverable. The time-delay count s is set to (1/f_Base)/Δt, where f_Base is the lowest dynamically relevant frequency from the FFT (here the L4 longitudinal mode), which the authors argue makes the dictionary approximately Koopman-invariant for periodi

What would settle it

Take the published algorithm and run it on a synthetic dataset with two known counter-rotating traveling waves of known amplitude plus a prescribed nonlinear burst at the collision point (e.g., a Gaussian bump that grows then decays); if the recovered R does not match the burst's amplitude and decay within a few percent, the claim that R measures nonlinear amplification quantitatively is falsified.

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Extended reading notes

Core claim

The central demonstration is that by first shifting the data into the moving frame of a traveling wave, embedding the state with a physics-based number of time delays (set by the period of the lowest dynamically relevant frequency, here the L4 longitudinal mode), and using an optimized DMD to remove sensor-noise bias, the resulting spectrum cleanly separates DMD modes into those belonging to the clockwise and counterclockwise waves and their stationary harmonics. Reconstructing only those traveling-wave modes, scaling them via a least-squares fit, and subtracting them from the full data leaves a remainder R that the authors identify as the nonlinear amplification. Across the three operating

Load-bearing premise

The claim that the residual R quantifies nonlinear amplification assumes that everything not explained by the scaled traveling-wave reconstructions is nonlinear interaction; but the paper itself notes longitudinal modes remain in R, so the residual is not purely nonlinearity.

Editorial extensions

If this is right

  • Researchers can use this workflow to decompose RDC luminosity or pressure data into traveling waves and a nonlinear remainder, enabling quantitative mode-by-mode comparison of nonlinear amplification.
  • The decay rate of the remainder after a collision gives a data-driven marker for how strongly collisions re-energize secondary waves, which could be linked to detonation stability and mode-transition onset.
  • The same sequence should transfer to pressure-probe data (higher temporal resolution), offering a complementary view of the same nonlinear interactions.
  • The physics-based formula for the number of time-delay embeddings removes a heuristic choice from EDMD and may simplify the method for other periodic-flow experiments.
  • Because the remainder is separated from the linear waves, it becomes a target for further system identification (e.g., sparse regression on the remainder) to build simplified models of the nonlinear interaction.

Reading between the lines

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

  • The reported percentages are likely sensitive to the choice of scaling coefficients and to leftover longitudinal modes in R; a conservative reading is that they bound nonlinear amplification from above until the longitudinal content is removed (the paper itself flags this).
  • One testable extension: apply the identical sequence to a synthetic signal of two known traveling waves plus a prescribed collision burst; if the recovered remainder does not match the injected burst in amplitude and decay shape, the residual-based quantification should be reinterpreted.
  • The remainder R offers a natural observable for precursor detection: if its collision-peak amplitude grows over successive collisions as the inlet mass flow changes, it may flag an approaching mode transition before the frequency spectrum changes.
  • Because the decomposition is model-free, it can serve as a benchmark for reduced-order combustion models: a good model should reproduce not only the traveling waves but also the R remainder's amplitude and post-collision decay.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper proposes an algorithmic sequence for applying EDMD/DMD variants to high-speed flame-luminosity video of a rotating detonation combustor. The pipeline shifts data into moving frames, selects a time-delay count from an FFT-identified low-frequency mode, truncates by an optimal singular-value threshold, uses ExactDMD with OptDMD refinement, reconstructs CW/CCW traveling-wave components by selecting spectral subsets, fits scaling coefficients α1, α2 on the training data, and defines R = Z − α1Z_CW − α2Z_CCW as the nonlinear remainder. The paper reports that this sequence extracts the traveling-wave structure, quantifies nonlinear amplification at collision (e.g., 54.47% for the 2CR mode), and compares decay behavior across three operating modes.

Significance. If the decomposition were clean, the workflow would give RDC practitioners a practical way to separate counter-rotating detonation waves from other features and to compare interaction amplitudes across modes. The manuscript's strengths include a clear assembly of existing DMD methods, a physical motivation for time-delay embeddings, a sensitivity analysis of the parameters s and r (Appendix B), and open data availability through Zenodo (Ref. [33]). However, the central quantitative claims rest on interpreting a fitting residual as nonlinear amplification, and this interpretation is not established. With validation, a precise definition of the reported percentages, and an explicit treatment of the remaining longitudinal modes, the workflow could be a valuable data-analysis contribution; in its current form the quantitative separation claims outrun the evidence.

major comments (4)
  1. [Section 4.4 and Section 5, Eq. (28)] The central quantitative claim is not supported. R is defined as the residual after subtracting truncated, denoised reconstructions of two traveling-wave subsets whose coefficients α are fitted to the training data. Any omitted spectral content—L4 longitudinal modes, harmonics, noise, interaction modes—is automatically included in R. The conclusion (Section 5) explicitly states that longitudinal modes 'are still present in the data' and proposes future DMD on R to separate them. Therefore, the percentages in Section 4.4 (54.47%, 61.57%, 55.79%) should not be presented as contributions of nonlinear amplification alone. Either remove/reword these quantitative claims or demonstrate, by decomposing R, that L4 and noise are negligible within the reported windows.
  2. [Appendix B, Eqs. (38)–(39)] All reported error metrics are training errors. The same 500-snapshot subset is used to select s, r, the mode subsets in Step 6, and to fit α1, α2 via Eq. (27); no holdout segment, cross-validation, or bootstrap is reported. Since OptDMD solves a nonlinear least-squares problem on this exact data (Eq. 17), the reconstruction accuracy and spectral agreement do not establish that the extracted Z_CW/Z_CCW are physical rather than overfit. Please add validation on a disjoint time interval (or bootstrap) and report the resulting spread in the Section 4.4 percentages.
  3. [Section 4.4, Figure 11] The reported percentages are not defined. The text says 'how much the nonlinear signal contributes to the total luminosity,' but no equation specifies whether this is a spatial-point ratio, a spatial-norm ratio, or a time-averaged quantity. The labels in Figure 11 (e.g., 45.33%, 54.47%) appear to be snapshot-wise ratios of ||R|| to ||Z||, but this is never stated. The same ambiguity applies to the curves in Figure 12. Add an explicit definition of the percentage and the sampling procedure; otherwise the comparison across modes cannot be reproduced.
  4. [Section 4.3, Steps 6–7] The separation relies on constant scalar coefficients α1, α2 fitted globally over the training interval. A physically meaningful decomposition of a nonlinear interaction should allow amplitude modulation near the collision; by forcing a single scaling per component, the method necessarily pushes any time-localized growth or decay of the traveling waves into R. This strengthens the concern that R conflates nonlinear amplification with unmodeled linear amplitude variation. Please discuss this assumption and, if possible, test per-time scaling or windowed validation.
minor comments (4)
  1. [Section 3.1.1, Eq. (1)] The displayed equation 'Kg(z) = g(F(z_k)) = F(z_k)' is mathematically confusing. The Koopman operator acts on the observable g, not on the state; if g(z)=z, the correct statement is K g(z_k) = g(F(z_k)) = z_{k+1} = F(z_k), followed by the expansion. Please correct the notation.
  2. [Section 3.1.3, Eq. (14) and Eq. (29)] The rounding/ceiling operator used to define s is not clear from the typeset equation. Please specify whether s is the floor or ceiling of T/Δt and define T consistently.
  3. [Figure 11] The figure captions and panel labels should define the red/yellow/blue markers and state how the percentage labels under each snapshot were computed. The current caption refers only to the blue markers.
  4. [Appendix B] The statement that certain (r,s) combinations 'lead to poor conditioning ... appear as sudden peaks' would benefit from a concrete example or a threshold; otherwise the sensitivity curves are hard to interpret.

Circularity Check

1 steps flagged · score 6.0 of 10

The residual R is a least-squares fitting residual relabeled as nonlinear amplification; Section 5 concedes longitudinal modes remain in R, so the reported percentages are not an independent measure of nonlinearity.

  1. fitted input called prediction [Section 4.2, Step 7 (Eqs. 27–28) and Section 5]
    "In Step 7 the scaling coefficients for the reconstructed subset are computed by solving the least square problem [α1 α2]^T = argmin ||Z_train − (α̃1 Z_CW + α̃2 Z_CCW)||_F^2. (27) Subsequently, the remainder containing the nonlinear interactions is computed as R = Z − α1 Z_CW − α2 Z_CCW. (28) ... the algorithmic sequence should be extended such that the remainder itself is subject to DMD analysis. This may help to separate the longitudinal modes that are still present in the data."

    By Eq. (27), α1 and α2 are fit by least squares to the same training signal Z_train. Hence R in Eq. (28) is by construction the portion of that training signal not captured by the two fitted traveling-wave reconstructions. Any signal would yield such a residual. The paper then identifies R with 'nonlinear amplification' and reports snapshot-wise percentages such as 54.47% for the 2CR mode (Section 4.4). Section 5 explicitly concedes that longitudinal modes are still present in R, so the residual also contains linear content. Thus the reported 'contribution due to nonlinear amplification' is a renamed fitting residual, not an independently measured nonlinear effect.

full rationale

The core DMD/EDMD machinery is standard external methodology (ExactDMD, OptDMD, optimal thresholding), and the paper does not rely on a load-bearing self-citation chain or an imported uniqueness theorem. The circularity is narrower but central: the decomposition Z ≈ α1 Z_CW + α2 Z_CCW is fitted to the training data, and the remainder R is then interpreted as a clean measurement of nonlinear amplification. That interpretation is not derived; it is an assumption, and the paper's own Section 5 undermines it by acknowledging that longitudinal modes remain in R. The spectral-agreement and reconstruction-error metrics are also computed on the training data, so they do not provide out-of-sample validation that would distinguish physical nonlinearity from omitted linear content, harmonics, or noise. Because the headline quantitative claims (percentage contributions and decay behavior of the 'nonlinear remainder') reduce, by construction, to properties of a least-squares residual, a score of 6 is appropriate. This is partial circularity rather than a fully tautological derivation, since the identification of the two traveling-wave subspaces and the DMD approximations themselves are legitimate, independently defined computations.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claims rest on a small number of explicitly fitted parameters (s, r, α, f_CW/f_CCW) and on several domain assumptions: periodicity, radial uniformity, validity of the noise-threshold criterion, and—most critically—the interpretation of the residual as pure nonlinear amplification, which the authors themselves qualify at the end of the paper. No new physical entities are introduced.

free parameters (4)
  • s (number of time-delay embeddings) = s=43 for 2CR; in general s = ceil(1/(f_Base Δt))
    Hyperparameter chosen from the FFT-determined lowest relevant frequency (L4 mode); the dictionary and the DMD model depend directly on s (Eqs. 12-14, 29).
  • r (truncation rank) = r_CCW,s=90, r_CW,s=86 for 2CR
    Rank chosen via Gavish-Donoho optimal threshold (Eq. 21); determines how many modes are retained and which dynamics are considered noise.
  • α1, α2 (scaling coefficients) = not reported numerically
    Fitted by least squares in Eq. (27) to match the full training signal; the nonlinear remainder R is computed from these fitted coefficients (Eq. 28).
  • f_CW, f_CCW (wave frequencies for moving frame) = f_CW = f_CCW = 3.989e3 Hz for 2CR
    Obtained from FFT of the same luminosity/pressure data (Section 2.3); used to set the moving frame and circular shifts (Eqs. 24-25).
assumptions (6)
  • standard math Koopman operator theory provides a valid linear representation of the nonlinear dynamics in the space of observables.
    Background theory used to justify DMD; invoked in Section 3.1.1 and Appendix A.
  • domain assumption The RDC dynamics are sufficiently periodic that the time-delay dictionary is approximately Koopman invariant (Eq. 11).
    The formula for s (Eq. 14/29) assumes a single low-frequency period T = 1/f_Base; the data includes multiple interacting modes, so the invariance is only approximate.
  • domain assumption Radial variations in luminosity are negligible, and the mirror distortion is adequately corrected by the data pipeline.
    Used to reduce 329×329 images to a 1D radial average z_k ∈ R^819 (Section 2.2).
  • ad hoc to paper The remainder R = Z - α1 Z_CW - α2 Z_CCW isolates nonlinear wave amplification.
    Central interpretative step in Section 4.4, yet Section 5 admits that longitudinal modes are still present in R, so this assumption is not satisfied.
  • domain assumption The Gavish-Donoho singular value threshold is valid for this noise setting.
    Used in Step 3 of the sequence; the threshold assumes a specific noise structure that may not match the experimental images.
  • domain assumption Wave speeds are constant over the 500-snapshot subset, and the rounding in the circular shift has a minor effect.
    The moving-frame transform uses integer shifts h = round(Δθ_c/Δθ_n) (Eq. 24); the paper notes this rounding introduces oscillations.

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Pith. "Pith review of Data Driven Modeling of Nonlinear Dynamics in a Rotating Detonation Combustor via Finite Dimensional Approximations of the Koopman Operator." pith.science (2026). https://pith.science/paper/KPTDX7N6

@misc{pith2026260722457,
  author       = {Pith},
  title        = {Pith review of: Data Driven Modeling of Nonlinear Dynamics in a Rotating Detonation Combustor via Finite Dimensional Approximations of the Koopman Operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPTDX7N6}},
  note         = {Machine review of arXiv:2607.22457}
}
read the original abstract

A Rotating Detonation Combustor (RDC) is a promising technology for increasing efficiency in propulsion and power generation applications. The dynamics of the RDC are governed by continuously propagating detonation waves within an annular combustion chamber. Multiple operating modes can be observed, including nonlinear interactions between counter-rotating waves and the emergence of standing wave patterns. Koopman operator theory provides a framework to globally linearize nonlinear dynamical systems by representing their evolution in the space of observables rather than states. In this work, finite-dimensional approximations of the Koopman operator are constructed using variants of Dynamic Mode Decomposition (DMD) applied to high-speed video data capturing the natural flame luminosity of the detonation waves in the RDC at the Technical University (TU) Berlin. By introducing time-delay embeddings as a dictionary of observables, this approach overcomes the limitations of standard DMD methods, particularly for accurate reconstruction of standing wave patterns and for capturing nonlinear interactions. In addition, a technique is presented to mitigate the influence of sensor noise in the luminosity measurements. Finally, it is shown that the DMD-based models provide insight into the dynamics of different operating modes by decomposing the reconstructed signal into its characteristic features.

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.