{"id":"4bae7b45-a624-4255-98db-80fd9573aac8","arxiv_id":"2607.22457","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A time-delay-embedded DMD workflow with a moving frame of reference decomposes RDC luminosity data into counter-rotating traveling waves and a nonlinear interaction remainder.","lead":"Researchers used a sequence of Dynamic Mode Decomposition variants on high-speed flame videos to separate the traveling detonation waves in a rotating detonation combustor from the nonlinear 'remainder' they leave behind. The method lets them quantify how much luminosity comes from wave-collision amplification across three operating modes.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Residual R is not established as a clean measure of nonlinear amplification: Section 5 concedes longitudinal modes remain in R, so the reported percentages (e.g., 54.47%) may conflate unmodeled linear content with nonlinear interaction.","rationale":"The reader's weakest_assumption correctly identifies the most load-bearing concern. The paper's contribution is not merely the DMD pipeline—it is the physical interpretation of the residual as nonlinear amplification. That interpretation is undermined by the authors' own admission in Section 5 that longitudinal modes remain in R, and by the fact that R is defined as a residual from truncated DMD reconstructions rather than as an independently validated nonlinear signal. I considered whether a more fundamental issue exists, such as the constant scaling coefficients α1, α2 in Eq. 27 forcing time-independent traveling-wave amplitudes; however, this is largely the intended separation, and the real problem is that R is not shown to contain only nonlinear content. I do not see an internal inconsistency in the DMD sequence itself; the moving-frame transformation, time-delay embedding, and OptDMD steps are standard and appear mutually consistent. The reported training-set reconstruction agreement and spectral match are useful but in-sample evidence. Because the quantitative claims are conditional on the residual interpretation, the reader's CONDITIONAL verdict remains appropriate—no stronger rejection is warranted, since the qualitative extraction of traveling waves is plausible and the limitation is explicitly acknowledged.","tokens_in":21422,"tokens_out":4270,"duration_ms":52426,"concrete_test":"Recompute R after explicitly including the identified L4 mode(s) in the linear reconstruction (or projecting R onto the complement of the L4 subspace) and recalculate the collision percentages in Section 4.4. If the 54.47% (2CR), 61.57% (2CRT), and 55.79% (DS2) values drop substantially—or if the maximum of R no longer localizes at the collision point—then the claim that R isolates nonlinear amplification is unsupported. A supporting check: fit α1, α2 on a 250-snapshot training window and evaluate residual on the held-out 250 snapshots; large errors or shifted percentages would indicate the decomposition is an in-sample fit rather than a physical separation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative claim—that R = Z - α1 Z_CW - α2 Z_CCW (Eq. 28) isolates nonlinear amplification—requires that everything not captured by the two reconstructed traveling-wave subspaces be nonlinear interaction. This condition is not established. Section 5 explicitly states that longitudinal modes 'are still present in the data' and would need to be separated by running DMD on R. Further, Z_CW and Z_CCW are truncated, denoised DMD reconstructions, so R necessarily contains any spectral content excluded in Steps 4–6: harmonics, L4 modes, residual noise, and interaction modes. The percentages in Section 4.4 appear to be snapshot-wise ratios of ||R|| to ||Z||, so even an approximately constant L4 contribution inflates them. The lack of held-out validation or error bars means one cannot distinguish physical nonlinear amplification from systematic decomposition error. The paper's own conclusion overstates the certainty of the decomposition: the remainder is better described as 'unmodeled content' unless longitudinal modes are explicitly removed.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":21687,"tokens_out":8279,"duration_ms":98498,"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":[{"comment":"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.","section":"Section 4.4 and Section 5, Eq. (28)"},{"comment":"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.","section":"Appendix B, Eqs. (38)–(39)"},{"comment":"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.","section":"Section 4.4, Figure 11"},{"comment":"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.","section":"Section 4.3, Steps 6–7"}],"minor_comments":[{"comment":"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.","section":"Section 3.1.1, Eq. (1)"},{"comment":"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.","section":"Section 3.1.3, Eq. (14) and Eq. (29)"},{"comment":"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.","section":"Figure 11"},{"comment":"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.","section":"Appendix B"}],"recommendation":"major_revision","confidential_remarks":"The methodological novelty is modest, and the authors acknowledge it. The value of the manuscript hinges on whether the decomposition can be made quantitative. If the authors can separate the longitudinal modes from R and add validation on a holdout time segment, I would support acceptance. Otherwise, the quantitative claims about nonlinear amplification should be removed or drastically qualified. The paper fits an applied data-driven modeling venue better than a pure mathematics journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read of arXiv:2607.22457. The paper does something genuinely useful: it assembles a sequence of existing DMD variants—moving-frame transform, time-delay embedding, ExactDMD, OptDMD, and a threshold-based truncation—into a workflow that actually works on difficult, noisy RDC image data. The moving-frame idea and the physics-based estimate for the number of time-delay embeddings (s = 1/(f Δt)) are sensible and well-motivated. The authors are honest that the novelty is the composition, not new DMD theory. On the data side, the reconstructions are convincing: eigenvalues land on the unit circle, spectral agreement is strong, and the decomposition cleanly separates CW/CCW traveling waves across all three operating modes. The comparison showing standard exactDMD failing is a useful cautionary tale for practitioners. Credit where due: the writing is clear, the algorithmic sequence is reproducible, and the data are openly available. That is real value for the RDC diagnostics community. The soft spots are real but not fatal. The central quantitative claim—that the residual R = Z - α1 Z_CW - α2 Z_CCW isolates nonlinear amplification—is not established. As the authors themselves concede in Section 5, longitudinal modes are still present in R; the percentages (54.47%, 61.57%, 55.79%) therefore mix unmodeled linear content with genuine nonlinear interaction. The lack of holdout validation compounds this: every error metric (etrain, spectral agreement) is computed on training data, and the α coefficients are fit to the same data, so overfitting cannot be ruled out. That said, the paper's qualitative conclusions—that the primary wave decays faster than secondary waves, and that interactions differ across modes—are plausible and consistent with the physics and with prior work by Barnouin et al. I would not treat the percentages as quantitative measurements of pure nonlinearity. That is a significant over-claim, but it is fixable: run DMD on R to remove L4 modes, validate on held-out snapshots, and report error bars. The sensitivity analysis in Appendix B stops short of these steps. Overall, this is a serious, well-executed engineering contribution that deserves a proper referee, not a desk reject. The main load-bearing assumption should be challenged in review. I would bring it to the reading group: it is a good example of how to adapt DMD to transport-dominated experimental data, and it prompts a useful discussion about what a 'nonlinear remainder' really means. I would cite it for the methodological pipeline and the comparison of operating modes. Recommendation: send it to peer review with a request for holdout validation and a more careful interpretation of the residual. The contribution is there, but the central quantitative claim needs to be reined in or supported with additional analysis.","headline":"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.","tokens_in":22191,"tokens_out":700,"would_cite":true,"duration_ms":9764,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["rotating detonation combustor","Koopman operator","dynamic mode decomposition","time-delay embedding","counter-rotating waves","nonlinear amplification","flame luminosity","data-driven modeling"],"falsifier":"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.","tokens_in":21272,"feed_emoji":"🔥","tokens_out":6150,"duration_ms":66034,"temperature":0.7,"pith_summary":"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.","feed_headline":"One workflow splits detonation luminosity into waves and nonlinearity","feed_subtitle":"Koopman-based DMD isolates counter-rotating waves and quantifies their collision-driven amplification.","key_machinery":"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","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Koopman DMD teases waves apart from nonlinearity in detonation","Time-delay Koopman model splits detonation waves and nonlinearity","Data-driven model separates detonation wave modes and collisions","Koopman method unveils wave and nonlinearity in rotating detonation","Physics-aware DMD isolates wave interactions in combustor"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Koopman DMD teases waves apart from nonlinearity in detonation","Time-delay Koopman model splits detonation waves and nonlinearity","Data-driven model separates detonation wave modes and collisions","Koopman method unveils wave and nonlinearity in rotating detonation","Physics-aware DMD isolates wave interactions in combustor"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000168,"raw_usage":{"total_tokens":1097,"prompt_tokens":740,"completion_tokens":357,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":484,"completion_tokens_details":{"reasoning_tokens":268}},"tokens_in":484,"tokens_out":357,"duration_ms":5037,"temperature":1.0,"reasoning_tokens":268,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T04:39:54.641102+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}