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REVIEW 3 major objections 5 minor 30 references

Koopman-based upsampling, used as a preprocessing step before sparse regression, improves equation recovery when the learned representation matches the dynamics.

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2026-08-03 14:52 UTC pith:JZYYUSOK

load-bearing objection Honest, careful benchmark of DMD/EDMD upsampling before SINDy/PDE-FIND; gains are real but modest and system-dependent, and the paper overstates its mechanistic claim slightly. the 3 major comments →

arxiv 2607.29036 v1 pith:JZYYUSOK submitted 2026-07-31 cs.LG math.DS

Dynamics-aware identification of governing equations from sparse and noisy data

Pongpisit Thanasutives , Yoshinobu Kawahara This is my paper
classification cs.LG math.DS
keywords sparse identification of nonlinear dynamicsPDE functional identificationKoopman operatordynamic mode decompositionextended dynamic mode decompositionderivative estimationsparse regressionupsampling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper asks whether fitting a Koopman-style linear model to sparse, noisy snapshots and using it to insert intermediate time points before derivative estimation improves the equations recovered by SINDy and PDE-FIND. Its answer is yes under a stated condition: when the model's observable dictionary and low-rank structure match the dynamics, upsampling reduces derivative error. The strongest evidence is that polynomial EDMD preprocessing cuts median Lorenz-63 coefficient error from 0.911 to 0.526 and Van der Pol from 0.107 to 0.015, and low-rank DMD/EDMD improves Burgers and advection-diffusion discovery. For Fisher-KPP with a smooth initial condition the raw baseline is competitive, which the paper reads as confirming the conditionality rather than the failure of the method. If the claim holds, Koopman-based upsampling offers a practical preprocessing layer for sparse, noisy equation discovery, with clear gains when the dictionary is chosen for the dynamics.

Core claim

On its own terms, the paper establishes that finite-dimensional Koopman approximations (DMD, EDMD, optimized DMD, POD-EDMD) can act as a dynamics-aware interpolation-denoiser that hands SINDy or PDE-FIND a lower-error derivative-estimation problem. The strongest gain is ODE coefficient accuracy: polynomial EDMD cuts median coefficient error from 0.911 to 0.526 (Lorenz-63) and 0.107 to 0.015 (Van der Pol). The PDE gain is conditional: low-rank DMD/EDMD improves Burgers and advection-diffusion; smooth Fisher-KPP keeps the baseline competitive, while a sharper initial condition reverses that. DMD-assisted upsampling also stabilizes non-oracle EBIC support-size selection, recovering exact Fisher

What carries the argument

The load-bearing object is the finite-dimensional Koopman/DMD model M, a matrix that advances a vector of observables by one sampling interval. Fractional steps M^(l/q)=exp((l/q) log M) move it by a fraction of an interval, and a local-reset interpolation scheme anchors each interval at the next measured snapshot, filling q-1 intermediate points and preventing long-horizon drift. The observable dictionary (polynomial or radial-basis features for ODEs, POD-compressed coordinates for PDEs) determines whether a linear M can represent the nonlinear evolution; the positive results come when that dictionary is aligned with the target equation.

Load-bearing premise

The load-bearing premise is that a finite-dimensional linear (Koopman/DMD) model fitted to the sparse noisy snapshots interpolates the hidden states accurately enough inside the observed window that derivatives from the reconstructed trajectory beat direct finite differences of the raw samples.

What would settle it

Take a clean dense reference trajectory of a polynomial ODE with known coefficients, subsample and add noise as in the paper, then compare the derivative error (against the clean reference) of raw finite differences versus derivatives of the EDMD-polynomial reconstructed trajectory at q=5 over many seeds. If the reconstructed trajectory's derivative error is not lower than the raw estimate for a system whose dictionary matches the dynamics, the central claim fails. A direct run of Fisher-KPP with the sharp front-type initial condition at the stated settings would also serve: the paper predicts

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Polynomial EDMD preprocessing can substantially reduce coefficient error for polynomial ODE discovery under sparse noisy sampling, improving the practical score in roughly 80% of matched records.
  • For PDEs, the gain is conditional: low-rank DMD/EDMD reconstructions improve Burgers and advection-diffusion, while the raw baseline remains competitive for smooth Fisher-KPP transients; using a front-type initial condition can flip the comparison.
  • DMD-assisted upsampling can make non-oracle information-criterion support-size selection reliable enough to recover the exact Fisher-KPP support in all tested seeds, where the unassisted baseline never did.
  • Because Koopman-based preprocessing outperforms tuned linear and smoothing-spline interpolation at the same inserted point count, its value does not come from densification alone but from the learned low-rank dynamics.
  • In practice, the method should be applied as a conditional preprocessing choice, selected when the dictionary and POD rank are known to fit the data, rather than unconditionally.

Where Pith is reading between the lines

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

  • Editorial inference: the conditionality of the gain suggests a dictionary-selection or model-validation step could make the preprocessing adaptive; the paper does not test that.
  • Editorial inference: the non-monotone Lorenz-63 response to the upsampling factor hints that a sparse linear model is trustworthy only within a limited interpolation horizon; a residual-based or Lyapunov-time-informed rule for picking the upsampling factor is a natural next step.
  • Editorial inference: because derivative estimation is the bottleneck, combining Koopman upsampling with weak-form or integral SINDy, which avoids temporal derivatives, may yield larger robustness gains than either alone; the paper lists weak-form methods as future work but does not combine them.
  • Editorial inference: the non-oracle information-criterion experiment uses derivative-regression residuals as the criterion, so extending the evaluation to forecast-based validation on experimental data would test whether the selected equations generalize outside the fitting window.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper proposes and evaluates a preprocessing strategy for SINDy/PDE-FIND in which finite-dimensional Koopman approximations (DMD, EDMD, optimized DMD, and POD-based variants) are fitted to sparse, noisy snapshots and then used to interpolate and denoise the trajectory inside the observed time window before derivative estimation and sparse regression. The benchmark includes two ODE systems (Lorenz–63, Van der Pol) and three periodic PDE systems (Burgers, Fisher–KPP, advection–diffusion) across several sparsity and noise levels. The main results are that polynomial EDMD substantially reduces median coefficient error for the ODE systems, and that low-rank DMD-assisted preprocessing gives system-dependent gains for the PDE systems, with the raw baseline remaining competitive for Fisher–KPP. The paper also reports an ablation of interpolation strategies, a comparison against linear and smoothing-spline interpolation, and a non-oracle EBIC/Pareto model-selection experiment.

Significance. If the empirical claims are correct, the paper provides a useful, carefully bounded contribution: it identifies when dynamics-aware upsampling helps and when it does not, and it gives concrete guidance for practitioners. The manuscript is unusually transparent about the oracle nature of the main diagnostic score, and it includes reproducible code, sensitivity analyses, an interpolation-strategy ablation, and a non-oracle selection experiment. The main weakness is that the central mechanism asserted in the abstract—that Koopman-based upsampling reduces derivative-estimation error—is never directly measured; only downstream coefficient and support metrics are reported. In addition, the headline comparisons are presented without uncertainty quantification or significance tests, and the 'valid evaluation records' filter is not quantified. These issues are fixable within the scope of the manuscript, but they currently leave the causal claim and the strength of the comparative conclusions under-supported.

major comments (3)
  1. [§3.1, Table 2, Abstract] The abstract and conclusion state that Koopman-based upsampling 'can reduce derivative-estimation error,' but no experiment directly measures the error between the estimated derivatives used in Eqs. (2)–(3) and the true derivatives. The reported metrics (support F1, coefficient error, practical score) are downstream outcomes. Because the DMD/EDMD interpolant is trained on the same sparse noisy snapshots and then used to produce the derivative target, the coefficient-error improvement could in principle arise from changes in library conditioning or from the polynomial observable alignment rather than from more accurate derivatives. The true derivatives are available in these synthetic benchmarks, so a direct comparison (e.g., relative L2 derivative error as a function of q and noise, correlated with coefficient error) is feasible and should be added. This is load-bearing because the causa
  2. [§3.1–§3.2, Tables 2–4] The comparative claims are reported as point estimates without uncertainty. For example, 'EDMD-polynomial improves the practical score in 80.3% of cases' and 'improves each quantity in 74.3% of matched cases' are given without confidence intervals or paired significance tests. Tables 2 and 3 report only means and medians, and Table 4 uses only five seeds. The 'valid evaluation records' filter mentioned in Section 5 and the Table 3 footnote is not quantified; if a substantial number of records are excluded for computational-budget reasons, the comparison may be biased. Please report the number of valid records per cell, standard errors or bootstrap confidence intervals, and paired tests for the matched-record percentages and for the non-oracle selection results.
  3. [§2.4, Eq. (17)] The local-reset interpolation uses fractional powers M^{ℓ/q} computed by eigendecomposition with eigenvalues clamped away from zero, and the resulting step matrices are projected to real form. The manuscript does not report the clamping threshold, the frequency of non-diagonalizable or problematic M, or any condition-number diagnostics. This is not necessarily fatal, but it is relevant to the robustness of the central mechanism, especially because Table 5 shows non-monotone behavior in q for Lorenz-63. Please state the clamping value, report any failure cases or diagnostics, and ideally include a sensitivity check on the clamping threshold.
minor comments (5)
  1. [§2.3, Eq. (11)] The EDMD regression is written in row-stacked form with K = (Ψ_X^T Ψ_X + α_K I)^{-1} Ψ_X^T Ψ_Y, and later M = K^T is used as the column-action map. The row/column convention is explained, but it is easy to misread. A one-sentence clarification of which shape applies to each matrix would help.
  2. [Table 2 vs Eq. (7)] The Score column is the mean of per-record S values, while F1 is a mean and coefficient error is a median. The authors correctly state that the Score should not be recomputed from the displayed aggregate components, but this makes the table hard to verify. Consider reporting per-cell standard deviations for F1 and Score, or a supplementary table with per-record values.
  3. [Appendix C, Table 9] The Burgers row in Appendix C uses POD-EDMD-RBF with r=4, while the main text (§2.3) states that the main Burgers experiments use r=8. The appendix says the rank is validation-selected, but the inconsistency is not explicitly flagged for Burgers as it is for advection–diffusion. Please clarify that the Appendix-C rank differs from the main benchmark and explain the effect on comparability.
  4. [§5, Limitations] The sentence 'Accuracy metrics are computed over valid evaluation records under the prescribed computational budget' is vague. Please specify what makes a record invalid, how many records were invalid per table, and whether the conclusions change if invalid records are included.
  5. [Appendix C] The smoothing-spline parameter is validation-tuned, but the interpolation factor q is fixed across methods because the authors argue that held-out observations cannot identify a preferred dense-grid spacing. This is a reasonable choice, but it means the comparison is not fully 'fair' in the sense that the spline is not given the same tuning freedom. A brief acknowledgment of this asymmetry would be useful.

Circularity Check

0 steps flagged

No circularity: the fitted DMD/EDMD preprocessors are evaluated against external synthetic ground-truth equations; oracle thresholds and self-citations are disclosed and not load-bearing.

full rationale

The fitted DMD/EDMD/optDMD models (Eqs. 9–14) are used only to insert intermediate snapshots (Eq. 17) before derivative estimation; they are not the evaluation target. The reported F1/coefficient-error/score metrics are computed against the true synthetic systems in Table 1, which are external to the preprocessor fit. The practical score S in Eq. (24) uses true coefficients, but §2.2 labels it 'a simple synthetic-benchmark diagnostic' and 'paper-specific,' so threshold selection by S is a disclosed oracle control, not a fitted parameter renamed as a prediction. The non-oracle EBIC/Pareto experiment (§2.5, Table 4) uses ground truth only after selection. The citations to the authors' prior work (Thanasutives et al. 2023–2026) are background references to software and model-selection practices that are re-implemented in this paper; no load-bearing claim is imported through self-citation. The main weakness—that the derivative-error-reduction mechanism is only indirectly evidenced by downstream coefficient error, and that PDE preprocessing mixes temporal upsampling with POD denoising (§3.2)—is an evidentiary limitation, not a definitional reduction. No equation or fitted quantity is equivalent by construction to the reported results.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 0 invented entities

The method rests on the empirical adequacy of finite-dimensional Koopman/POD approximations and the synthetic benchmark design. The listed free parameters are disclosed hyperparameters rather than hidden fitted constants; no new physical or mathematical entities are introduced.

free parameters (5)
  • upsampling factor q = q=5 (main benchmarks)
    Fixed interpolation factor. Appendix A shows q=5 is not uniformly optimal (e.g., Burgers best at q=1), so results depend on this hand-chosen value.
  • POD rank r = r=8 Burgers, r=2 Fisher-KPP, r=4 advection-diffusion
    System-specific fixed hyperparameters chosen by the authors; Fisher-KPP low rank is used to limit noise amplification. Sensitivity shows defaults are competitive but not globally optimal.
  • ridge parameter alpha = 1e-8 for STLSQ and EDMD fits
    Small ridge used to condition solves; chosen by convention rather than tuned.
  • EBIC gamma = 0.5
    Combinatorial penalty weight in Eq. (22) for non-oracle model selection; a default choice from the EBIC literature that affects selected support sizes.
  • STLSQ threshold grids = Lambda_ODE and Lambda_PDE grids listed in Section 2.2
    All oracle results are best-case over these discrete grids, so the grid resolution and spacing are free choices that affect reported scores.
axioms (5)
  • domain assumption A finite-dimensional Koopman/DMD map (Eq. 1) produces plausible intermediate states for interpolation inside the observed window.
    The entire preprocessing idea rests on this; if the dictionary or low-rank structure misrepresents the dynamics, upsampling can inject model error.
  • domain assumption The candidate libraries in Section 2.2 contain the true active terms for every benchmark system.
    The libraries are constructed to include u, u^2, u_x, uu_x, u_xx, etc.; real applications may require richer or unknown libraries.
  • standard math The synthetic ground-truth trajectories are accurate numerical solutions of the stated ODEs and PDEs.
    The benchmark assumes the reference simulations correctly represent the target dynamics.
  • domain assumption POD rank-r subspace retains the dynamics-relevant field information despite noise.
    PDE preprocessing jointly performs spatial POD denoising and temporal upsampling; if the rank is too low, dynamics are discarded.
  • domain assumption EBIC with gamma=0.5 and a Kneedle-style elbow selects the correct support size in the non-oracle experiment.
    This model-selection rule is a methodological choice validated only empirically on the benchmark systems.

pith-pipeline@v1.3.0-daily-deepseek · 17796 in / 11947 out tokens · 122269 ms · 2026-08-03T14:52:21.956981+00:00 · methodology

0 comments
read the original abstract

Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.

Figures

Figures reproduced from arXiv: 2607.29036 by Pongpisit Thanasutives, Yoshinobu Kawahara.

Figure 1
Figure 1. Figure 1: Representative trajectories and fields for Lorenz–63, Van der Pol, periodic Burgers, and Fisher–KPP. Markers in the ODE panels indicate sparse noisy observations at 3% relative noise. White vertical markers in the PDE panels indicate sparse observation times; the heatmaps show the clean reference fields before noise injection. 3. Empirical results The accuracy metrics in Tables 2 and 3 are oracle-tuned ove… view at source ↗
Figure 2
Figure 2. Figure 2: Representative field for the periodic advection–diffusion benchmark, 𝑢𝑡 = −𝑢𝑥 + 0.02𝑢𝑥 𝑥. White vertical markers in the left panel indicate sparse observation times; the heatmap shows the clean reference field before noise injection. The right panel shows three periodic snapshots. selects the correct median support size 𝑘 = 3, but only three of five seeds select the exact support size; those three seeds ch… view at source ↗
Figure 3
Figure 3. Figure 3: Pareto plots for non-oracle model selection on Van der Pol, Burgers, and Fisher–KPP with a steep periodic front-type initial condition. Each curve summarizes the median ΔEBIC over seeds after collapsing duplicate supports by support size. The selected model is chosen by elbow detection on the EBIC/support-size curve, not by the true support. when a learned linear observable model is rolled forward from the… view at source ↗
Figure 4
Figure 4. Figure 4: and [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: PDE sensitivity of POD-EDMD-RBF preprocessing to the interpolation factor 𝑞 and POD rank 𝑟. Each heatmap reports median coefficient error at relative noise 0.03 and PDE sparse factor 8; white stars mark the default values used in the main benchmark. The 𝑞 = 1 column corresponds to POD-only denoising without temporal insertion [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

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