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REVIEW 4 major objections 7 minor 26 references

Data-Driven Model Identification Using Time Delayed Nonlinear Maps for Systems with Multiple Attractors

T0 review · 4 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A single learned linear operator on time-delayed polynomial features can reproduce nonlinear dynamics with multiple attractors, if training samples every basin.

desk verdict A useful, clearly written hybrid of EDMD and HODMD whose real novelty is the phase-space-informed training strategy; the manual choice of polynomial order is the main soft spot, but the paper is honest about it. read the letter →

arxiv 2411.10910 v1 pith:W7IT7UWB submitted 2024-11-16 math.DS physics.data-an

Athanasios P. lliopoulos , Evelyn Lunasin , John G. Michopoulos , Steven N. Rodriguez , Stephen Wiggins This is my paper · ORCID
classification math.DSphysics.data-an MSC 37M0537M1093B2893B30
keywords systemidentificationnonlineardelayedmapshigherordertimedelaymultipleattractorsbasinsofattractiondynamicmodedecompositionphasespaceinformedsamplingKoopmanoperator
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 proposes the Nonlinear Delayed Map (NLDM) algorithm: instead of fitting a linear operator directly to raw states, it first lifts each state into a feature vector containing time-delayed states and polynomial nonlinear terms, then learns one linear map from those features to the next state. The central claim is that this one operator can accurately reproduce the dynamics of nonlinear systems with multiple attractors, including the chaotic Lorenz system, provided the training data include trajectories from every basin of attraction. The authors show that training on a single basin produces an operator that fails across the basin boundary, while stacking trajectories from both basins yields a single operator that predicts well on both sides and can approximate the boundary by iteration. They demonstrate this on a hierarchy of systems: damped oscillators, a two-attractor planar system, a double-well oscillator, a periodic attractor in three dimensions, a mixed equilibrium/limit-cycle system, and the Lorenz system. A sympathetic reader would care because the method offers a practical route to data-driven models that respect phase-space geometry without requiring derivatives or explicit equations.

What carries the argument

The central object is the compact linear map $\Lambda \in \mathbb{R}^{S \times L}$ defined by $x_k = \Lambda \upsilon_k$, where $\upsilon_k$ is a feature vector containing the $d$ most recent states along with polynomial combinations up to order $o$, such as $x_k, y_k, x_{k-1}, y_{k-1}, x_k^2, x_k x_{k-1}, \dots$. This combines the nonlinear-observable lifting of EDMD with the multiple-delay structure of HODMD, but in a compact form $A = [A_1, \dots, A_d]$ that avoids the structural zero constraints of the standard companion matrix. The feature matrix $\Upsilon$ can be stacked across trajectories from different basins, so a single least-squares solve yields an operator that encodes all basins at once, and iterating that operator from new initial conditions is the prediction step. This formulation is what lets users add trajectories incrementally and visualize approximate basin boundaries from the learned map.

What would settle it

Take the damped cubic system of Section 9.2, withhold the true polynomial order from the algorithm, and ask it to choose $(d, o)$ from data alone; if no choice achieves an RRMSE below a few times $10^{-2}$ on a held-out trajectory in the same basin while the known-order choice reaches roughly $10^{-6}$, then the method as described is not a data-driven identifier for unknown systems.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that a finite-dimensional linear map $\Lambda$ acting on a feature space built from $d$ delayed states and polynomial observables up to order $o$ can serve as a global iterator for nonlinear dynamics with multiple attractors, as long as the training data cover the relevant basins. The learned relation is $X = \Lambda \Upsilon$, where columns of $\Upsilon$ stack delayed states and nonlinear terms, and $\Lambda$ is found by least squares or the Moore-Penrose pseudoinverse and then iterated to predict trajectories from arbitrary initial conditions. The paper's results show that single-basin training gives an operator that only works inside that basin, whereas training with data from both sides of a basin boundary yields a single operator that generalizes across it, reproduces stable and unstable limit cycles, and tracks the Lorenz attractor for short horizons. The authors also use the iterated operator on scattered initial conditions to approximate basin boundaries, closing a feedback loop that guides further data collection.

Load-bearing premise

The user must know the correct polynomial order $o$ and a suitable feature dictionary in advance, because the paper's examples use prior knowledge of the true dynamics and the method is not shown to select this order automatically.

Editorial extensions

If this is right

  • A single linear operator trained on stacked trajectories from both basins can predict test trajectories on either side of a basin boundary, where single-basin training fails.
  • Iterating the learned operator from a fine grid of initial conditions approximates the basins and their boundaries without integrating the true system.
  • Because the method avoids derivative estimation, it remains usable when training data carry modest Gaussian noise (about 0.1% of the signal range) without additional smoothing.
  • The minimum number of training trajectories scales with the number of basins and relevant phase-space features, not with the state dimension alone.
  • For a chaotic system with one global attractor, one operator trained on a few trajectories can track both lobes of the Lorenz system and extend its prediction horizon when the delay order and sampling rate are increased.

Reading between the lines

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

  • The paper's Remark 2 means the method as presented is not yet a standalone identifier for genuinely unknown systems: automatic selection of the polynomial order $o$, or a data-driven dictionary-selection criterion, is a natural next step and the examples show performance degrades sharply when $o$ is wrong.
  • The success of stacking trajectories from disjoint basins suggests that active learning could replace manual phase-space-informed sampling: train a first operator, use its approximate basin boundaries to pick new initial conditions, and retrain, following the feedback loop the paper describes qualitatively.
  • The compact delayed-feature formulation could be tested as a cheap approximate-basin-boundary tool for control, where knowing which attractor a proposed input will reach matters more than an exact trajectory match.
  • If the single-operator claim extends to chaotic systems, it hints that finite-dimensional linear representations of multistable dynamics may be more practically reachable than Koopman-embedding theory currently suggests.
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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 / 7 minor

Summary. The paper introduces NLDM, a system identification method that combines time-delayed state augmentation (as in HODMD) with nonlinear polynomial feature lifting (as in EDMD). The unknown map Lambda is obtained by least squares from stacked trajectory data, and predictions are made by iterating Lambda on feature vectors built from the last d states. The authors propose using phase-space information, specifically attractors and their basins, to choose training trajectories; they demonstrate the method on damped oscillators, a bistable system, a double-well oscillator, a periodic-attractor system, a mixed equilibrium/limit-cycle system, and the Lorenz equations, reporting RRMSE values for training and test trajectories. The central claim is that a single linear operator, trained on trajectories from all basins, can represent nonlinear dynamics with multiple attractors and can be iterated to approximate basin boundaries.

Significance. If the claims were fully established, the method would be a practically useful extension of EDMD and HODMD, particularly for multistable systems, with a clean least-squares formulation and a sensible train/test protocol that separates initial conditions. The paper deserves credit for explicitly measuring generalization on unseen initial conditions, for testing robustness to Gaussian noise, and for making the phase-space-informed training strategy concrete through a feedback loop. However, the significance is currently limited by three gaps: no automatic selection of the polynomial order (Remark 2), no comparison with baseline methods, and only qualitative support for the basin-boundary approximation claim. The first gap is the most serious because Section 9.2 shows that a wrong order cannot be fixed by adding data.

major comments (4)
  1. [Section 9.2 and Remark 2] The choice of polynomial order o is load-bearing and is made using information that a data-driven identifier would not have. For Eq. (25), the paper reports that (d,o)=(2,1) and (2,2) fail, that enriching the training data does not help, and that increasing the delay order d does not help; only o=3, which matches the cubic term in the true equations, succeeds. Remark 2 states: 'for the order of the basis functions, we leveraged prior knowledge of the system dynamics used to generate the synthetic data.' No automatic model-selection criterion for o is given anywhere in the paper. Since the failure mode is not cured by more data, the general claim in the Conclusions that the algorithm 'can accurately learn the dynamics of nonlinear systems' is not established for unknown systems unless an automatic order-selection procedure is added.
  2. [Section 9] The paper does not include any baseline comparison. The introduction and Section 2.6 motivate NLDM as a hybrid of EDMD and HODMD and claim improved tractability and accuracy, but none of the numerical experiments in Section 9 compares NLDM against EDMD, HODMD, or SINDy on the same training and test sets. The reported RRMSE values are informative as absolute measures, but they cannot substantiate the claimed advantages over existing methods. Adding at least one baseline (e.g., EDMD and HODMD with the same feature and delay choices) is necessary to support the comparative statements.
  3. [Section 9.4 and Conclusions] The claim that NLDM can approximate basin boundaries is only supported by qualitative visual comparison. In Figures 6 and 7, the basins obtained by iterating the learned operator are compared visually with those produced by ODE45, and the text acknowledges differences in 'thickness and continuity'. No quantitative error measure (e.g., distance between predicted and true basin boundaries or classification accuracy of a grid of initial conditions) is reported. Moreover, the feedback loop that improves the basin-boundary approximation requires the user to inspect the learned operator and choose additional training trajectories, so the procedure does not currently constitute an automatic method. A quantitative evaluation is needed before the Conclusions' statement about approximating basin boundaries can be accepted.
  4. [Section 9.3, Table 2] The test RRMSE values for initial conditions near the basin boundary are orders of magnitude larger than the training RRMSE (e.g., 4.81e-2 for TS-ic [0.025,1] versus 8.53e-3 for training), and the text reports divergence for boundary initial conditions. The claim of cross-basin generalization needs a precise statement of the region of validity around the basin boundary, rather than a binary 'near vs far' discussion.
minor comments (7)
  1. [Eqs. (22)-(23)] The index K is already used for the number of snapshots; using it to enumerate training trajectories causes confusion. Should be Q.
  2. [Figure 2 caption and Section 9.2] The notation '(d, 0) = (2, 1)' should read '(d, o) = (2, 1)'.
  3. [Section 4, Eq. (6)] The notation '˜xk−' is undefined; write '˜x_{k-1}' or a similar explicitly defined symbol.
  4. [Figure 10 caption] The reference to 'Eq. (9.6)' should point to the actual system introduced in Section 9.6.
  5. [Title page] The MSC line reads '2000 MSC'; this should be '2020 MSC'.
  6. [Section 9.3] There is a typo: 'The the system of ODEs' should be 'The system of ODEs'.
  7. [Table 1] The entry 'NaN' for RRMSE is not explained; state whether this indicates divergence to infinity or an undefined computation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: test predictions are independent of the fitted operator, and the acknowledged oracle choice of polynomial order is a model-selection limitation, not a circular step.

full rationale

The paper's derivation chain is self-contained: it defines a feature map (Eq. 15), builds time-delayed nonlinear feature matrices (Eqs. 18-20), solves for the linear operator by least squares / pseudoinverse (Eq. 13 applied to Eq. 20), and then iterates that operator from test initial conditions (Section 6.1). The reported RRMSEs are computed on test trajectories whose initial conditions were not used to fit Λ (e.g., Section 9.1, TS-ic = (0,2); Section 10.1, test IC [5,1,6]), so the predictions are not fitted values by construction. The main caveat is stated explicitly in Remark 2: 'for the order of the basis functions, we leveraged prior knowledge of the system dynamics used to generate the synthetic data.' This is a genuine limitation of the data-driven identification claim, because a wrong polynomial order causes failure (Section 9.2), but it is an oracle model-selection assumption rather than a circular derivation: the paper does not define the predicted dynamics in terms of the selected order, and the test evaluations are independent of the fitting step. There is no load-bearing self-citation, no imported uniqueness theorem, and no equation in the prediction step reduces to the training target by definition. The iterative basin-refinement procedure in Section 9.4 uses the learned operator to guide additional training data, but the final comparison is against ODE45 reference solutions, so the claimed improvement is empirically checked rather than assumed. Overall, no circular step meets the standard of Eq. X = Eq. Y by construction or a fitted parameter renamed as a prediction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The ledger shows the method's central claim depends on several manually chosen hyperparameters (o, d, K, training IC set) and on assumptions that the feature dictionary and basin sampling are sufficient. No new physical entities are introduced.

free parameters (5)
  • Polynomial feature order o = o=1 (Sec 9.1), o=3 (Sec 9.2-9.4), o=2 (Sec 9.5-9.6 and Sec 10)
    Chosen per experiment using prior knowledge of the system's equations; wrong order fails (Fig. 2).
  • Time delay order d = d=2 (Sec 9.1-9.3), d=4 (Sec 9.4), d=5 (Sec 9.6), d=3 or 4 (Sec 10)
    Tuned to obtain low RRMSE; larger d is said to help with noise.
  • Number of snapshots K = K=1000 (most examples), K=2000 (Table 1), K=4000 or 12000 (Lorenz)
    Sampling density chosen to keep training RRMSE small; no sensitivity analysis.
  • Training trajectory set (initial conditions and basin coverage) = Varies per example: 4 ICs for oscillator, 5 ICs for Lorenz, inside/outside MFCD cylinder, both basins for…
    Selected with knowledge of the phase space geometry; central to the claimed success of the method.
  • Gaussian noise level sigma = 0.1% of signal range in most examples
    Set by hand; robustness claims are tied to this low noise level.
assumptions (4)
  • domain assumption The learned finite-dimensional linear map on the feature space approximates the true nonlinear dynamics well enough to be iterated over long prediction horizons.
    Invoked in Eq. (20) and Section 6.1; no error bounds or convergence guarantee is provided.
  • ad hoc to paper Polynomial features of order o can represent the vector field or its Koopman closure.
    Remark 2 admits the order is chosen from known system equations; Section 9.2 shows wrong orders fail.
  • domain assumption A finite set of trajectories, one per basin, provides sufficient sampling for a least-squares operator that generalizes inside those basins.
    Used throughout Sections 8-9; no formal sampling condition is stated.
  • domain assumption Iterating the learned operator on a grid of initial conditions yields a valid approximation of the true basin boundaries.
    Used in the Section 9.4 feedback loop to select additional training data; accuracy is only visualized, not quantified.

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Cite this review

Pith. "Pith review of Data-Driven Model Identification Using Time Delayed Nonlinear Maps for Systems with Multiple Attractors." pith.science (2026). https://pith.science/paper/W7IT7UWB

@misc{pith2026241110910,
  author       = {Pith},
  title        = {Pith review of: Data-Driven Model Identification Using Time Delayed Nonlinear Maps for Systems with Multiple Attractors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7IT7UWB}},
  note         = {Machine review of arXiv:2411.10910}
}
read the original abstract

This study presents a method, along with its algorithmic and computational framework implementation, and performance verification for dynamical system identification. The approach incorporates insights from phase space structures, such as attractors and their basins. By understanding these structures, we have improved training and testing strategies for operator learning and system identification. Our method uses time delay and non-linear maps rather than embeddings, enabling the assessment of algorithmic accuracy and expressibility, particularly in systems exhibiting multiple attractors. This method, along with its associated algorithm and computational framework, offers broad applicability across various scientific and engineering domains, providing a useful tool for data-driven characterization of systems with complex nonlinear system dynamics.

Figures

Figures reproduced from arXiv: 2411.10910 by the authors.

Figure 1
Figure 1. Comparison of true trajectories (solid lines) with [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Importance of specifying the correct order of basis functions in the [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Illustration of NLDM performance for the system expressed by Eqs.(26). Panel (a) shows the performance of the NLDM algorithm when the training phase involves data solely from the left basin. The absence of trajectories in the test data, with initial condition indicated by the red dots, means that the predictions diverge. Panel (b) demonstrates the performance of the NLDM algorithm when the training phase includes da… view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Visualization of the basins of attraction and basin boundaries for the system [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: (a) Performance of the NLDM algorithm on the system expressed by Eq. (27) when training trajectory contains zero noise. Multiple trajectories are shown, with initial conditions indicated by red dots, uniformly distributed around the top half of a unit circle. The true …
Figure 6
Figure 6. Figure 6: Performance of NLDM for the system of Eqs. (27) with training data contam￾inated by small noise (σ = 0.1). Panel (a) shows initial conditions in the red and green regions converging to their respective attractors with trajectories obtained by iterating the learned oper…
Figure 7
Figure 7. Figure 7: Performance of NLDM algorithm for the system of ODEs shown in Eqs. (27) with training data contaminated by a small amount of noise (σ = 0.1%). Panel (a) illustrates the effect of strategically adding training data from the other basin of attraction. Panel (b) demonstra…
Figure 8
Figure 8. Figure 8: Failing case for the NLDM algorithm applied to Eq. (28) when the projection in the xy- plane of the initial condition for the training data is inside the projection of the limit cycle onto the xy-plane, while that for the test trajectory is outside. The transient part …
Figure 9
Figure 9. Figure 9: Performance of the NLDM algorithm applied to Eq. (28) for multiple test cases when training data involves two trajectories, inside and outside the projection of the limit cycle in two dimensional plane. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]
Figure 10
Figure 10. Figure 10: Phase plane representation for Eq. (9.6) and performance of the [PITH_FULL_IMAGE:figures/full_fig_p032_10.png]
Figure 11
Figure 11. Figure 11: Performance of the NLDM algorithm on the Lorenz system. The five figures in (a) shows training data corresponding to initial conditions with varying trajectory behaviors, such as one trajectory staying only on the left lobe, one only on the right lobe, and one visitin…
Figure 12
Figure 12. Figure 12: Extending the prediction horizon for the Lorenz system. The five figures in (a) [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]

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