{"id":"417e709c-a2c1-4780-b369-7ba239c7aac6","arxiv_id":"2411.10910","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A hybrid of extended and higher-order dynamic mode decomposition, trained with trajectories from every basin of attraction, can identify nonlinear systems with multiple attractors and approximate boundaries between them.","lead":"This paper proposes a variant of dynamic mode decomposition that feeds time-delayed, nonlinearly transformed state data into a linear operator, and tests it on systems with multiple coexisting attractors. The main practical message is that training data should be drawn from all basins of attraction if the learned model is to generalize across the whole phase space.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Manual choice of polynomial order o is load-bearing: Sec. 9.2 shows wrong o fails even with more data, and Remark 2 admits oracle knowledge of o; without automatic order selection, the claims of data-driven identification and cross-basin generalization are not established.","rationale":"The Reader's CONDITIONAL verdict is based on the same weakest assumption: the polynomial order o must be known in advance. I find this concern load-bearing rather than incidental. The paper's own Section 9.2 is the decisive evidence: wrong o fails, and more data or larger d does not fix it; Remark 2 admits the use of prior knowledge of the generating system. Since the central claim in the abstract and Conclusions is about a data-driven algorithm for unknown nonlinear systems, an oracle-supplied model class is a genuine scope restriction. I do not see an internal inconsistency in the algorithm or a reason to reject the paper: within the regime where o and d are correctly specified and training covers all basins, the numerical examples support the claimed behavior. The missing baselines and single-run RRMSEs are secondary weaknesses that reinforce CONDITIONAL but do not change it. Therefore the appropriate action is to keep the Reader's verdict unchanged, with the condition that the paper clearly frame the method as requiring a user-supplied feature order (or an automatic model-selection step) and soften the 'data-driven identification' claim accordingly.","tokens_in":18492,"tokens_out":14489,"duration_ms":159288,"concrete_test":"Run a model-selection experiment on the Sec. 9.2 system (or a suite of polynomial systems with unknown degree): create a held-out set of test initial conditions, then select (d,o) automatically from training snapshots using cross-validated one-step prediction RRMSE on a validation split, without using the known vector field. Report the selected pair and test RRMSE versus the oracle-o result. If selection recovers o=3 and matches the oracle test RRMSE, the concern is mitigated; if only o=3 works and no automatic rule finds it, then the paper's standalone data-driven identification claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that NLDM is a data-driven identifier that learns dynamics, including systems with multiple attractors, from trajectories. The weakest link is the required a priori choice of polynomial order o. Section 9.2 shows that for the damped nonlinear sink (25), (d,o)=(2,1) and (2,2) both fail, that enriching the training set does not help, and that increasing the delay order does not help; only o=3 works. Remark 2 explicitly concedes: 'for the order of the basis functions, we leveraged prior knowledge of the system dynamics used to generate the synthetic data.' Moreover, the benchmark values of o are hand-supplied and not derived by a stated rule: Sec. 9.2 needs o=3 for a cubic field, while Sec. 9.6 reports (d,o)=(5,2) for a degree-5 radial system. Since the abstract and Conclusions present NLDM as a general data-driven method and claim that a single learned operator can approximate basin boundaries by iteration, this is not a cosmetic tuning issue: for a genuinely unknown system the user does not know which o makes the method work, and Sec. 9.2 rules out data enrichment as a remedy for a wrong o.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18774,"tokens_out":5187,"duration_ms":47519,"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":[{"comment":"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.","section":"Section 9.2 and Remark 2"},{"comment":"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.","section":"Section 9"},{"comment":"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.","section":"Section 9.4 and Conclusions"},{"comment":"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.","section":"Section 9.3, Table 2"}],"minor_comments":[{"comment":"The index K is already used for the number of snapshots; using it to enumerate training trajectories causes confusion. Should be Q.","section":"Eqs. (22)-(23)"},{"comment":"The notation '(d, 0) = (2, 1)' should read '(d, o) = (2, 1)'.","section":"Figure 2 caption and Section 9.2"},{"comment":"The notation '˜xk−' is undefined; write '˜x_{k-1}' or a similar explicitly defined symbol.","section":"Section 4, Eq. (6)"},{"comment":"The reference to 'Eq. (9.6)' should point to the actual system introduced in Section 9.6.","section":"Figure 10 caption"},{"comment":"The MSC line reads '2000 MSC'; this should be '2020 MSC'.","section":"Title page"},{"comment":"There is a typo: 'The the system of ODEs' should be 'The system of ODEs'.","section":"Section 9.3"},{"comment":"The entry 'NaN' for RRMSE is not explained; state whether this indicates divergence to infinity or an undefined computation.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"The main novelty is not the operator construction, which is a straightforward combination of EDMD and HODMD, but the phase-space-informed training protocol. The missing automatic order selection and the absence of baselines are fixable within a revision. I would not reject on scientific grounds, but the abstract and Conclusions overstate generality relative to the evidence, and the authors should be asked to either add an automatic selection procedure for o or substantially soften the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should read this as a methods paper with a modest but real contribution: the authors combine EDMD-style polynomial observables with HODMD-style time delays into a single linear map, and then show that training on stacked trajectories from multiple basins produces a single operator that generalizes across basins. That training strategy, plus the basin-feedback loop (iterate the learned operator, visualize approximate basins, add trajectories near boundaries), is the genuinely new element. The compact matrix formulation in Section 4 is also clean and avoids the structural constraints of the usual HODMD companion form. The examples cover a useful hierarchy, from linear oscillators to a double well, a system with coexisting limit cycles, and the Lorenz system. The paper is well written, reproducible, and the linear algebra is sound.\n\nThe soft spot is the one you flagged: the polynomial order o is chosen using knowledge of the true equations. Remark 2 admits this explicitly, and Section 9.2 shows that a wrong o fails even when you add data or increase the delay order. That matters because the abstract and conclusions claim a general \"data-driven\" identification method. For a genuinely unknown system, the user does not know which o will work, and the paper provides no automatic selection rule. However, the paper does not hide this; the limitation is stated plainly, and the rest of the examples use o that matches the known nonlinearity. I would not call this a load-bearing flaw that sinks the paper, but it does cap the strength of the central claim.\n\nTwo other issues are worth naming, both minor. First, there are no baseline comparisons against EDMD, HODMD, or SINDy, so the reader cannot judge whether the multi-basin training strategy helps beyond what a standard EDMD with a good dictionary would achieve. Second, the reported RRMSEs are single-run values without uncertainty estimates; given the randomness of noise and initial conditions, some repeatability would strengthen the results. The basin-boundary approximation claim in Section 9.4 is plausible but only demonstrated qualitatively.\n\nOverall: this is a solid, honest paper that makes a modest but useful contribution to data-driven modeling of multistable systems. It deserves a serious referee, not a desk rejection. With baselines and some discussion of automatic order selection, it could be a good publication.\n\nRecommendation: send it to peer review, and ask the authors to add comparisons to standard EDMD and to address the polynomial-order selection problem directly.","headline":"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.","tokens_in":19305,"tokens_out":1541,"would_cite":true,"duration_ms":18637,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M05","37M10","93B28","93B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single learned linear operator on time-delayed polynomial features can reproduce nonlinear dynamics with multiple attractors, if training samples every basin.","keywords":["system identification","nonlinear delayed maps","higher order time delay","multiple attractors","basins of attraction","dynamic mode decomposition","phase space informed sampling","Koopman operator"],"falsifier":"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.","tokens_in":18301,"feed_emoji":"🌀","tokens_out":6225,"duration_ms":65224,"temperature":0.7,"pith_summary":"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.","feed_headline":"Trained across basins, one linear map learns nonlinear dynamics","feed_subtitle":"Sampling every basin of attraction lets one iterative map reproduce nonlinear, even chaotic, trajectories.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the stacked-snapshot DMD formulation and linear operator approximation that NLDM starts from.","marker":"[5]"},{"why":"Provides the Koopman-theoretic justification for representing nonlinear dynamics by a linear operator.","marker":"[9]"},{"why":"Contributes the dictionary of nonlinear observables used to lift data before learning the linear map.","marker":"[11]"},{"why":"Introduces higher order DMD and the multiple-delay representation that NLDM compacts.","marker":"[20]"},{"why":"Motivates the use of time-delayed coordinates for reconstructing dynamics from time series.","marker":"[23]"},{"why":"Provides the Lorenz system used as the chaotic benchmark for the method.","marker":"[26]"}],"fun_headline_variants":["Basin-wide training yields a single iterated linear map","One linear map reproduces multiple attractors from basin data","Time-delay features make linear maps learn nonlinear dynamics","Iterated linear map tracks chaotic trajectories across basins","Sampling all basins teaches one map global dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Basin-wide training yields a single iterated linear map","One linear map reproduces multiple attractors from basin data","Time-delay features make linear maps learn nonlinear dynamics","Iterated linear map tracks chaotic trajectories across basins","Sampling all basins teaches one map global dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00013,"raw_usage":{"total_tokens":1075,"prompt_tokens":847,"completion_tokens":228,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":152}},"tokens_in":463,"tokens_out":228,"duration_ms":4081,"temperature":1.0,"reasoning_tokens":152,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T19:10:55.313658+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Koopman-theoretic justification for representing nonlinear dynamics by a linear operator."},{"cited_title":"Takens, Detecting strange attractors in turbulence, in: Dynamical Systems and Turbulence, Warwick 1980, Springer, 1981, pp","cited_arxiv_id":null,"evidence_quote":"Motivates the use of time-delayed coordinates for reconstructing dynamics from time series."},{"cited_title":"Sparrow, The Lorenz equations: bifurcations, chaos, and strange attractors, Vol","cited_arxiv_id":null,"evidence_quote":"Provides the Lorenz system used as the chaotic benchmark for the method."}],"review_version":1}