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REVIEW 4 major objections 6 minor 58 references

Data-driven nonlinear modal identification of nonlinear dynamical systems with physics-constrained Normalizing Flows

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Normalizing Flows can identify nonlinear normal modes from raw response data.

desk verdict Promising idea, but the losses don't enforce the NNM invariance property, and the validation is too weak to support the central claim; still worth a serious referee. read the letter →

arxiv 2501.14124 v1 pith:6VPBLORJ submitted 2025-01-23 nlin.CD

Abdolvahhab Rostamijavanani , Shanwu Li , Yongchao Yang This is my paper · ORCID
classification nlin.CD
keywords nonlinearnormalmodesNormalizingFlowsdata-drivenmodalanalysissystemidentificationProperOrthogonalDecompositioninvariantmanifoldsDuffingoscillatorflowpastcylinder
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 Normalizing Flows network, trained with physics-based losses and on measured response trajectories only, can identify the nonlinear normal modes (NNMs) of an unknown nonlinear dynamical system together with the exact invertible transformation between physical and modal coordinates. The authors argue this matters because closed-form equations are usually unavailable, and linear tools such as POD fail for strongly nonlinear dynamics. On a two-degree-of-freedom Duffing oscillator the method reconstructs the in-phase and out-of-phase invariant manifolds at several energy levels and makes recursive long-horizon predictions. On laminar flow past a cylinder, two flow-derived coordinates reconstruct the velocity field far more accurately than two POD modes. The identified coordinates are presented as a nonlinear generalization of POD modes that carry more dynamical information per coordinate.

What carries the argument

The central object is a masked autoregressive flow: a stack of invertible bijectors that map a simple independent Gaussian distribution onto the data distribution, with the exact inverse serving as both encoder and decoder. Five physics losses carry the argument: negative log-likelihood for reconstruction, a correlation penalty driving latent coordinates toward independence, a dynamics-block residual enforcing that advancing in latent space matches the observed next state, a prediction loss in original coordinates, and a velocity loss pairing each modal displacement with its time derivative. In the flow-field case, POD supplies a low-dimensional input so the one-to-one flow can be applied, and the flow then removes the remaining nonlinear dependence between POD coordinates.

What would settle it

For the 2-DOF Duffing oscillator, compute the invariant manifolds numerically from the known equations, for example by solving for the Shaw–Pierre manifold, and compare them with the NF-identified modal coordinates; if the inferred manifolds differ by more than the reconstruction error, the identification claim fails.

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Extended reading notes

Core claim

The paper claims that a masked autoregressive Normalizing Flow, trained on measured state trajectories with losses enforcing reconstruction, independence of latent coordinates, displacement–velocity consistency, and future-state evolution, identifies the nonlinear normal modes of an unknown nonlinear system together with an exact invertible transformation between physical and modal coordinates. In the flow-field case, applying the flow to the first ten POD coordinates yields two nonlinear modal coordinates whose two-coordinate reconstruction is orders of magnitude more accurate than POD with two coordinates. For the Duffing oscillator, transferring each latent pair back to physical space individually recovers the in-phase and out-of-phase invariant manifolds, whose curvature grows with energy as expected for NNMs.

Load-bearing premise

The load-bearing premise is that the latent coordinates, forced to be independent and to obey the displacement–velocity relation, actually coincide with the nonlinear normal modes of the underlying system, and, in the flow-field case, that the ten retained POD modes preserve enough information for two latent coordinates to reconstruct the full field.

Editorial extensions

If this is right

  • Nonlinear normal modes can be identified from measured responses alone, without a closed-form model, for weakly nonlinear multi-degree-of-freedom systems.
  • A single invertible network serves as both encoder and decoder, so the nonlinear modal transformation is exact rather than approximated, enabling accurate single-mode reconstructions.
  • With the same number of retained coordinates, the learned flow coordinates reconstruct cylinder wake fields with orders-of-magnitude lower error than POD.
  • Embedding the dynamics in latent modal coordinates supports recursive long-horizon prediction of the original response.

Reading between the lines

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

  • If the latent coordinates are true NNMs, the same framework could build reduced-order models and controllers directly from sensor data, bypassing equation discovery.
  • A direct testable extension is to compare the identified manifolds against spectral submanifold theory for a benchmark oscillator, which would settle whether the independence and velocity losses pin down the physical NNMs uniquely.
  • The one-to-one mapping limitation suggests that for very high-dimensional systems the POD preprocessing step may control accuracy; the paper's results do not yet establish how many retained POD modes are needed.
  • The stated assumption of no internal resonance means the method may conflate modes in resonant regimes; a natural extension is to add frequency-separation or interaction terms to the loss.
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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 / 6 minor

Summary. The paper presents a data-driven framework based on normalizing flows (NF) for identifying nonlinear normal modes (NNMs) and the nonlinear modal transformation directly from measured response data. The method trains a masked autoregressive flow to map original state coordinates to latent coordinates, with additional loss terms enforcing statistical independence, the displacement-velocity kinematic relation, and consistency with a learned dynamics block. The framework is demonstrated on a 2-DOF Duffing oscillator and on flow past a cylinder (with POD pre-processing). The authors claim that the identified latent coordinates are NNMs and that they provide more accurate representation than POD with the same number of coordinates.

Significance. The idea of leveraging the invertibility and independence properties of normalizing flows for nonlinear modal identification is interesting and potentially useful for black-box system identification. The use of a dynamics block for recursive prediction is a positive feature, as is the attempt to quantify nonlinear dependence via mutual information in the flow case. However, the central claim that the latent coordinates are NNMs is not rigorously supported: the loss functions do not enforce the invariant-manifold property that defines NNMs, and the validation is mostly qualitative. The quantitative comparison with POD is also not like-for-like, and the hyperparameters were tuned on test error. If the authors can provide independent verification against analytical NNMs and fair baselines, the method could be a valuable contribution to nonlinear modal analysis.

major comments (4)
  1. [Section 3.1, Eq. (7) and Eq. (11), Section 4.2.3] The loss functions used in the framework do not enforce the defining property of NNMs, namely the invariance of the two-dimensional (p_i, q_i) modal subspaces under the flow. The correlation loss Lcorr only enforces statistical independence of the instantaneous latent variables, and the dynamics block G is a dense MLP with no block-diagonal constraint, so it is free to mix all latent coordinates. Consequently, the identified latent coordinates can be any invertible nonlinear coordinate system that factorizes the data distribution, not necessarily the Shaw-Pierre NNMs. The manuscript provides only qualitative validation (Fig. 8 and the remark that Fig. 10 is 'in agreement with analytical results [51]') with no quantitative comparison against analytically computed NNMs or invariant manifolds. To support the central claim that the method identifies NNMs, the authors should compare the identified latent coordinates and their dynamics with reference NNMs computed from the known equations (e.g., via continuation or shooting) and report a quantitative error measure.
  2. [Section 5.2, Table 3] The comparison between NF and POD reconstruction is not a like-for-like comparison. The NF is trained on 10 POD modal coordinates (Section 4.2.2) and then evaluated using 2 latent coordinates with the remaining 8 set to zero, whereas the POD reconstruction uses only 2 POD modes. The NF latent coordinates are nonlinear functions of all 10 POD coordinates, so the comparison conflates the effect of the nonlinear transformation with the additional information provided by the 10 modes. To fairly assess representation power with the same number of coordinates, the NF should be trained on the first 2 POD coordinates only, or the POD reconstruction should be given the same information (e.g., 10 modes then truncated to 2 via a linear projection). Without such a controlled comparison, the claim that NF 'achieves much higher reconstruction accuracy than POD' (Section 5.2) is not established.
  3. [Section 4.2.3 and Table 1] The hyperparameters, including the loss weights in Table 1, were selected to minimize testing error, as stated in Section 4.2.3. This makes the reported performance (e.g., Table 3 and Fig. 11) an in-sample evaluation rather than a clean out-of-sample test, and the quantitative claims are subject to selection bias. The authors should describe a proper validation procedure (e.g., a separate validation set or cross-validation) and report performance on a held-out test set that was not used for any hyperparameter selection. In particular, the prediction accuracy shown in Fig. 11(b) should be reported with quantitative error metrics over the prediction horizon, not only by visual inspection.
  4. [Section 3.2 and Section 5.2] The abstract and introduction claim that the approach identifies NNMs 'using measured response data only' for a wide range of physical systems, but the flow-field case study only performs nonlinear mode decomposition; no dynamics block, velocity constraint, or NNM identification is attempted for the flow (Section 3.2 explicitly states that the focus is on decomposition rather than prediction and dynamics). This overstates the generality of the central claim. The authors should either restrict the claim to the Duffing-type systems for which dynamics are identified, or extend the flow experiment to include dynamics and verify that the latent coordinates of the flow satisfy the defining properties of NNMs.
minor comments (6)
  1. [Section 4.2.1] In the sentence describing the dynamics block, 'predicate' should be 'predict'; also, the notation for the recursive loss in Eq. (13) is unclear about how the composition of G is implemented in practice and what value of m is used.
  2. [Section 5.1] There are typos: 'presentd' should be 'presented' and 'invertable' should be 'invertible'. In addition, the text in Section 5.1 refers to Fig. 7 before it is introduced; please check the figure ordering.
  3. [Section 4.2.2 and Eq. (11)] The notation in Eq. (11) is ambiguous: the sum over sample indices uses both 's' and 'ns' inconsistently, and the definition of the correlation matrix Corr (e.g., Pearson correlation over time) is not explicitly given. Please clarify how Corr is computed for the displacement and velocity matrices.
  4. [Section 5.2] The mutual information values are mentioned qualitatively ('the mutual information value indicates that they are dependent') but no numerical values are reported. Consider reporting the MI before and after the NF layers to quantitatively demonstrate the reduction in nonlinear dependence.
  5. [Section 2.1] The term 'Jacobean' should be 'Jacobian'; also, the paper alternates between 'NF' and 'Normalizing Flows' without a consistent abbreviation style, which is a minor readability issue.
  6. [Section 6] The limitations paragraph acknowledges training instability, computational cost, limited energy range, and the absence of internal resonance, but it does not acknowledge the lack of independent verification that the latent coordinates are NNMs or the unfairness of the POD comparison. These limitations should be stated explicitly.

Circularity Check

2 steps flagged · score 6.0 of 10

NNM identification is partly self-definitional: the latent coordinates are trained to satisfy the imposed independence and velocity-pairing losses and then declared to be NNMs on the basis of those same imposed properties; test-error-based hyperparameter selection additionally makes the reported prediction a circular validation.

  1. self definitional [Section 3.1 (Eq. 7); Section 4.2.1, Eq. (9), Eq. (11), Eq. (15)]
    "The intrinsic modal coordinates are denoted as Z0 which are identified as the NNMs when integrating their physics constraints with the presented deep learning-based data-driven system identification framework. ... modal-uncorrelated loss functions are presented as follows ( Lcorr): ... ||Corr (q) , In×n|| ... To enforce NNMs-NF-DNN to learn under this constraint, a corresponding loss function is integrated: Lvel = ... || ∆pi/∆t , qi|| ..."

    Z0 is trained with Lcorr (decorrelating p and q) and Lvel (forcing q = Δp/Δt), so these properties hold by construction. The paper then presents exactly those imposed properties as the identification of NNMs. No term in Eq. (9) enforces the Shaw-Pierre invariant-manifold/decoupling condition that defines an NNM, so the identification of Z0 as NNMs is a definitional relabeling of the constrained latent coordinates, not a result derived from the data. The 'physics constraints' used to validate the NNM label are the same constraints imposed during training.

  2. fitted input called prediction [Section 4.2.3 and Section 5.1 (Fig. 11)]
    "We analyze the performance of DNN across a variety of training sessions (hyperparameters-tuning). It has been examined different sets of hyperparameters (weights of loss functions) and the results are based on the hyper-parameters associated with minimum testing errors (Table. 1)."

    The loss-function weights are selected to minimize errors on the testing data, and the same testing data then provide the reported reconstruction and 499-step prediction errors (Fig. 11, Table 3). The 'prediction' is therefore evaluated on the data used for model selection; the hyperparameters are effectively fitted to the test set, so the reported predictive accuracy is not an independent, out-of-sample validation.

full rationale

The method is not globally circular: the Normalizing Flow is trained to reconstruct the response, the dynamics block G is trained to advance latent coordinates, and the 499-step recursive prediction is a real algorithmic task whose success is not guaranteed merely by the losses. The POD comparison in the flow case is also an external benchmark. However, the central claim that the learned latent coordinates are NNMs is self-definitional. The paper asserts that NNM invariance is characterized by NF independence and imposes independence and velocity-pairing through Lcorr and Lvel; it then presents those imposed properties as the identification of NNMs. The defining invariant-manifold property of NNMs is never enforced or independently verified; the only external check is a qualitative statement that Fig. 10 'is in agreement with analytical results [51]' with no quantitative error measure. In addition, hyperparameters are chosen by testing error, so the reported test-set prediction is contaminated. These issues make the NNM-identification claim partially circular, while the method itself retains independent predictive content; hence a score of 6 rather than 8-10.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The method relies on several adjustable hyperparameters and unverified assumptions about the correspondence between NF latent coordinates and true NNMs. No new physical entities are introduced.

free parameters (4)
  • Loss weights (alpha_N, alpha_evol, alpha_prd, alpha_corr, alpha_vel) = 1, 1000, 1000, 1, 1
    Tuned by minimizing testing error (Table 1, Section 4.2.3).
  • Learning rate = 1e-5
    Set to stabilize training; no sensitivity study provided.
  • NF architecture (layers, dense layers, neurons) = Duffing: 6x3x256; flow: 8x3x512
    Chosen by hyperparameter tuning (Table 2).
  • Number of POD modes retained for flow = 10
    Chosen by hand to reduce dimension, not justified by an energy criterion.
assumptions (3)
  • ad hoc to paper The latent coordinates of a Normalizing Flow, when constrained by the correlation and velocity losses, correspond to the nonlinear normal modes of the underlying system.
    This is the central identification claim; it is not proven or compared to analytic NNMs.
  • domain assumption The dynamics block G can be approximated by a 4-layer MLP with ReLU activation.
    Assumes the modal dynamics are a finite-dimensional map; typical for discrete-time state evolution.
  • domain assumption For the flow case, 10 POD modes contain enough information for the NF to learn the nonlinear dependencies.
    The truncation is not validated with an energy criterion or convergence check.

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Cite this review

Pith. "Pith review of Data-driven nonlinear modal identification of nonlinear dynamical systems with physics-constrained Normalizing Flows." pith.science (2026). https://pith.science/paper/6VPBLORJ

@misc{pith2026250114124,
  author       = {Pith},
  title        = {Pith review of: Data-driven nonlinear modal identification of nonlinear dynamical systems with physics-constrained Normalizing Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6VPBLORJ}},
  note         = {Machine review of arXiv:2501.14124}
}
read the original abstract

Identifying the intrinsic coordinates or modes of the dynamical systems is essential to understand, analyze, and characterize the underlying dynamical behaviors of complex systems. For nonlinear dynamical systems, this presents a critical challenge as the linear modal transformation, which is universal for linear systems, does not apply to nonlinear dynamical systems. As natural extensions to linear normal modes,the nonlinear normal modes (NNMs) framework provides a comprehensive representation of nonlinear dynamics. Theoretically, NNMs may either be computed numerically or analytically from the closed-form models or equations of dynamical systems, or experimentally identified from controllable input-output tests, both of which, however, are typically unknown or unavailable practically. In this study, we present a physics-integrated Normalizing Flows deep learning-based data-driven approach which identifies the NNMs and the nonlinear modal transformation function of NNMs using measured response data only. Specifically, we leverage the unique features of the Normalizing Flows model: 1) the independent latent spaces, naturally spanned by the Normalizing Flows, are exploited to facilitate nonlinear modal decomposition; 2) the invertible transformation through the Normalizing Flows, enabling efficient and accurate nonlinear transformation between original and modal coordinates transformation. Therefore, our framework leverages the independency feature and invertibility of Normalizing Flows to create a model that captures the dynamics of unknown nonlinear dynamical systems.

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.