{"id":"ac08f58e-8957-493d-a065-fac2a41b6f24","arxiv_id":"2501.14124","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A normalizing-flows network identifies nonlinear normal modes and the invertible modal mapping directly from response data, with demonstrations on Duffing oscillators and laminar flow past a cylinder.","lead":"This paper trains a normalizing-flows neural network to identify the hidden modes of nonlinear vibrating systems from measurement data alone. It demonstrates the approach on simulated oscillators and on flow past a cylinder, reporting more accurate reconstruction than the standard POD method.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The loss terms do not enforce the invariance property that defines NNMs, and no independent comparison to analytical NNMs is provided, so the central claim that latent coordinates are NNMs is unsupported.","rationale":"I reviewed the reader's verdict and the manuscript. The reader's weakest assumption—that independent, velocity-consistent latent coordinates do not necessarily coincide with NNMs—is indeed the crux. I sharpen it: the loss formulation in Section 4.2.1 (Eqs. 9–15) contains no term that enforces the invariance/decoupling property that defines NNMs; the dynamics block G is a dense MLP, so it can learn arbitrary coupling in latent coordinates even if the instantaneous marginals factorize. The paper's only validation of modal identity is qualitative. A quantitative test comparing the learned transformation to analytically computed NNMs for the same Duffing parameters, and checking whether single-mode trajectories remain confined to the learned invariant manifold, would settle the concern. This is more load-bearing than the POD comparison issue in Section 5.2, since if the coordinates are not NNMs the central claim fails regardless of reconstruction accuracy. I therefore agree with the reader's conditional verdict; additional independent verification is required. The concrete test described above would resolve the concern. Secondary issues noted by the reader (test-set hyperparameter selection, POD comparison asymmetry) reinforce the need for revision but are not the central conceptual gap.","tokens_in":15935,"tokens_out":7201,"duration_ms":66482,"concrete_test":"For the 2-DOF Duffing system in Eq. (17), compute the analytical NNMs (e.g., via the Shaw-Pierre invariant manifold method or a normal-form/spectral-submanifold computation) and compare the learned latent coordinates and the learned dynamics block G. Concretely: (1) verify the invariance condition by initializing the system on the learned single-mode manifold (set only one (p_i, q_i) pair nonzero and the other latent coordinates to their mean or zero) and checking whether the subsequent predicted trajectory remains on that manifold; if not, the coordinates are not NNMs. (2) Compute the Jacobian of G at representative states and check whether cross-pair blocks (∂G_j/∂(p_i, q_i) for j ≠ i) are negligible. If they are not, the modal dynamics are coupled and the identified coordinates are not NNMs.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the learned latent coordinates Z0 = N^{-1}(X) of the masked autoregressive flow are the nonlinear normal modes. The training losses in Eq. (9) enforce reconstruction (LN), zero cross-correlation among modal coordinates (Lcorr), a velocity constraint (Lvel), and consistency with the dynamics block G (Levol, Lprd). None of these imposes the defining property of an NNM: that the two-dimensional (p_i, q_i) modal subspace is invariant under the flow, i.e., that the modal dynamics decouple across i. The dynamics block G is a dense MLP (Section 4.2.3) and is free to mix all latent coordinates. Lcorr only forces statistical independence of the instantaneous latent variables; it does not force the latent-space dynamics to be block-diagonal. Therefore the identified coordinates can be an arbitrary invertible nonlinear coordinate system in which the data distribution factorizes, not necessarily the Rosenberg/Shaw-Pierre NNMs. The only validation offered is qualitative wavelet plots (Fig. 8) and a remark that Fig. 10 'is in agreement with analytical results [51]' with no quantitative error measure or comparison to analytically computed invariant manifolds/backbone curves. Thus the central claim that the method 'identifies NNMs' is not established by the reported experiments.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16274,"tokens_out":5494,"duration_ms":47881,"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":[{"comment":"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.","section":"Section 3.1, Eq. (7) and Eq. (11), Section 4.2.3"},{"comment":"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.","section":"Section 5.2, Table 3"},{"comment":"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.","section":"Section 4.2.3 and Table 1"},{"comment":"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.","section":"Section 3.2 and Section 5.2"}],"minor_comments":[{"comment":"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.","section":"Section 4.2.1"},{"comment":"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.","section":"Section 5.1"},{"comment":"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.","section":"Section 4.2.2 and Eq. (11)"},{"comment":"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.","section":"Section 5.2"},{"comment":"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.","section":"Section 2.1"},{"comment":"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.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of a nonlinear dynamics journal, and the use of normalizing flows for nonlinear modal decomposition is a fresh idea. However, the central identification claim is not yet supported by the evidence: the loss functions do not enforce the invariant-manifold property, and the POD comparison in the flow experiment is misleading because the NF is trained on 10 modes while POD is evaluated with 2. I recommend major revision, with a request for quantitative comparison against analytical NNMs on the Duffing case, a fair POD baseline, and a proper train/validation/test protocol."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take on Rostamijavanani et al.\n\nThe genuinely new piece is the use of normalizing flows with physics-constrained losses for data-driven NNM identification. The exact invertibility of NF is a real advantage over autoencoder decoders, which are only approximate inverses. The idea is sensible, and the loss design (reconstruction, independence, velocity, dynamics, prediction) covers the main ingredients you'd want. On the Duffing examples the method produces plausible-looking modal coordinates and decent reconstructions, and the paper is candid about training instability and the cost of high-dimensional flows.\n\nThe soft spots are serious, and they center on the gap between what the losses enforce and what NNMs actually are. An NNM is defined by an invariant manifold: the two-dimensional (p_i, q_i) subspace should be invariant under the flow, meaning the modal dynamics decouple from the other coordinates. None of the five losses imposes that. The dynamics block G is a dense MLP that is free to mix all latent coordinates. Lcorr only forces instantaneous statistical independence of the latent variables, which is not dynamical decoupling. So the identified latent space could simply be an invertible, factorized coordinate system for the data distribution, not the Shaw-Pierre/Rosenberg NNMs. The only evidence against that reading is a qualitative wavelet plot and a hand-wavy remark that Fig. 10 'is in agreement with analytical results [51]'. No quantitative error measure, no comparison to analytically computed invariant manifolds or backbone curves. That gap is load-bearing: it is the paper's central claim that the method 'identifies NNMs'.\n\nThe flow-field comparison is also unfair. The NF is trained on 10 POD coordinates and then evaluated with 2 latent coordinates (the other 8 set to zero), while POD is evaluated with its first 2 modes. Obviously a nonlinear function of 10 coordinates can represent more with 2 numbers than a linear projection onto the first 2 modes. That result shows the NF provides a nonlinear compression, which is fine, but it doesn't validate the 'mode' interpretation.\n\nThere are also standard but real methodological problems: hyperparameters were selected to minimize testing error, so the reported prediction performance is not a clean out-of-sample result; there are no error bars; no code or data; and the 'agreement with analytical results' is not backed by numbers.\n\nAll of this is correctable. If the authors test on a system with known analytical NNMs, quantify the error, demonstrate dynamical decoupling (e.g., block-diagonal G or zero cross-coupling), and fix the POD comparison, the method could be a solid contribution. As it stands, the paper is a promising idea with weak validation.\n\nI'd send it to peer review, but only with the expectation of substantial revision. The idea is new enough and the problem important enough in structural dynamics that a serious referee should see it and push on the exact points above. I'd bring it to a reading group discussion, mainly as an example of the gap between loss-based regularization and true physical invariance.","headline":"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.","tokens_in":16720,"tokens_out":4208,"would_cite":false,"duration_ms":36482,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Normalizing Flows can identify nonlinear normal modes from raw response data.","keywords":["nonlinear normal modes","Normalizing Flows","data-driven modal analysis","nonlinear system identification","Proper Orthogonal Decomposition","invariant manifolds","Duffing oscillator","flow past cylinder"],"falsifier":"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.","tokens_in":15777,"feed_emoji":"🌀","tokens_out":5390,"duration_ms":48753,"temperature":0.7,"pith_summary":"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.","feed_headline":"Invertible AI network finds hidden nonlinear modes from raw data","feed_subtitle":"A physics-constrained normalizing flow identifies nonlinear normal modes and beats POD at equal coordinate count.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces NICE, the nonlinear independent component estimation model that supplies the independence property of latent spaces used for modal decomposition.","marker":"[42]"},{"why":"Introduces RealNVP, the invertible bijector architecture used as the Normalizing Flows model.","marker":"[43]"},{"why":"Defines nonlinear normal modes as invariant manifolds via a nonlinear transformation, the physical target the losses are designed to enforce.","marker":"[26]"},{"why":"Prior physics-integrated deep learning identification of NNMs, the approach this paper extends with Normalizing Flows.","marker":"[37]"},{"why":"Provides the POD method and theory used as the linear baseline for comparison in the flow-field case.","marker":"[54]"},{"why":"Supplies the POD modal decomposition formulation for fluid flows used in the preprocessing step.","marker":"[55]"},{"why":"Cited as the analytical or experimental source with which the identified NNM manifold curvature agrees.","marker":"[51]"}],"fun_headline_variants":["Normalizing flows surface hidden nonlinear modes from raw data","Invertible AI maps raw data to nonlinear normal modes","Physics-constrained flow identifies NNMs from measured data","Data-only deep learning reveals nonlinear modal coordinates","Invertible normalizing flow maps raw data to nonlinear modes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Normalizing flows surface hidden nonlinear modes from raw data","Invertible AI maps raw data to nonlinear normal modes","Physics-constrained flow identifies NNMs from measured data","Data-only deep learning reveals nonlinear modal coordinates","Invertible normalizing flow maps raw data to nonlinear modes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000285,"raw_usage":{"total_tokens":1662,"prompt_tokens":914,"completion_tokens":748,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":681}},"tokens_in":530,"tokens_out":748,"duration_ms":6779,"temperature":1.0,"reasoning_tokens":681,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:21:12.465014+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines nonlinear normal modes as invariant manifolds via a nonlinear transformation, the physical target the losses are designed to enforce."},{"cited_title":"Data-driven iden- tification of nonlinear normal modes via physics- integrated deep learning","cited_arxiv_id":null,"evidence_quote":"Prior physics-integrated deep learning identification of NNMs, the approach this paper extends with Normalizing Flows."},{"cited_title":"Turbulence, coherent struc- tures, dynamical systems and symmetry","cited_arxiv_id":null,"evidence_quote":"Provides the POD method and theory used as the linear baseline for comparison in the flow-field case."},{"cited_title":"Modal analysis of fluid flows: An overview","cited_arxiv_id":null,"evidence_quote":"Supplies the POD modal decomposition formulation for fluid flows used in the preprocessing step."},{"cited_title":"Modal testing of nonlinear vibrat- ing structures based on nonlinear normal modes: Experimental demonstration","cited_arxiv_id":null,"evidence_quote":"Cited as the analytical or experimental source with which the identified NNM manifold curvature agrees."}],"review_version":1}