{"id":"12db2e65-6e69-49fd-8ca2-e626d835d38d","arxiv_id":"2501.06118","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A structure-preserving neural network method identifies nonlinear port-Hamiltonian systems from input-state-output data, improving long-term forecasting over physics-free baselines.","lead":"This paper presents a machine-learning framework that identifies nonlinear port-Hamiltonian systems from input-state-output data by learning the energy function and system matrices as neural networks. Enforcing the physical structure improves long-term prediction and data efficiency compared with unstructured baselines, especially when prior knowledge is incorporated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central claim rests on exact state-derivative labels from the true model (Section 3.1); since practical input-state-output data require derivative estimation, the long-term advantage may not transfer without a noisy-derivative experiment.","rationale":"The reader's weakest-assumption analysis identifies exactly the same load-bearing gap: the method is trained and evaluated with state derivatives computed from the true model equations, and no experiment tests derivative estimation from noisy sampled states. I agree that this is the most important threat to the central claim, because the claim is about identification from input-state-output data, and the training loss relies directly on x_dot. The paper is honest about this choice, and the issue is empirically addressable, so conditional acceptance remains appropriate. I do not see a more fundamental internal inconsistency: the parametrizations in Equations (4)-(5) do enforce J skew-symmetric and R symmetric positive semidefinite, the examples cover nonlinearities in J, R, and H, and the comparison to a matched-parameter black-box MLP is a reasonable baseline. The missing noisy-derivative experiment is not a reason to reject, but it is the condition that must be verified before the abstract claim can be stated without qualification. Therefore the reader's CONDITIONAL verdict should stand unchanged.","tokens_in":15030,"tokens_out":4257,"duration_ms":45425,"concrete_test":"On the mass-spring example (Section 4.1), generate training states at Delta t=0.01, add Gaussian measurement noise to x at SNR levels of 30 dB and 20 dB, estimate x_dot by central finite differences with optional smoothing, and retrain the default pH model and the matched-parameter baseline for Ntra=30 and 300. Report the 100 s long-term NMAE using the same protocol as Figure 3 and the low-data NMAE curves. If the pH advantage over the baseline is not preserved or reverses at either noise level, the abstract claim should be restricted to exact-derivative settings.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline claim is that the port-Hamiltonian structure 'helps to make long-term predictions compared to baselines that do not incorporate physics' (Abstract). The evidence for this claim is produced under an idealized supervision condition. In Section 3.1 the authors state that, although in practice one estimates x_dot from observed trajectories, 'For simplicity, we used instead the values given by the system equation with true coefficients.' The loss in Equation (13) is a supervised regression on x_dot and y, so every reported NMAE curve (Figures 2, 5, 6, 8) and every long-term trajectory plot (Figures 3, 7, 9) is obtained with oracle derivative labels. The baseline MLP is trained on the same oracle labels, so the comparison isolates the effect of structure under ideal supervision, not under the practical input-state-output setting advertised in the title and abstract. The only noise study (Section 4.3) adds noise to inputs and outputs while keeping states and derivatives noise-free, so it does not probe the sensitivity to derivative estimation error. If derivative estimates are noisy or biased, the advantage of the pH model over the baseline could shrink or disappear, because the structured hypothesis class is also fitted to corrupted targets. The paper is transparent about this limitation, but the central claim as stated is not yet supported for realistic data.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a framework for identifying nonlinear port-Hamiltonian (pH) systems from input-state-output data. The method represents the state-dependent matrices J, R, B and the Hamiltonian H by neural networks (MLPs or KANs) using parametrizations that enforce J = -J^T and R = R^T >= 0 by construction. Training minimizes a normalized supervised loss on state derivatives and outputs (Eq. 13). Experiments on a mass-spring system, a magnetically levitated ball, and a permanent-magnet synchronous motor compare the pH models with a parameter-matched unstructured MLP baseline, showing lower normalized mean absolute error (NMAE) and better long-horizon trajectory predictions. The paper also explores incorporating prior knowledge about which components are constant and compares MLP and KAN baselines.","tokens_in":15251,"tokens_out":2569,"duration_ms":27491,"significance":"If the central claim holds for realistic identification settings, the contribution is valuable: it provides a systematic, structure-preserving neural approach for nonlinear pH system identification, with a clear mechanism for enforcing passivity-related structural constraints and for integrating prior knowledge. The paper is transparent about its assumptions, ships code, and evaluates on three systems with distinct nonlinearity locations. However, the reported experiments rely on oracle state-derivative labels derived from the true system equations, which is not the advertised input-state-output setting. The significance of the contribution is therefore conditional on demonstrating that the advantage over unstructured baselines persists when derivatives must be estimated from sampled, possibly noisy state data.","major_comments":[{"comment":"The training loss (13) is supervised on state derivatives x_dot, and Section 3.1 states that 'For simplicity, we used instead the values given by the system equation with true coefficients.' Consequently, every reported NMAE curve (Figures 2, 5, 6, 8) and long-term trajectory (Figures 3, 7, 9) is obtained with oracle derivative labels. The baseline is trained on the same oracle labels, so the comparison isolates structure under ideal supervision, not under the practical input-state-output setting claimed in the title and abstract. A concrete fix is to repeat the main experiments with derivative estimates obtained from noisy state observations, e.g., via finite differences with smoothing or via a learned derivative estimator, and to report whether the pH model still outperforms the baseline.","section":"Section 3.1, Eq. (13)"},{"comment":"The noise study is described as adding 'uniform white noise ... to the inputs and outputs during training, while the states and derivatives of these were kept noise-free.' This setup does not probe the scenario that is most likely to affect the method's practical utility: when the state trajectories themselves are noisy and the derivatives x_dot must be estimated from them. Since the central claim depends on the advantage of the structured model under realistic data conditions, the paper should include an experiment where state measurements are corrupted and derivatives are estimated, rather than only corrupting inputs and outputs.","section":"Section 4.3, Figure 9"}],"minor_comments":[{"comment":"The element labeled 'Figure 10' is a table of NMAE values, not a figure; it should be referenced and numbered as a table.","section":"Section 4.4, Figure 10"},{"comment":"The text says 'we did not use KANs to approximate the parameters θ• in the pH representation (4),' but the abstract and introduction state that the paper explores KAN architectures for the pH framework. Clarify that KANs were tested only as baselines, not for the structure-preserving pH models.","section":"Section 4.4"},{"comment":"The normalization in the loss uses σ_{ẋ_i}² and σ_{y_i}², but the definition provided immediately after the equation defines only σ_{ẋ_i}; the analogous definition for σ_{y_i} should be stated.","section":"Section 3.2, Eq. (13)"},{"comment":"The statement that 'all the choices of γ considered resulted in the same order of magnitude in the error measures' is vague; specify the values of γ that were tested and the observed error ranges.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is transparent about the oracle-derivative assumption, which is commendable, but the central claim about practical identification is not yet supported. The requested experiments (derivative estimation from noisy state data, and a noise study on states) are within the scope of the manuscript and would directly address the gap. I do not see a fundamental flaw in the method itself; the issue is the mismatch between the stated claim and the experimental conditions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, incremental paper on learning nonlinear port-Hamiltonian (pH) models from state data, but the headline claim is only demonstrated under an idealized supervision condition. Worth sending to review, but I'd want a noisy-derivative experiment before trusting the abstract.\n\nThe genuinely new part is the combination, not any single ingredient. They take the structured parametrization from Schwerdtner et al. (J skew, R = A A^T/n, B via vtf), apply it to the nonlinear case with dedicated MLPs for each matrix and the Hamiltonian, add a KAN variant and prior-knowledge ansatz functions, and then run a systematic data-size study. The long-term forecasting test (train on 10 s, predict to 100 s) is the right evaluation for a structure-preserving method, and the wrong-prior example in Figure 5 is a nice sanity check. The pH models beat a parameter-matched MLP baseline in all three examples, and the gap grows with more data, so the qualitative claim has real support within this setup.\n\nThe soft spot is exactly what the stress test flagged. Section 3.1 says they used x_dot from the true system equations rather than estimating it from observed trajectories. The baseline gets the same oracle labels, so the comparison is fair as a proof of concept, but it does not validate the practical input-state-output setting advertised in the title and abstract. The noise study in Section 4.3 only corrupts inputs and outputs, leaving states and derivatives clean, so it does not address the most likely failure mode. If derivative estimates are noisy, the structured model's advantage could shrink or vanish, because it is also fitted to corrupted targets. I don't see anything dishonest here - the limitation is stated plainly - but the central claim outruns the evidence.\n\nMinor issues: no error bars or seed variance (matters when training with 10-30 trajectories); the GitHub link appears malformed; and the closest pH-NN baselines [39, 53] are cited but not compared numerically. A small table against those would make the contribution much clearer.\n\nWho this is for: people in structure-preserving system identification and physics-informed learning. It's a useful data point, and the prior-knowledge variants are practical. My call: send to peer review, expect major revision. The method is sound and the authors are honest; the missing experiment is easy to specify.","headline":"Useful but conditional: the structural advantage is real in an idealized simulation, but the abstract overclaims until noisy-derivative experiments are added.","tokens_in":15819,"tokens_out":3443,"would_cite":true,"duration_ms":35008,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B30","93B15","93C10","68T07","93B99"],"pacs":[],"model":"deepseek-v4-flash","headline":"Enforcing port-Hamiltonian structure in learned dynamics improves long-term prediction over physics-free neural models.","keywords":["port-Hamiltonian systems","system identification","neural networks","structure-preserving learning","Hamiltonian","dissipation","long-term prediction","Kolmogorov-Arnold networks"],"falsifier":"Train the same architecture on a benchmark nonlinear pH system, for example the magnetically levitated ball, but replace exact derivatives with finite-difference or filtered estimates from noisy sampled trajectories at realistic noise levels, and compare long-horizon NMAE against the black-box baseline; the claim that structure aids long-term prediction is falsified if pH models no longer outperform once actual observation noise is present.","tokens_in":14807,"feed_emoji":"⚙️","tokens_out":5769,"duration_ms":52518,"temperature":0.7,"pith_summary":"This paper proposes a way to identify nonlinear port-Hamiltonian systems from input-state-output trajectories while keeping the defining structure intact. The dynamics are written as $\\dot{x} = (J(x)-R(x))\\nabla H(x)+B(x)u$, with $J$ skew-symmetric and $R$ symmetric positive semidefinite, and each component is represented by a neural network whose output is reshaped so the constraints hold by construction. The central claim is that preserving this structure gives models that extrapolate beyond the training horizon better than physics-free neural baselines, and that prior knowledge about which parts are constant or quadratic reduces the data needed. The framework is tested on systems with nonlinearities in the interconnection matrix, in the dissipation matrix, and in a non-quadratic Hamiltonian, and on noisy training inputs at moderate levels. A comparison with Kolmogorov-Arnold networks in the same role finds the MLP implementation more accurate in these examples.","feed_headline":"Structure-preserving neural nets beat black-box long-term forecasts","feed_subtitle":"Enforcing skew-symmetric and dissipative structure in learned dynamics extends reliable prediction far past the training window.","key_machinery":"The central object is the structured parametrization of the port-Hamiltonian data, equation (4): a vector-to-matrix reshaping map $\\operatorname{vtf}$ stacks an $n^2$-vector into an $n \\times n$ matrix, and the system matrices are built as $\\operatorname{vtf}(\\theta_J)^\\top - \\operatorname{vtf}(\\theta_J)$ for $J$ and $\\frac{1}{\\sqrt{n}}\\operatorname{vtf}(\\theta_R)\\operatorname{vtf}(\\theta_R)^\\top$ for $R$. This makes $J(x) = -J(x)^\\top$ and $R(x) = R(x)^\\top \\succeq 0$ true for every state by construction. The trained networks approximate $\\theta_J, \\theta_R, \\theta_B, \\theta_H$; an optional prior parametrization replaces a network output by a learned combination of prescribed ansatz functions, which can encode statements such as \"$J$ and $B$ are constant\" or \"$H$ is quadratic.\" The same machinery supplies the state derivative through automatic differentiation of $\\theta_H$, so the Hamiltonian need not be known in closed form.","core_discovery":"The paper's core discovery is that enforcing the port-Hamiltonian geometry in the network output is itself the inductive bias that makes long-term prediction work. Rather than learning the map $(x,u) \\mapsto (\\dot{x},y)$ freely, the network learns the parameter functions $\\theta_J, \\theta_R, \\theta_B, \\theta_H$ and assembles $J = \\operatorname{vtf}(\\theta_J)^\\top - \\operatorname{vtf}(\\theta_J)$, $R = \\frac{1}{\\sqrt{n}}\\operatorname{vtf}(\\theta_R)\\operatorname{vtf}(\\theta_R)^\\top$, $B = \\operatorname{vtf}(\\theta_B)$, $H = \\theta_H$; the skew-symmetry of $J$ and the positive semidefiniteness of $R$ follow from the parametrization, so no constraint penalty is needed. The predicted derivative is $(J(x)-R(x))\\nabla H(x)+B(x)u$, with $\\nabla H$ obtained by automatic differentiation, and the loss is a normalized mean-squared error over state derivatives and outputs. In experiments on a nonlinear mass-spring system, a magnetically levitated ball, and a permanent-magnet synchronous motor, the structure-preserving models consistently match or beat a parameter-matched black-box MLP, especially in low-data and long-horizon regimes; with correct prior assumptions the same error is reached with far fewer training trajectories. The paper also reports that the long-term advantage is visible even when the one-step validation errors of the baseline and the pH model look similar.","pith_inferences":["Editorial extension: the paper computes derivatives using the true system equations; a natural next test is replacing them with finite-difference or filtered estimates from noisy sampled states, which would reveal how much of the long-horizon advantage survives realistic observation noise.","Editorial extension: the same structured parametrization could be combined with structure-preserving time integration or used to produce passive models for controller design, so the identified $J,R,B,H$ could serve as a surrogate in energy-based control synthesis rather than only for prediction.","Editorial extension: if the long-term advantage is caused by enforcing passivity and dissipation, the effect should be strongest when test inputs drive the system into regimes with significant dissipation; the sinusoidal input families used in the paper may under-sample such regimes."],"forward_implications":["With the port-Hamiltonian structure enforced, validation error falls as the number of training trajectories increases and, once enough data are available, clearly beats a parameter-matched black-box MLP on the same one-step metric.","Correct prior information about which components are constant or quadratic cuts the data requirement: in the mass-spring example the pH-prior model reaches a prescribed error with 30 trajectories while the baseline needs more than 1000.","Identified systems trained on only the first 10 seconds continue to track true trajectories over 100 seconds, whereas baselines that fit the training window deviate once forecasting begins.","The framework tolerates moderate noise in training inputs and outputs in the PMSM study: predictions stay close to clean-data quality up to 30 dB and only start to deviate around 25 dB.","KANs used in the same role gave one to two orders of magnitude larger NMAE than the MLP implementation in the three examples considered, so the paper's main results use MLPs."],"supporting_citations":[{"why":"Contributes the vector-to-matrix reshaping parametrization of linear port-Hamiltonian systems that this paper adapts to state-dependent J, R, B and H.","marker":"[45]"},{"why":"Provides the recent nonlinear pH identification approach this work extends with input-state-output data, and the sinusoidal input sampling used for training.","marker":"[53]"},{"why":"Supplies the mass-spring benchmark and the compositional training-data generation scheme the experiments follow.","marker":"[39]"},{"why":"Motivates the derivative-estimation problem and the use of ansatz-function priors in nonlinear system identification, against which the pH parametrization is contrasted.","marker":"[4]"},{"why":"Introduces Kolmogorov-Arnold networks, the alternative architecture the paper compares with the MLP in its final experiments.","marker":"[30]"},{"why":"Supplies the magnetically levitated ball example and represents the Gaussian-process port-Hamiltonian identification baseline.","marker":"[2]"}],"fun_headline_variants":["Physics-steered nets outpredict black boxes","Skew-symmetry and dissipation boost forecasts","Structure beats black-box for long-horizon","Port-Hamiltonian nets: better beyond training"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method's training uses exact state derivatives, computed in the experiments from the true system equations, rather than derivatives estimated from noisy sampled measurements; if that idealization is removed, the reported advantage over physics-free baselines is not guaranteed.","fun_headline_variants_meta":{"raw":{"variants":["Physics-steered nets outpredict black boxes","Skew-symmetry and dissipation boost forecasts","Structure beats black-box for long-horizon","Port-Hamiltonian nets: better beyond training"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000226,"raw_usage":{"total_tokens":1465,"prompt_tokens":937,"completion_tokens":528,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":470}},"tokens_in":553,"tokens_out":528,"duration_ms":5392,"temperature":1.0,"reasoning_tokens":470,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:05:31.748335+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the same architecture on a benchmark nonlinear pH system, for example the magnetically levitated ball, but replace exact derivatives with finite-difference or filtered estimates from noisy sampled trajectories at realistic noise levels, and compare long-horizon NMAE against the black-box baseline; the claim that structure aids long-term prediction is falsified if pH models no longer outperform once actual observation noise is present.","supporting_citations":[{"cited_title":"Schwerdtner","cited_arxiv_id":null,"evidence_quote":"Contributes the vector-to-matrix reshaping parametrization of linear port-Hamiltonian systems that this paper adapts to state-dependent J, R, B and H."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the recent nonlinear pH identification approach this work extends with input-state-output data, and the sinusoidal input sampling used for training."},{"cited_title":"Neary and U","cited_arxiv_id":null,"evidence_quote":"Supplies the mass-spring benchmark and the compositional training-data generation scheme the experiments follow."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Motivates the derivative-estimation problem and the use of ansatz-function priors in nonlinear system identification, against which the pH parametrization is contrasted."},{"cited_title":"Neon tetra fish (Paracheirodon innesi) as farm-to-optical-table Bragg reflectors","cited_arxiv_id":"2406.07819","evidence_quote":"Introduces Kolmogorov-Arnold networks, the alternative architecture the paper compares with the MLP in its final experiments."},{"cited_title":"Beckers, J","cited_arxiv_id":null,"evidence_quote":"Supplies the magnetically levitated ball example and represents the Gaussian-process port-Hamiltonian identification baseline."}],"review_version":1}