Nonlinear port-Hamiltonian system identification from input-state-output data

Achraf El Messaoudi; Hannes Gernandt; Karim Cherifi; Marco Roschkowski

REVIEW 2 major objections 4 minor 1 cited by

Nonlinear port-Hamiltonian system identification from input-state-output data

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Enforcing port-Hamiltonian structure in learned dynamics improves long-term prediction over physics-free neural models.

desk verdict Useful but conditional: the structural advantage is real in an idealized simulation, but the abstract overclaims until noisy-derivative experiments are added. read the letter →

arxiv 2501.06118 v2 pith:SWZTV34Z submitted 2025-01-10 eess.SY cs.SYmath.DSmath.OCnlin.CD

Karim Cherifi , Achraf El Messaoudi , Hannes Gernandt , Marco Roschkowski This is my paper · ORCID

classification eess.SYcs.SYmath.DSmath.OCnlin.CD MSC 93B3093B1593C1068T0793B99

keywords port-Hamiltoniansystemssystemidentificationneuralnetworksstructure-preservinglearningHamiltoniandissipationlong-termpredictionKolmogorov-Arnold

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

The 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.

Minor 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.

Who 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[Section 3.1, Eq. (13)] 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 4.3, Figure 9] 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.

minor comments (4)

[Section 4.4, Figure 10] The element labeled 'Figure 10' is a table of NMAE values, not a figure; it should be referenced and numbered as a table.
[Section 4.4] 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 3.2, Eq. (13)] 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 3.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the pH parametrization enforces structure by construction, and the long-term advantage claim is tested against an external parameter-matched MLP baseline on out-of-sample trajectories.

full rationale

The paper's central claim—that enforcing port-Hamiltonian structure improves long-term prediction—does not reduce to its own inputs. Equation (4) is an explicit parametrization: J is formed as vtf(theta_J)^T - vtf(theta_J), which is skew-symmetric by construction, and R is formed as (1/sqrt(n)) vtf(theta_R) vtf(theta_R)^T, which is symmetric positive semidefinite by construction. This is a modelling constraint, not a prediction derived from the data. The loss (13) is a supervised regression on state derivatives and outputs, but the paper's headline evidence consists of long-term trajectory forecasts over 100 s (Figures 3, 7, 9) using trajectories not present in the training set, compared against a baseline MLP whose width is scaled to match the pH network's parameter count (Section 3.3). Thus the 'structure helps long-term prediction' conclusion is an out-of-sample, externally benchmarked empirical finding rather than a tautology. The authors' self-citations ([7]-[10], [17], [18]) are background and methodological references; none is invoked as a load-bearing uniqueness theorem or as the sole justification for the pH ansatz, and the parametrization builds on the independent works [45,46,48]. Section 3.1 honestly discloses that derivatives x_dot are computed from the true system equations rather than estimated, which is a realism limitation for practical input-state-output data and a possible threat to external validity, but it is not circularity: the baseline receives the same oracle derivative labels, and the long-term predictions are still not contained in the training loss. No self-definitional step, fitted-input-renamed-as-prediction, or imported uniqueness argument is present in the derivation chain.

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

The central claim rests on the assumption that the true system is in the port-Hamiltonian class and that accurate derivative data is available. The method introduces no new physical entities.

free parameters (3)

output loss weight gamma = 0.25
Chosen by hand to balance state derivative and output losses; the paper notes validation error is insensitive to the exact value.
MLP architecture = 3 hidden layers, 64 neurons, swish
Default architecture chosen without systematic search; depth and training time shown to affect long-term prediction quality (Figure 3).
KAN configuration = grid size 5, spline order 3, hidden dimension 3
Chosen because KANs are more expensive; not tuned for each example.

assumptions (3)

domain assumption The data-generating system is exactly a port-Hamiltonian system of the form (1) with J skew-symmetric, R symmetric positive semidefinite, and output y = B(x)^T grad H(x).
This is the model class the entire method is built to identify; it is stated in the introduction and used throughout.
domain assumption The state derivative x_dot and output y are exactly available from the system equations during training.
Section 3.1 states the derivatives are computed from the true coefficients, which is a strong data-access assumption.
standard math Universal approximation theorem for MLPs and Kolmogorov-Arnold representation theorem.
Invoked in Section 2.2 to justify the representational power of the networks.

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Cite this review

Pith. "Pith review of Nonlinear port-Hamiltonian system identification from input-state-output data." pith.science (2026). https://pith.science/paper/SWZTV34Z

@misc{pith2026250106118,
  author       = {Pith},
  title        = {Pith review of: Nonlinear port-Hamiltonian system identification from input-state-output data},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWZTV34Z}},
  note         = {Machine review of arXiv:2501.06118}
}

read the original abstract

A framework for identifying nonlinear port-Hamiltonian systems using input-state-output data is introduced. The framework utilizes neural networks' universal approximation capacity to effectively represent complex dynamics in a structured way. We show that using the structure helps to make long-term predictions compared to baselines that do not incorporate physics. We also explore different architectures based on MLPs, KANs, and using prior information. The technique is validated through examples featuring nonlinearities in either the skew-symmetric terms, the dissipative terms, or the Hamiltonian.

Figures

Figures reproduced from arXiv: 2501.06118 by the authors.

**Figure 2.** Average validation error for the spring model as a function of varying [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗

**Figure 3.** The trajectory of the momentum p2 of the spring for different pH training setups. The vertical line at 10 s indicates the final time using in the training of the trajectories. An interesting question is how well our method performs in identifying the true system parameters. One cannot expect to fully recover the system’s coefficients since one could, for example, add a constant to the Hamiltonian or move a scaling … view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Comparison between identified system parameters and true system [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: NMAE in function of number of trajectories used for training for [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]

**Figure 6.** Figure 6: Average validation error for the ball example as a function of varying [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]

**Figure 7.** Figure 7: State trajectory of the ball example for the ground truth, baseline, [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]

**Figure 8.** Figure 8: Average validation error for the motor example as a function of varying [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

**Figure 9.** Figure 9: Flux trajectory of the ball example based on the pH model with [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]

**Figure 10.** Figure 10: Comparison of NMAE for KAN and MLP. As a summary, from our experiments, we noticed that incorporating physics using the port-Hamiltonian framework yields better results than baseline MLP and KAN for the system identification of nonlinear systems. If additional inform…

Discussion (0). Continue with ORCID to comment.

Forward citations

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