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arxiv: 2606.12163 · v1 · pith:4C2BDBE2 · submitted 2026-06-10 · math.OC

Kernel-based identification of nonlinear port-Hamiltonian systems

Brayan M. Shali , Henk J. van Waarde This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-27 08:43 UTCgrok-4.3pith:4C2BDBE2record.jsonopen to challenge →

classification math.OC
keywords port-Hamiltonian systemssystem identificationkernel methodsrepresenter theoremnonlinear systemsreproducing kernel Hilbert spacesoptimization
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The pith

A representer theorem reduces kernel-based identification of nonlinear port-Hamiltonian systems to a finite-dimensional optimization problem with a convergent algorithm.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper sets out a data-driven method to recover the structure of nonlinear port-Hamiltonian systems from input-state-output trajectories. Instead of assuming specific parametric forms, the energy, dissipation, and interconnection maps are placed in reproducing kernel Hilbert spaces, yielding an infinite-dimensional optimization. The central result is a representer theorem that collapses the search to a finite set of coefficients. An iterative algorithm is then supplied and shown to converge for the resulting non-convex problem.

Core claim

By representing the defining maps of a port-Hamiltonian system inside suitably chosen reproducing kernel Hilbert spaces, the identification task admits a representer theorem that converts the original infinite-dimensional problem into a finite-dimensional non-convex program; a convergent algorithm is provided for solving this reduced program from measured data.

What carries the argument

The representer theorem that reduces the infinite-dimensional optimization over reproducing kernel Hilbert spaces to a finite-dimensional problem in the coefficients of the kernel expansions.

If this is right

  • Identification of port-Hamiltonian models becomes possible from data without prior parametric assumptions on the system maps.
  • The learned models automatically inherit the energy-balance structure of port-Hamiltonian systems.
  • The finite-dimensional reduction makes numerical solution feasible with standard optimization tools.
  • Convergence of the supplied algorithm guarantees that stationary points of the reduced problem can be reached reliably.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same kernel-representer strategy could be applied to other structured classes of dynamical systems whose maps satisfy similar reproducing-property constraints.
  • Choice of kernel may be tuned to encode prior physical knowledge such as positivity or symmetry of energy functions.
  • The finite-dimensional problem could serve as a building block for subsequent data-driven controller synthesis that respects the port-Hamiltonian geometry.

Load-bearing premise

The maps that define the port-Hamiltonian system can be well approximated inside the chosen reproducing kernel Hilbert spaces.

What would settle it

Apply the procedure to input-state-output data generated by a known nonlinear port-Hamiltonian system and check whether the recovered maps reproduce the true state trajectories on fresh inputs to within a small error.

Figures

Figures reproduced from arXiv: 2606.12163 by Brayan M. Shali, Henk J. van Waarde.

Figure 1
Figure 1. Figure 1: Sampled state trajectory corresponding to a 30 second piecewise [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example output trajectories corresponding to a 30 second piecewise [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sections of the Hamiltonians: true (black, solid), approximated (teal, [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

Port-Hamiltonian systems provide a structured framework for modeling physical systems by explicitly capturing their energy storage, dissipation, and exchange. However, deriving such models often requires detailed physical insight and precise knowledge of system parameters, which may not be available in practice. In this paper, we propose a kernel-based framework for the identification of port-Hamiltonian systems from input-state-output data. In contrast to conventional parametric approaches, the maps defining the port-Hamiltonian system are represented in suitably chosen reproducing kernel Hilbert spaces. This leads to an infinite-dimensional optimization problem over the corresponding function spaces. Our main result establishes a representer theorem that reduces this problem to a tractable finite-dimensional one. Since the reduced problem is non-convex, we further provide an algorithm for its solution and prove its convergence.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a kernel-based framework for identifying nonlinear port-Hamiltonian systems from input-state-output data. The maps defining the system (Hamiltonian, dissipation, and input/output ports) are represented in suitably chosen reproducing kernel Hilbert spaces, yielding an infinite-dimensional optimization problem. The central result is a representer theorem reducing this to a finite-dimensional problem; an algorithm is then given for the resulting non-convex optimization together with a convergence proof.

Significance. If the representer theorem and convergence result hold under the stated assumptions, the work supplies a non-parametric, structure-preserving route to data-driven pH modeling that avoids explicit parametric forms. The combination of a representer theorem with a provably convergent algorithm for the reduced problem constitutes a clear technical contribution in the intersection of kernel methods and port-Hamiltonian systems theory.

major comments (2)
  1. [Abstract] Abstract, paragraph 3: the representer theorem is asserted to reduce the identification problem to a tractable finite-dimensional one, yet this reduction is valid only when the true maps belong to (or can be approximated in) the chosen RKHS. No universality, density, or approximation-error bounds are supplied for the kernels, rendering the theorem's applicability to general nonlinear pH systems an unverified assumption that is load-bearing for the main claim.
  2. [Representer theorem section] The section presenting the representer theorem (likely §3): while the finite-dimensional reduction is derived, the subsequent non-convex problem's dependence on kernel hyperparameters is not analyzed; without this, the convergence proof applies only to the reduced problem and does not automatically transfer to the original infinite-dimensional identification task when the RKHS assumption is relaxed.
minor comments (2)
  1. Notation for the port-Hamiltonian structure (J, R, etc.) should be introduced once and used consistently; cross-references to the original pH equations would improve readability.
  2. The algorithm description would benefit from an explicit statement of the stopping criterion and any regularization parameters introduced to handle the non-convexity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below, clarifying the scope of our results while acknowledging where additional remarks can improve the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph 3: the representer theorem is asserted to reduce the identification problem to a tractable finite-dimensional one, yet this reduction is valid only when the true maps belong to (or can be approximated in) the chosen RKHS. No universality, density, or approximation-error bounds are supplied for the kernels, rendering the theorem's applicability to general nonlinear pH systems an unverified assumption that is load-bearing for the main claim.

    Authors: The representer theorem is derived under the explicit modeling assumption that the Hamiltonian, dissipation, and port maps belong to the chosen RKHSs. The manuscript presents a kernel-based, structure-preserving identification method within this framework and does not assert universality, density, or approximation guarantees for arbitrary nonlinear maps. This is consistent with standard practice in kernel methods. To prevent misinterpretation, we will revise the abstract to state the assumption more explicitly. revision: yes

  2. Referee: [Representer theorem section] The section presenting the representer theorem (likely §3): while the finite-dimensional reduction is derived, the subsequent non-convex problem's dependence on kernel hyperparameters is not analyzed; without this, the convergence proof applies only to the reduced problem and does not automatically transfer to the original infinite-dimensional identification task when the RKHS assumption is relaxed.

    Authors: The convergence guarantee applies to the finite-dimensional non-convex problem obtained after the representer theorem, for any fixed choice of kernel hyperparameters. Hyperparameter selection is treated as a separate, standard step (e.g., via cross-validation), as is conventional in kernel learning. The analysis does not extend to the infinite-dimensional problem when the RKHS assumption is dropped, which is outside the theorem's stated scope. We will add a clarifying sentence in the revised manuscript to make this scope explicit. revision: partial

Circularity Check

0 steps flagged

No circularity; representer theorem is an independent mathematical reduction

full rationale

The paper formulates an infinite-dimensional optimization over RKHS for the port-Hamiltonian maps and invokes a representer theorem to obtain a finite-dimensional problem. This follows the standard kernel-method pattern (solution lies in span of kernel evaluations at data points) and does not reduce by construction to fitted parameters renamed as predictions, self-citations, or definitional equivalence. The RKHS membership assumption is stated explicitly as a modeling choice rather than derived from the result itself, leaving the central claim self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; full details of assumptions and any fitted quantities unavailable.

axioms (1)
  • domain assumption The defining maps of the port-Hamiltonian system lie in suitably chosen reproducing kernel Hilbert spaces.
    Stated directly in the abstract as the modeling choice that enables the representer theorem.

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discussion (0)

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Reference graph

Works this paper leans on

52 extracted references · 8 canonical work pages · 1 internal anchor

  1. [1]

    Port-Hamiltonian systems theory: An introductory overview,

    A. van der Schaft and D. Jeltsema, “Port-Hamiltonian systems theory: An introductory overview,”Foundations and Trends in Systems and Control, vol. 1, no. 2, pp. 173–378, 2014

    2014

  2. [2]

    Multidomain modeling of nonlinear networks and systems,

    D. Jeltsema and J. M. Scherpen, “Multidomain modeling of nonlinear networks and systems,”IEEE Control Systems Magazine, vol. 29, no. 4, pp. 28–59, 2009

    2009

  3. [3]

    An introduction to inter- connection and damping assignment passivity-based control in process engineering,

    F. D ¨orfler, J. K. Johnsen, and F. Allg ¨ower, “An introduction to inter- connection and damping assignment passivity-based control in process engineering,”Journal of Process Control, vol. 19, no. 9, pp. 1413–1426, 2009

    2009

  4. [4]

    A. J. van der Schaft,L 2-Gain and Passivity Techniques in Nonlinear Control. Springer International Publishing, 2017

    2017

  5. [5]

    Numerical methods to compute a minimal realization of a port-Hamiltonian system

    K. Cherifi, V . Mehrmann, and K. Hariche, “Numerical methods to compute a minimal realization of a port-Hamiltonian system.” arXiv:1903.07042, 2019

    work page internal anchor Pith review Pith/arXiv arXiv 1903

  6. [6]

    Identification of port- Hamiltonian systems from frequency response data,

    P. Benner, P. Goyal, and P. Van Dooren, “Identification of port- Hamiltonian systems from frequency response data,”Systems & Control Letters, vol. 143, p. 104741, 2020

    2020

  7. [7]

    Port-Hamiltonian system identification from noisy frequency tesponse data

    P. Schwerdtner, “Port-Hamiltonian system identification from noisy frequency tesponse data.” arXiv:2106.11355, 2021

    work page arXiv 2021

  8. [8]

    Port-Hamiltonian dynamic mode decomposition,

    R. Morandin, J. Nicodemus, and B. Unger, “Port-Hamiltonian dynamic mode decomposition,”SIAM Journal on Scientific Computing, vol. 45, no. 4, pp. A1690–A1710, 2023

    2023

  9. [9]

    Learnability of linear port-Hamiltonian sys- tems,

    J.-P. Ortega and D. Yin, “Learnability of linear port-Hamiltonian sys- tems,”Journal of Machine Learning Research, vol. 25, no. 68, pp. 1–56, 2024

    2024

  10. [10]

    J. N. Kutz, S. L. Brunton, B. W. Brunton, and J. L. Proctor,Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems. SIAM, 2016

    2016

  11. [11]

    A new kernel-based approach for linear system identification,

    G. Pillonetto and G. De Nicolao, “A new kernel-based approach for linear system identification,”Automatica, vol. 46, no. 1, pp. 81–93, 2010

    2010

  12. [12]

    Kernel methods in system identification, machine learning and function estimation: A survey,

    G. Pillonetto, F. Dinuzzo, T. Chen, G. De Nicolao, and L. Ljung, “Kernel methods in system identification, machine learning and function estimation: A survey,”Automatica, vol. 50, no. 3, pp. 657–682, 2014

    2014

  13. [13]

    Kernels for linear time invariant system identification,

    F. Dinuzzo, “Kernels for linear time invariant system identification,” SIAM Journal on Control and Optimization, vol. 53, no. 5, pp. 3299– 3317, 2015

    2015

  14. [14]

    A shift in paradigm for system identification,

    L. Ljung, T. Chen, and B. Mu, “A shift in paradigm for system identification,”International Journal of Control, vol. 93, no. 2, pp. 173– 180, 2020

    2020

  15. [15]

    V . I. Paulsen and M. Raghupathi,An Introduction to the Theory of Reproducing Kernel Hilbert Spaces. Cambridge University Press, 2016

    2016

  16. [16]

    Universal kernels,

    C. A. Micchelli, Y . Xu, and H. Zhang, “Universal kernels,”Journal of Machine Learning Research, vol. 7, no. 95, pp. 2651–2667, 2006

    2006

  17. [17]

    A universal reproducing kernel Hilbert space for learning nonlinear systems operators

    M. Lazar, “A universal reproducing kernel Hilbert space for learning nonlinear systems operators.” arXiv:2412.18360, 2024

    work page arXiv 2024

  18. [18]

    Identification of stable linear systems via the sequential stabilizing spline algorithm,

    G. Pillonetto and A. Chiuso, “Identification of stable linear systems via the sequential stabilizing spline algorithm,”IFAC-PapersOnLine, vol. 51, no. 15, pp. 19–24, 2018

    2018

  19. [19]

    Identification of externally positive systems,

    C. Grussler, J. Umenberger, and I. R. Manchester, “Identification of externally positive systems,” inIEEE Conference on Decision and Control, pp. 6549–6554, 2017. 16

    2017

  20. [20]

    Kernel-based identification of positive systems,

    M. Khosravi and R. S. Smith, “Kernel-based identification of positive systems,” inProceedings of the IEEE Conference on Decision and Control, pp. 1740–1745, 2019

    2019

  21. [21]

    Kernel-based models for system analysis,

    H. J. van Waarde and R. Sepulchre, “Kernel-based models for system analysis,”IEEE Transactions on Automatic Control, vol. 68, no. 9, pp. 5317–5332, 2023

    2023

  22. [22]

    Kernel modeling of fading memory systems,

    Y . Huo, T. Chaffey, and R. Sepulchre, “Kernel modeling of fading memory systems,”IEEE Transactions on Automatic Control, vol. 71, no. 3, pp. 2046–2052, 2026

    2046

  23. [23]

    Towards a representer theorem for identification of passive systems,

    B. M. Shali and H. J. van Waarde, “Towards a representer theorem for identification of passive systems,” inProceedings of the IEEE Conference on Decision and Control, pp. 8760–8765, 2024

    2024

  24. [24]

    Kernel-based estimation of frequency response function of strictly passive systems,

    S. Ebrahimkhani and J. Lataire, “Kernel-based estimation of frequency response function of strictly passive systems,”IEEE Control Systems Letters, pp. 1–1, 2025

    2025

  25. [25]

    Non-parametric models for non-negative functions,

    U. Marteau-Ferey, F. Bach, and A. Rudi, “Non-parametric models for non-negative functions,” inProceedings of the 34th International Conference on Neural Information Processing Systems, 2020

    2020

  26. [26]

    Learning psd-valued functions using kernel sums-of-squares

    B. Muzellec, F. Bach, and A. Rudi, “Learning psd-valued functions using kernel sums-of-squares.” arXiv:2111.11306, 2022

    work page arXiv 2022

  27. [27]

    Identification for control: From the early achievements to the revival of experiment design*,

    M. Gevers, “Identification for control: From the early achievements to the revival of experiment design*,”European Journal of Control, vol. 11, no. 4–5, pp. 335–352, 2005

    2005

  28. [28]

    Learning controllers from data via kernel-based interpolation,

    Z. Hu, C. De Persis, and P. Tesi, “Learning controllers from data via kernel-based interpolation,” inProceedings of the IEEE Conference on Decision and Control, pp. 8509–8514, 2023

    2023

  29. [29]

    Robust and kernelized data- enabled predictive control for nonlinear systems,

    L. Huang, J. Lygeros, and F. D ¨orfler, “Robust and kernelized data- enabled predictive control for nonlinear systems,”IEEE Transactions on Control Systems Technology, vol. 32, no. 2, pp. 611–624, 2024

    2024

  30. [30]

    A kernelized operator approach to nonlinear data-enabled predictive control

    T. de Jong, S. Weiland, and M. Lazar, “A kernelized operator approach to nonlinear data-enabled predictive control.” arXiv:2501.17500, 2025

    work page arXiv 2025

  31. [31]

    Passive nonlinear FIR filters for data-driven control

    Z. Wang and F. Forni, “Passive nonlinear FIR filters for data-driven control.” arXiv:2508.05279, 2025

    work page arXiv 2025

  32. [32]

    Physics-informed machine learning,

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, “Physics-informed machine learning,”Nature Reviews Physics, vol. 3, no. 6, pp. 422–440, 2021

    2021

  33. [33]

    An overview on recent machine learning techniques for port-Hamiltonian systems,

    K. Cherifi, “An overview on recent machine learning techniques for port-Hamiltonian systems,”Physica D: Nonlinear Phenomena, vol. 411, p. 132620, 2020

    2020

  34. [34]

    On learning Hamiltonian systems from data,

    T. Bertalan, F. Dietrich, I. Mezi ´c, and I. G. Kevrekidis, “On learning Hamiltonian systems from data,”Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, no. 12, 2019

    2019

  35. [35]

    Learning Hamiltonian dynamics with repro- ducing kernel Hilbert spaces and random features,

    T. Smith and O. Egeland, “Learning Hamiltonian dynamics with repro- ducing kernel Hilbert spaces and random features,”European Journal of Control, vol. 80, p. 101128, 2024

    2024

  36. [36]

    Hamiltonian neural net- works,

    S. Greydanus, M. Dzamba, and J. Yosinski, “Hamiltonian neural net- works,” inProceedings of the 33rd International Conference on Neural Information Processing Systems, Curran Associates Inc., 2019

    2019

  37. [37]

    Symplectic ODE-Net: Learn- ing Hamiltonian dynamics with control,

    Y . D. Zhong, B. Dey, and A. Chakraborty, “Symplectic ODE-Net: Learn- ing Hamiltonian dynamics with control,” inInternational Conference on Learning Representations, 2020

    2020

  38. [38]

    Pseudo-Hamiltonian system identification,

    S. Holmsen, S. Eidnes, and S. Riemer-Sørensen, “Pseudo-Hamiltonian system identification,”Journal of Computational Dynamics, vol. 11, no. 1, pp. 59–91, 2024

    2024

  39. [39]

    Port-Hamiltonian neural networks for learning explicit time-dependent dynamical systems,

    S. A. Desai, M. Mattheakis, D. Sondak, P. Protopapas, and S. J. Roberts, “Port-Hamiltonian neural networks for learning explicit time-dependent dynamical systems,”Physical Review E, vol. 104, no. 3, p. 034312, 2021

    2021

  40. [40]

    Gaussian pro- cess port-Hamiltonian systems: Bayesian learning with physics prior,

    T. Beckers, J. Seidman, P. Perdikaris, and G. J. Pappas, “Gaussian pro- cess port-Hamiltonian systems: Bayesian learning with physics prior,” in Proceedings of the IEEE Conference on Decision and Control, pp. 1447– 1453, IEEE, 2022

    2022

  41. [41]

    Port-Hamiltonian neural networks with output error noise models

    S. Moradi, G. I. Beintema, N. Jaensson, R. T ´oth, and M. Schoukens, “Port-Hamiltonian neural networks with output error noise models.” arXiv:2502.14432, 2025

    work page arXiv 2025

  42. [42]

    Non- linear port-Hamiltonian system identification from input-state-output data,

    K. Cherifi, A. El Messaoudi, H. Gernandt, and M. Roschkowski, “Non- linear port-Hamiltonian system identification from input-state-output data,” 2025

    2025

  43. [43]

    Combining convex–concave decompositions and linearization approaches for solv- ing bmis, with application to static output feedback,

    Q. Tran Dinh, S. Gumussoy, W. Michiels, and M. Diehl, “Combining convex–concave decompositions and linearization approaches for solv- ing bmis, with application to static output feedback,”IEEE Transactions on Automatic Control, vol. 57, no. 6, pp. 1377–1390, 2012

    2012

  44. [44]

    Operator-valued positive definite kernels and differentiable universality,

    J. C. Guella, “Operator-valued positive definite kernels and differentiable universality,”Analysis and Applications, vol. 20, no. 04, pp. 681–735, 2022

    2022

  45. [45]

    Sufficient conditions for the convergence of monotonic math- ematical programming algorithms,

    R. Meyer, “Sufficient conditions for the convergence of monotonic math- ematical programming algorithms,”Journal of Computer and System Sciences, vol. 12, no. 1, pp. 108–121, 1976

    1976

  46. [46]

    First and second order analysis of nonlinear semidefinite programs,

    A. Shapiro, “First and second order analysis of nonlinear semidefinite programs,”Mathematical Programming, vol. 77, no. 1, pp. 301–320, 1997

    1997

  47. [47]

    On Learning Vector-Valued Functions,

    C. A. Micchelli and M. Pontil, “On Learning Vector-Valued Functions,” Neural Computation, vol. 17, no. 1, pp. 177–204, 2005

    2005

  48. [48]

    An introduction to kernel methods,

    Y . Kalnishkan, “An introduction to kernel methods,”Technical Report, available at https://cml.rhul.ac.uk/publications/files/tr0901.pdf, 2009

    2009

  49. [49]

    Theory of reproducing kernels,

    N. Aronszajn, “Theory of reproducing kernels,”Transactions of the American Mathematical Society, vol. 68, no. 3, pp. 337–404, 1950

    1950

  50. [50]

    Kernel-based models for system analysis

    H. J. van Waarde and R. Sepulchre, “Kernel-based models for system analysis.” arXiv:2110.11735, 2021

    work page arXiv 2021

  51. [51]

    Derivative reproducing properties for kernel methods in learning theory,

    D.-X. Zhou, “Derivative reproducing properties for kernel methods in learning theory,”Journal of Computational and Applied Mathematics, vol. 220, no. 1, pp. 456–463, 2008

    2008

  52. [52]

    J. B. Conway,A Course in Functional Analysis. Springer New York, 2007. Brayan M. Shalireceived the M.Sc. degree (summa cum laude) and the Ph.D. degree in applied mathe- matics from the University of Groningen, Gronin- gen, The Netherlands, in 2019 and 2023, respec- tively. He is currently a Postdoctoral Researcher with KU Leuven, Leuven, Belgium. His rese...

    2007