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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 Pith2026-06-27 08:43 UTCgrok-4.3pith:4C2BDBE2open 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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Reference graph

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