pith. sign in

arxiv: 2412.07301 · v2 · pith:22E2YQ4Tnew · submitted 2024-12-10 · 🧮 math.OC

Reconstructing the system coefficients for coupled harmonic oscillators

Jan Bartsch , Ahmed A. Barakat , Simon Buchwald , Gabriele Ciaramella , Stefan Volkwein , Eva M. Weig This is my paper

Pith reviewed 2026-05-23 07:15 UTC · model grok-4.3

classification 🧮 math.OC
keywords inverse problemsTikhonov regularizationcoupled harmonic oscillatorsparameter identificationoptimizationdamping coefficientscoupling coefficients
0
0 comments X

The pith

An iterative optimization strategy reconstructs the coupling and damping coefficients of coupled harmonic oscillators from limited experimental data.

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

The paper frames the identification of unknown functions in physical models as an inverse problem and applies optimization with Tikhonov regularization to recover coupling and damping coefficients in systems of coupled harmonic oscillators. It introduces an iterative strategy that works from an initial set of laboratory measurements and requires no further experiments. This is compared against a purely experimental alternative to show a reduction in the total number of tests needed. A sympathetic reader would care because the approach makes it practical to characterize the behavior of such systems when additional lab work is expensive or time-consuming.

Core claim

By employing an iterative strategy with Tikhonov regularization, the coupling and damping coefficients in a system of oscillators can be identified from initial experimental data, ensuring an efficient and experiment-free approach after the first measurements, as demonstrated by comparison with a purely experimental alternative.

What carries the argument

Iterative strategy with Tikhonov regularization for solving the inverse problem of coefficient identification in coupled oscillator systems.

If this is right

  • The number of laboratory experiments required to identify the coefficients is significantly reduced.
  • Coefficients are recovered accurately from the initial data set without further measurements.
  • The method performs comparably or better than a purely experimental identification approach.
  • System parameters in oscillator models become identifiable under conditions where repeated testing is impractical.

Where Pith is reading between the lines

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

  • The same iterative regularization structure could be tested on inverse problems for other mechanical or electrical coupled systems.
  • Embedding the reconstruction inside a feedback loop might allow ongoing model updates during operation.
  • The reduction in experiments could lower overall costs in applications that rely on accurate damping and coupling values.

Load-bearing premise

The iterative strategy can accurately recover the true coefficients from the initial experimental data alone without requiring additional measurements or violating the underlying physical model assumptions.

What would settle it

Obtaining new experimental data that measures the coupling and damping coefficients directly and observing a mismatch with the values reconstructed by the iterative method.

Figures

Figures reproduced from arXiv: 2412.07301 by Ahmed A. Barakat, Eva M. Weig, Gabriele Ciaramella, Jan Bartsch, Simon Buchwald, Stefan Volkwein.

Figure 2
Figure 2. Figure 2: (a) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: (a) Vibration modes of the nanostring. ; (b) Form of external driving (cf. ( [PITH_FULL_IMAGE:figures/full_fig_p005_2_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (b) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (b) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Experimental measurement setup. (a) Schematic of the measurement setup used to drive and [PITH_FULL_IMAGE:figures/full_fig_p007_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Microwave cavity and nanostring chip. (a) Photograph of the coaxial quarter-wave microwave [PITH_FULL_IMAGE:figures/full_fig_p007_3_2.png] view at source ↗
Figure 5
Figure 5. Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Schematic visualization of the proposed strategy. [PITH_FULL_IMAGE:figures/full_fig_p010_5_1.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Experimental data (solid, orange) and simulated data (dashed, blue) applying the driving fre [PITH_FULL_IMAGE:figures/full_fig_p012_7_1.png] view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Deviation plots of the result of Algorithm 5.1. The different marker shapes depict the different control pairs (cf [PITH_FULL_IMAGE:figures/full_fig_p012_7_2.png] view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: Convergence history of Algorithm 5.1. (a) Convergence of θ; (b) Convergence of averaged xλy; (c) Convergence of corrections ¯d1, ¯d2 to the damping parameters. 1 2 3 4 5 0.6472 0.6474 0.6476 Driving/Control pair m Coupling parameter (MHz) λm average ⟨λ⟩ (a) u 1 1 u 1 2 u 2 1 u 2 2 u 3 1 u 3 2 u 4 1 u 4 2 u 5 1 u 5 2 10−4 10−3 10−2 10−1 100 Driving frequencies Relative deviation E p Initial guess Mean ini… view at source ↗
Figure 7.4
Figure 7.4. Figure 7.4: Results of final identification for qr1 mode. The different marker shapes depict the different control pairs given in [PITH_FULL_IMAGE:figures/full_fig_p013_7_4.png] view at source ↗
Figure 7.5
Figure 7.5. Figure 7.5: Experimental data (solid, orange) and simulated data (dashed, blue) applying the driving fre [PITH_FULL_IMAGE:figures/full_fig_p014_7_5.png] view at source ↗
Figure 7.6
Figure 7.6. Figure 7.6: Deviation plots and coupling for qr2 mode. The different marker shapes depict the different control pairs (cf [PITH_FULL_IMAGE:figures/full_fig_p014_7_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a) [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
read the original abstract

Physical models often contain unknown functions and relations. In order to gain more insights into the nature of physical processes, these unknown functions have to be identified or reconstructed. Mathematically, we can formulate this research question within the framework of inverse problems. In this work, we consider optimization techniques to solve the inverse problem using Tikhonov regularization and data from laboratory experiments. We propose an iterative strategy that eliminates the need for further laboratory experiments. Our method is applied to identify the coupling and damping coefficients in a system of oscillators, ensuring an efficient and experiment-free approach. We present our results and compare them with those obtained from an alternative, purely experimental approach. By employing our proposed strategy, we demonstrate a significant reduction in the number of laboratory experiments required.

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

3 major / 2 minor

Summary. The paper formulates the reconstruction of coupling and damping coefficients in coupled harmonic oscillators as an inverse problem solved via Tikhonov regularization on laboratory data. It proposes an iterative strategy that uses initial experimental measurements to avoid additional lab experiments, applies the method to the oscillator system, and claims to demonstrate a significant reduction in the number of required experiments relative to a purely experimental alternative.

Significance. If the iterative recovery is shown to be accurate and stable, the approach would reduce experimental cost for parameter identification in linear oscillator networks, a common setting in control and mechanical engineering.

major comments (3)
  1. [Abstract] Abstract: the claim of 'a significant reduction in the number of laboratory experiments' and a comparison with 'an alternative, purely experimental approach' is asserted without any quantitative metrics, error tables, or data details. This prevents evaluation of the central claim.
  2. [Method / Results] The iterative strategy is stated to recover coefficients 'from the initial experimental data alone' (reader's weakest assumption), yet no convergence analysis, regularization-parameter selection procedure, or validation against known ground-truth coefficients is supplied to confirm that the recovered values satisfy the underlying oscillator model without further measurements.
  3. [Optimization formulation] No mention of the specific form of the Tikhonov functional, the choice of the regularization parameter (listed as the sole free parameter), or any a-posteriori error bounds that would justify the 'experiment-free' assertion.
minor comments (2)
  1. Notation for the oscillator equations and the precise definition of the coupling matrix should be introduced before the inverse-problem statement to make the mapping from data to coefficients explicit.
  2. [Abstract] The abstract refers to 'our results' and 'comparison' without indicating whether synthetic or real laboratory data were used; a brief statement on data provenance would improve clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major comment below and will revise the manuscript accordingly to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of 'a significant reduction in the number of laboratory experiments' and a comparison with 'an alternative, purely experimental approach' is asserted without any quantitative metrics, error tables, or data details. This prevents evaluation of the central claim.

    Authors: We agree that the abstract presents the claim without supporting quantitative details. In the revised manuscript we will augment the abstract with specific metrics (e.g., number of experiments reduced from N to M) and will add a results table that reports reconstruction errors for both the iterative and purely experimental approaches. revision: yes

  2. Referee: [Method / Results] The iterative strategy is stated to recover coefficients 'from the initial experimental data alone' (reader's weakest assumption), yet no convergence analysis, regularization-parameter selection procedure, or validation against known ground-truth coefficients is supplied to confirm that the recovered values satisfy the underlying oscillator model without further measurements.

    Authors: We will add an explicit convergence analysis, a subsection describing the regularization-parameter selection procedure (discrepancy principle), and additional numerical validation against known ground-truth coefficients to confirm that the recovered values satisfy the oscillator model. revision: yes

  3. Referee: [Optimization formulation] No mention of the specific form of the Tikhonov functional, the choice of the regularization parameter (listed as the sole free parameter), or any a-posteriori error bounds that would justify the 'experiment-free' assertion.

    Authors: We will expand the optimization section to write the precise Tikhonov functional, detail the a-posteriori choice of the regularization parameter, and include a brief discussion of a-posteriori error bounds from regularization theory that support the experiment-free claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The abstract and description frame the work as a standard inverse problem solved via Tikhonov regularization on external laboratory data to recover coupling and damping coefficients. The iterative strategy is positioned as a computational reduction in required experiments, compared against a purely experimental baseline. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text; the derivation chain relies on independent experimental inputs and regularization rather than reducing to its own assumptions by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; ledger entries are inferred at the level of standard inverse-problem assumptions because specific free parameters, axioms, or invented entities are not stated.

free parameters (1)
  • regularization parameter
    Standard in Tikhonov methods and must be chosen or tuned; abstract gives no value or selection procedure.
axioms (1)
  • domain assumption The physical system is accurately described by a model of coupled harmonic oscillators whose only unknowns are the coupling and damping coefficients.
    Implicit in framing the task as coefficient reconstruction from data.

pith-pipeline@v0.9.0 · 5667 in / 1207 out tokens · 31290 ms · 2026-05-23T07:15:17.223671+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

  1. [1]

    J., and Marquardt, F

    Aspelmeyer, M., Kippenberg, T. J., and Marquardt, F. Cavity optomechanics. Reviews of Modern Physics 86 , 4 (Dec. 2014), 1391–1452. Publisher: American Physical Society

    work page 2014

  2. [2]

    Bachtold, A., Moser, J., and Dykman, M. I. Mesoscopic physics of nanomechanical systems. Reviews of Modern Physics 94, 4 (2022), 045005

    work page 2022

  3. [3]

    A., Chowdhury, A., Le, A

    Bakarat, A. A., Chowdhury, A., Le, A. T., and Weig, E. M. Parametric normal mode splitting for coupling strength estimation. in preparation (2024)

    work page 2024

  4. [4]

    Reconstruction of unknown nonlinear operators in semilinear elliptic models using optimal inputs

    Bartsch, J., Buchwald, S., Ciaramella, G., and Volkwein, S. Reconstruction of unknown nonlinear operators in semilinear elliptic models using optimal inputs. arXiv:2405.12153, submitted to Math. Control. Relat. F. (2024)

    work page arXiv 2024

  5. [5]

    Adjoint-based calibration of nonlinear stochastic differential equations

    Bartsch, J., Denk, R., and Volkwein, S. Adjoint-based calibration of nonlinear stochastic differential equations. Appl. Math. Opt. 90 , 50 (2024)

    work page 2024

  6. [6]

    A., and Weig, E

    Barzanjeh, S., Xuereb, A., Gr ¨oblacher, S., Paternostro, M., Regal, C. A., and Weig, E. M. Optomechanics for quantum technologies. Nature Physics 18 , 1 (Jan. 2022), 15–24

    work page 2022

  7. [7]

    Modelling with Ordinary Differential Equations – A Comprehensive Approach

    Borz`ı, A. Modelling with Ordinary Differential Equations – A Comprehensive Approach . Chapman & Hall/CRC Numerical Analysis and Scientific Computing. CRC Press, Boca Raton, FL, 2020

    work page 2020

  8. [8]

    Analysis of a greedy reconstruction algorithm

    Buchwald, S., Ciaramella, G., and Salomon, J. Analysis of a greedy reconstruction algorithm. SIAM J. Control Optim. 59, 6 (2021), 4511–4537

    work page 2021

  9. [9]

    Iterative regularization of parameter identification problems by sequential quadratic programming methods

    Burger, M., and M¨uhlhuber, W. Iterative regularization of parameter identification problems by sequential quadratic programming methods. Inverse Probl. 18 , 4 (2002), 943–969

    work page 2002

  10. [10]

    A., Lehnert, K

    Clerk, A. A., Lehnert, K. W., Bertet, P., Petta, J. R., and Nakamura, Y. Hybrid quantum systems with circuit quantum electrodynamics. Nature Physics 16 , 3 (Mar. 2020), 257–267

    work page 2020

  11. [11]

    The classical Bloch equations

    Frimmer, M., and Novotny, L. The classical Bloch equations. Am. J. Phys. 82 , 10 (Oct. 2014), 947–954

    work page 2014

  12. [12]

    Gajo, K., Rastelli, G., and Weig, E. M. Tuning the nonlinear dispersive coupling of nanomechanical string res- onators. Phys. Rev. B 101 , 7 (Feb. 2020), 075420

    work page 2020

  13. [13]

    Y., Guo, G.-C., et al

    Guo, M.-L., Fang, J.-W., Chen, J.-F., Li, B.-L., Chen, H., Zhou, Q., Wang, Y., Song, H.-Z., Arutyunov, K. Y., Guo, G.-C., et al. Mode coupling in electromechanical systems: Recent advances and applications.Advanced Electronic Materials 9, 7 (2023), 2201305

    work page 2023

  14. [14]

    M., and Oldenburg, D

    Haber, E., Ascher, U. M., and Oldenburg, D. On optimization techniques for solving nonlinear inverse problems. Inverse Probl. 16 , 5 (2000), 1263–1280

    work page 2000

  15. [15]

    S., Rastelli, G., Seitner, M

    Huber, J. S., Rastelli, G., Seitner, M. J., K ¨olbl, J., Belzig, W., Dykman, M. I., and Weig, E. M. Spectral evidence of squeezing of a weakly damped driven nanomechanical mode. Phys. Rev. X 10 , 2 (Apr. 2020), 021066

    work page 2020

  16. [16]

    L., Kotz, S., and Balakrishnan, N

    Johnson, N. L., Kotz, S., and Balakrishnan, N. Continuous multivariate distributions , second ed., vol. 1. Wiley New York, 1994

    work page 1994

  17. [17]

    An introduction to the mathematical theory of inverse problems, third ed., vol

    Kirsch, A. An introduction to the mathematical theory of inverse problems, third ed., vol. 120 of Applied Mathematical Sciences. Springer, Cham, 2021

    work page 2021

  18. [18]

    Phase dynamics of coupled oscillators reconstructed from data

    Kralemann, B., Cimponeriu, L., Rosenblum, M., Pikovsky, A., and Mrowka, R. Phase dynamics of coupled oscillators reconstructed from data. Phys. Rev. E 77 , 6 (June 2008), 066205

    work page 2008

  19. [19]

    Iterative choices of regularization parameters in linear inverse problems

    Kunisch, K., and Zou, J. Iterative choices of regularization parameters in linear inverse problems. Inverse Probl. 14 , 5 (1998), 1247–1264

    work page 1998

  20. [20]

    Le, A. T. 3D Microwave Cavity-Controlled Nanoelectromechanical Systems: for Cavity Electromechanics and Quantum Sensing. PhD thesis, Technische Universit¨ at M¨ unchen, 2023

    work page 2023

  21. [21]

    Fundamentals of vibrations

    Meirovitch, L. Fundamentals of vibrations. McGraw-Hill, Boston, 2001

    work page 2001

  22. [22]

    Nocedal, J., and Wright, S. J. Numerical optimization , second ed. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2006

    work page 2006

  23. [23]

    S., Boneß, D

    Ochs, J. S., Boneß, D. K. J., Rastelli, G., Seitner, M., Belzig, W., Dykman, M. I., and Weig, E. M. Frequency comb from a single driven nonlinear nanomechanical mode. Phys. Rev. X 12 , 4 (Oct. 2022), 041019

    work page 2022

  24. [24]

    S., Rastelli, G., Seitner, M., Dykman, M

    Ochs, J. S., Rastelli, G., Seitner, M., Dykman, M. I., and Weig, E. M. Resonant nonlinear response of a nanomechanical system with broken symmetry. Phys. Rev. B 104 , 15 (Oct. 2021), 155434

    work page 2021

  25. [25]

    J., Ciocanel, M.-V., Lazarus, L., Topaz, C

    Panaggio, M. J., Ciocanel, M.-V., Lazarus, L., Topaz, C. M., and Xu, B. Model reconstruction from temporal data for coupled oscillator networks. Chaos 29, 10 (Oct. 2019), 103116

    work page 2019

  26. [26]

    Shores, T. S. Applied Linear Algebra and Matrix Analysis . Undergraduate Texts in Mathematics. Springer, New York, 2007

    work page 2007

  27. [27]

    R., et al

    Singiresu, S. R., et al. Mechanical Vibrations. Addison Wesley Boston, MA, 1995

    work page 1995

  28. [28]

    P., Weig, E

    Unterreithmeier, Q. P., Weig, E. M., and Kotthaus, J. P. Universal transduction scheme for nanomechanical systems based on dielectric forces. Nature 458, 7241 (2009), 1001–1004

    work page 2009