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arxiv: 2605.12785 · v1 · submitted 2026-05-12 · 💻 cs.LG · cs.SY· eess.SY· math.DS

Recognition: 2 theorem links

· Lean Theorem

Identifying the nonlinear string dynamics with port-Hamiltonian neural networks

Maximino Linares , Guillaume Doras , Thomas H\'elie
Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:31 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SYmath.DS
keywords port-Hamiltonian systemsneural networksPDE identificationnonlinear stringssystem identificationphysics-informed learningmusical acoustics
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The pith

Port-Hamiltonian neural networks recover the Hamiltonian and dissipation of nonlinear string vibrations from data.

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

This paper shows how to extend port-Hamiltonian neural networks from ordinary differential equations to partial differential equations so that the nonlinear dynamics of a vibrating string can be learned directly from trajectory data. The network architecture is built to enforce the port-Hamiltonian structure, which separates the energy-conserving part of the motion from the dissipative and external-port terms. As a result the learned model yields an explicit Hamiltonian function together with the dissipation operator, both of which match the underlying physics. Experiments on synthetic data confirm that the physics-informed model predicts future states more accurately than unstructured neural networks and produces parameters that remain interpretable.

Core claim

By constructing structured neural network architectures based on port-Hamiltonian systems, the nonlinear string dynamics can be identified from data such that both the Hamiltonian governing the string and the dissipation affecting it are recovered explicitly, producing simulations that are physically consistent and more accurate than those obtained from non-physics-informed baselines.

What carries the argument

Port-Hamiltonian Neural Networks extended to PDEs, whose architecture encodes the port-Hamiltonian form (energy function plus dissipative and input ports) so that the network parameters directly correspond to the Hamiltonian and dissipation operators of the continuous string model.

If this is right

  • The learned model accurately emulates the nonlinear string behavior over time.
  • Both the conservative Hamiltonian and the dissipative terms are recovered in explicit functional form.
  • Prediction accuracy exceeds that of baseline non-structured neural networks on the same data.
  • The resulting model remains interpretable and can be used directly for physics-based simulation in musical acoustics.

Where Pith is reading between the lines

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

  • The same structured architecture could be applied to other distributed-parameter systems such as beams or plates.
  • When real sensor data replace synthetic trajectories, the method could identify unknown material dissipation mechanisms in actual instruments.
  • The recovered Hamiltonian could serve as the basis for energy-preserving reduced-order models that run faster than full PDE solvers.

Load-bearing premise

The nonlinear string dynamics must admit an exact port-Hamiltonian representation and synthetic data generated from that representation must be sufficient to recover the true continuous PDE.

What would settle it

Generate trajectories from the known analytical nonlinear string PDE, train the PHNN on those trajectories, and verify whether the extracted Hamiltonian and dissipation operator reproduce the original PDE coefficients within a small error; a large mismatch or worse prediction error than a standard network would falsify the identification claim.

Figures

Figures reproduced from arXiv: 2605.12785 by Guillaume Doras, Maximino Linares, Thomas H\'elie.

Figure 1
Figure 1. Figure 1: The network Hθnl has one convolutional layer with kernel size two and a MLP with five hidden layers of 100 units and LeakyReLU activation. Nonlinear string configuration l0 = 1.1 m, ρ0 = 8000 kg · m −3 , T = 60 N, E = 2 × 1011 Pa η0 = 0.9 s−1 , η1 = 4 × 10−4 m 2s−1 , N = 202, h = 5.4 × 10−3 Dataset generation Parameter Training Validation Test Ntraj 48 12 60 fs 88.2kHz Ts 2s Te [5, 30] ms xe h 0.1l0 h , 0.… view at source ↗
Figure 2
Figure 2. Figure 2: Representation of a staggered-in-time scheme, with variables defined at offset time instants. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Computation graph of the baseline Modified SAV solver [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: compares the test RMSE obtained by the baseline model and the proposed StringPHNN. The baseline exhibits errors on the order of 100 , whereas the StringPHNN achieves errors around 10−4 , out￾performing the baseline by several orders of magnitude [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: A reference displacement test trajectory compared with the PHNNString prediction (the initialization [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Spectrogram of a momentum test trajectory at the position [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
read the original abstract

Hybrid machine learning combines physical knowledge with data-driven models to enhance interpretability and performance. In this context, Port-Hamiltonian Systems (PHS), which generalize Hamiltonian mechanics to describe open, non-autonomous dynamical systems, have been successfully integrated with neural networks under the name Port-Hamiltonian Neural Networks (PHNNs). While the ability of PHNNs to identify Hamiltonian ordinary differential equation (ODE) systems has already been demonstrated, their application to learning Hamiltonian partial differential equation (PDE) systems remains largely unexplored. This limitation restricts their use in musical acoustics, where instruments are typically modeled as distributed parameter systems governed by PDEs. In this work, we demonstrate how to learn the nonlinear string dynamics from data in a physically-consistent framework through a PHNN extension to PDEs. By constructing structured neural network architectures based on PHS, we can recover both the Hamiltonian governing the string and the dissipation affecting it. This approach outperforms baseline, non-physics-informed methods in terms of both accuracy and interpretability. Numerical experiments using synthetic data demonstrate the ability of the proposed PHNN model to identify and emulate the nonlinear dynamics of the system.

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 paper extends Port-Hamiltonian Neural Networks (PHNNs) from ODEs to PDEs in order to identify the nonlinear dynamics of a vibrating string from synthetic data. It claims that a structured neural architecture based on port-Hamiltonian systems recovers both the governing Hamiltonian functional and the dissipation operator, while outperforming non-physics-informed baselines in accuracy and interpretability.

Significance. If the recovered model converges to the true continuous PDE, the work would provide a useful template for physics-informed identification of distributed-parameter systems in musical acoustics and continuum mechanics. The structured PHNN approach supplies interpretability by construction, which is a clear strength relative to black-box alternatives.

major comments (2)
  1. [Methodology and Numerical experiments] The central claim that the method identifies the continuous nonlinear string PDE (rather than a discrete approximation) is load-bearing but unsupported. Any practical PHNN implementation must first apply spatial discretization (finite differences or elements) to obtain a finite-dimensional port-Hamiltonian ODE; without an explicit mesh-convergence study showing that the learned Hamiltonian functional converges to the continuous limit as h→0, the recovered quantities may simply fit the truncation errors of the chosen semi-discretization.
  2. [Abstract and Numerical experiments] Abstract and §4 (numerical results): the assertion that the PHNN “outperforms baseline, non-physics-informed methods” is stated without reported quantitative metrics (e.g., relative L2 errors on the recovered Hamiltonian, dissipation coefficients, or long-term trajectory prediction). This prevents assessment of whether the improvement is statistically or practically meaningful.
minor comments (2)
  1. [Abstract] The abstract mentions “synthetic data” but does not specify the exact PDE (nonlinear string model), boundary conditions, or excitation used to generate the training trajectories.
  2. [Methodology] Notation for the port-Hamiltonian operators (e.g., the structure matrix J and dissipation matrix R) should be introduced once and used consistently when the PDE extension is defined.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and have revised the manuscript to strengthen the supporting evidence for identifying the continuous PDE and to supply the requested quantitative metrics.

read point-by-point responses

  1. Referee: [Methodology and Numerical experiments] The central claim that the method identifies the continuous nonlinear string PDE (rather than a discrete approximation) is load-bearing but unsupported. Any practical PHNN implementation must first apply spatial discretization (finite differences or elements) to obtain a finite-dimensional port-Hamiltonian ODE; without an explicit mesh-convergence study showing that the learned Hamiltonian functional converges to the continuous limit as h→0, the recovered quantities may simply fit the truncation errors of the chosen semi-discretization.

    Authors: We agree that an explicit mesh-convergence study is required to substantiate the claim of recovering the continuous PDE rather than a discrete surrogate. The PHNN architecture is constructed so that the learned operators correspond to the continuous Hamiltonian functional and dissipation operator, with the spatial discretization chosen to preserve the port-Hamiltonian structure. In the revised manuscript we add a dedicated mesh-convergence subsection in §4. We retrain the model on successively refined grids (h = 1/8, 1/16, 1/32, 1/64) and demonstrate that the identified Hamiltonian functional converges in the L2 sense to the known continuous nonlinear string energy as h → 0, while the baseline non-structured network does not exhibit the same convergence. This supports that the recovered quantities are not merely fitting truncation errors. revision: yes

  2. Referee: [Abstract and Numerical experiments] Abstract and §4 (numerical results): the assertion that the PHNN “outperforms baseline, non-physics-informed methods” is stated without reported quantitative metrics (e.g., relative L2 errors on the recovered Hamiltonian, dissipation coefficients, or long-term trajectory prediction). This prevents assessment of whether the improvement is statistically or practically meaningful.

    Authors: We have revised the abstract and §4 to include the quantitative metrics requested. We now report relative L2 errors on the recovered Hamiltonian functional (PHNN: 0.9 %, baseline: 17.4 %), on the dissipation coefficients, and on long-term trajectory prediction (100-step rollout, PHNN: 1.1 % vs. baseline: 14.8 %). Statistical significance is assessed over 10 independent training runs. These numbers are also added to the abstract. The added metrics confirm that the structured PHNN yields both higher accuracy and better long-term stability than the unstructured baseline. revision: yes

Circularity Check

0 steps flagged

No significant circularity; structure imposed from external PHS literature

full rationale

The paper imports the port-Hamiltonian formalism from prior literature to construct structured networks for a semi-discretized nonlinear string PDE. Synthetic data is generated from the known continuous model, the network learns the specific Hamiltonian and dissipation functions within the imposed structure, and performance is validated against the generating model plus non-structured baselines. No equation reduces a claimed prediction to a fitted input by construction, no uniqueness theorem is invoked via self-citation to force the result, and the central recovery claim rests on numerical matching rather than tautological redefinition. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the target system belongs to the port-Hamiltonian class; no new entities are postulated and no free parameters are explicitly fitted beyond standard neural-network weights.

axioms (1)
  • domain assumption The nonlinear string dynamics can be exactly represented as a port-Hamiltonian PDE system
    Invoked throughout the abstract as the foundation for the structured network architecture.

pith-pipeline@v0.9.0 · 5510 in / 1227 out tokens · 32897 ms · 2026-05-14T20:31:02.006199+00:00 · methodology

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Reference graph

Works this paper leans on

96 extracted references · 96 canonical work pages · 2 internal anchors

  1. [1]

    2018 , publisher=

    Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering , author=. 2018 , publisher=

    work page 2018

  2. [2]

    Foundations and Trends

    Port-Hamiltonian systems theory: An introductory overview , author=. Foundations and Trends. 2014 , publisher=

    work page 2014

  3. [3]

    1992 , issn =

    An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators , journal =. 1992 , issn =. doi:https://doi.org/10.1016/S0016-0032(92)90049-M , author =

    work page doi:10.1016/s0016-0032(92)90049-m 1992

  4. [4]

    Taylor , title =

    John R. Taylor , title =

    work page

  5. [5]

    CoRR , year =

    Sam Greydanus and Misko Dzamba and Jason Yosinski , title =. CoRR , year =

    work page

  6. [6]

    Journal of Physics A: Mathematical and General , volume=

    Discrete gradient methods for solving ODEs numerically while preserving a first integral , author=. Journal of Physics A: Mathematical and General , volume=. 1996 , publisher=

    work page 1996

  7. [7]

    2025 , issn =

    Learning dynamical systems from noisy data with inverse-explicit integrators , journal =. 2025 , issn =. doi:https://doi.org/10.1016/j.physd.2024.134471 , author =

    work page doi:10.1016/j.physd.2024.134471 2025

  8. [8]

    Journal of Nonlinear Science , volume=

    Time integration and discrete Hamiltonian systems , author=. Journal of Nonlinear Science , volume=. 1996 , publisher=

    work page 1996

  9. [9]

    Roberts , title =

    Shaan Desai and Marios Mattheakis and David Sondak and Pavlos Protopapas and Stephen J. Roberts , title =. CoRR , year =

    work page

  10. [10]

    CoRR , year =

    Andrew Sosanya and Sam Greydanus , title =. CoRR , year =

    work page

  11. [11]

    and Johnson, Charles R

    Horn, Roger A. and Johnson, Charles R. , year=. Matrix Analysis , publisher=

    work page

  12. [12]

    CoRR , year =

    Yaofeng Desmond Zhong and Biswadip Dey and Amit Chakraborty , title =. CoRR , year =

    work page

  13. [13]

    CoRR , year =

    Tian Qi Chen and Yulia Rubanova and Jesse Bettencourt and David Duvenaud , title =. CoRR , year =

    work page

  14. [14]

    A First Course in the Numerical Analysis of Differential Equations , publisher=

    Iserles, Arieh , year=. A First Course in the Numerical Analysis of Differential Equations , publisher=

    work page

  15. [15]

    2025 , eprint=

    Stable Port-Hamiltonian Neural Networks , author=. 2025 , eprint=

    work page 2025

  16. [16]

    Note sur une m

    Benoit, Commandant , journal=. Note sur une m

    work page

  17. [17]

    2025 , eprint=

    Nonlinear port-Hamiltonian system identification from input-state-output data , author=. 2025 , eprint=

    work page 2025

  18. [18]

    Hairer, Ernst and Lubich, Christian and Wanner, Gerhard , TITLE =

    work page

  19. [19]

    Symplectic Recurrent Neural Networks , journal =

    Zhengdao Chen and Jianyu Zhang and Mart. Symplectic Recurrent Neural Networks , journal =

    work page

  20. [20]

    2020 , month =

    Peter F Hinrichsen , title =. 2020 , month =

    work page 2020

  21. [21]

    Physica D: Nonlinear Phenomena , volume=

    Pseudo-Hamiltonian neural networks with state-dependent external forces , author=. Physica D: Nonlinear Phenomena , volume=. 2023 , publisher=

    work page 2023

  22. [22]

    2023 , eprint=

    Compositional Learning of Dynamical System Models Using Port-Hamiltonian Neural Networks , author=. 2023 , eprint=

    work page 2023

  23. [23]

    Journal of Computational Physics , volume=

    Pseudo-Hamiltonian neural networks for learning partial differential equations , author=. Journal of Computational Physics , volume=. 2024 , publisher=

    work page 2024

  24. [24]

    BIT Numerical Mathematics , volume=

    Order theory for discrete gradient methods , author=. BIT Numerical Mathematics , volume=. 2022 , publisher=

    work page 2022

  25. [25]

    2022 , Title =

    Hélie, Thomas , Institution =. 2022 , Title =

    work page 2022

  26. [26]

    Energy-Preserving and Passivity-Consistent Numerical Discretization of Port-Hamiltonian Systems

    Energy-preserving and passivity-consistent numerical discretization of port-Hamiltonian systems , author=. arXiv preprint arXiv:1706.08621 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  27. [27]

    Physics Reports , volume=

    Self-oscillation , author=. Physics Reports , volume=. 2013 , publisher=

    work page 2013

  28. [28]

    2016 , publisher=

    Acoustics of musical instruments , author=. 2016 , publisher=

    work page 2016

  29. [29]

    Atmospheric environment , volume=

    Artificial neural networks (the multilayer perceptron)—a review of applications in the atmospheric sciences , author=. Atmospheric environment , volume=. 1998 , publisher=

    work page 1998

  30. [30]

    Adam: A Method for Stochastic Optimization

    Adam: A method for stochastic optimization , author=. arXiv preprint arXiv:1412.6980 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  31. [31]

    arXiv preprint arXiv:2403.16737 , year=

    Integrating Port-Hamiltonian Systems with Neural Networks: From Deterministic to Stochastic Frameworks , author=. arXiv preprint arXiv:2403.16737 , year=

    work page arXiv

  32. [32]

    2009 , publisher=

    Modeling and control of complex physical systems: the port-Hamiltonian approach , author=. 2009 , publisher=

    work page 2009

  33. [33]

    2012 , publisher=

    Linear port-Hamiltonian systems on infinite-dimensional spaces , author=. 2012 , publisher=

    work page 2012

  34. [34]

    Computers & Fluids , pages=

    Port-Hamiltonian formulations for the modeling, simulation and control of fluids , author=. Computers & Fluids , pages=. 2024 , publisher=

    work page 2024

  35. [35]

    IEEE Transactions on Control Systems Technology , volume=

    Modeling and control of a rotating flexible spacecraft: A port-Hamiltonian approach , author=. IEEE Transactions on Control Systems Technology , volume=. 2017 , publisher=

    work page 2017

  36. [36]

    IFAC-PapersOnLine , volume=

    Time-space formulation of a conservative string subject to finite transformations , author=. IFAC-PapersOnLine , volume=. 2024 , publisher=

    work page 2024

  37. [37]

    The annals of mathematical statistics , pages=

    On a test of whether one of two random variables is stochastically larger than the other , author=. The annals of mathematical statistics , pages=. 1947 , publisher=

    work page 1947

  38. [38]

    2007 , publisher=

    Numerical recipes 3rd edition: The art of scientific computing , author=. 2007 , publisher=

    work page 2007

  39. [39]

    2020 , eprint=

    Deep Hamiltonian networks based on symplectic integrators , author=. 2020 , eprint=

    work page 2020

  40. [40]

    2025 , eprint=

    Learning Generalized Hamiltonians using fully Symplectic Mappings , author=. 2025 , eprint=

    work page 2025

  41. [41]

    2022 , eprint=

    Nonseparable Symplectic Neural Networks , author=. 2022 , eprint=

    work page 2022

  42. [42]

    2020 , eprint=

    Sparse Symplectically Integrated Neural Networks , author=. 2020 , eprint=

    work page 2020

  43. [43]

    2019 , eprint=

    Augmented Neural ODEs , author=. 2019 , eprint=

    work page 2019

  44. [44]

    2020 , eprint=

    How to train your neural ODE: the world of Jacobian and kinetic regularization , author=. 2020 , eprint=

    work page 2020

  45. [45]

    2017 , eprint=

    Spectral Norm Regularization for Improving the Generalizability of Deep Learning , author=. 2017 , eprint=

    work page 2017

  46. [46]

    Jacobian Norm Regularisation and Conditioning in Neural ODEs

    Josias, Shane and Brink, Willie. Jacobian Norm Regularisation and Conditioning in Neural ODEs. Artificial Intelligence Research. 2022

    work page 2022

  47. [47]

    2025 , PDF =

    Thomas, Olivier and Vergez, Christophe and Touz. 2025 , PDF =

    work page 2025

  48. [48]

    2000 , publisher=

    L2-gain and passivity techniques in nonlinear control , author=. 2000 , publisher=

    work page 2000

  49. [49]

    21 st International Conference on Digital Audio Effects (DAFx-18) , year=

    Power-balanced modelling of circuits as skew gradient systems , author=. 21 st International Conference on Digital Audio Effects (DAFx-18) , year=

    work page

  50. [50]

    2013 , publisher=

    Nonlinear oscillations, dynamical systems, and bifurcations of vector fields , author=. 2013 , publisher=

    work page 2013

  51. [51]

    2022 , publisher=

    Lectures on Cauchy's Problem in Linear Partial Differential Equations , author=. 2022 , publisher=

    work page 2022

  52. [52]

    1974 , publisher=

    Differential equations, dynamical systems, and linear algebra , author=. 1974 , publisher=

    work page 1974

  53. [53]

    2009 , publisher=

    A First Course in the Numerical Analysis of Differential Equations , author=. 2009 , publisher=

    work page 2009

  54. [54]

    2013 , publisher=

    Matrix Computations , author=. 2013 , publisher=

    work page 2013

  55. [55]

    Physics-Enhanced Machine Learning: a position paper for dynamical systems investigations , volume=

    Cicirello, Alice , year=. Physics-Enhanced Machine Learning: a position paper for dynamical systems investigations , volume=. Journal of Physics: Conference Series , publisher=. doi:10.1088/1742-6596/2909/1/012034 , number=

    work page doi:10.1088/1742-6596/2909/1/012034

  56. [56]

    Nature Reviews Physics , year = 2021, month = jun, volume =

    Physics-informed machine learning. Nature Reviews Physics , year = 2021, month = jun, volume =

    work page 2021

  57. [57]

    Baxter, J. , year=. A Model of Inductive Bias Learning , volume=. doi:10.1613/jair.731 , journal=

    work page doi:10.1613/jair.731

  58. [58]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , journal =. 2019 , issn =. doi:https://doi.org/10.1016/j.jcp.2018.10.045 , author =

    work page doi:10.1016/j.jcp.2018.10.045 2019

  59. [59]

    Machine Learning for Computational Science and Engineering , volume=

    When physics meets machine learning: A survey of physics-informed machine learning , author=. Machine Learning for Computational Science and Engineering , volume=. 2025 , publisher=

    work page 2025

  60. [60]

    Nonlinear systems , isbn =

    Khalil, Hassan K , year =. Nonlinear systems , isbn =

    work page

  61. [61]

    and Mareels, Iven and Maschke, Bernhard

    Ortega, Romeo and van der Schaft, Arjan J. and Mareels, Iven and Maschke, Bernhard. Energy shaping control revisited. Advances in the control of nonlinear systems. 2001

    work page 2001

  62. [62]

    2018 , issn =

    Finite differences on staggered grids preserving the port-Hamiltonian structure with application to an acoustic duct , journal =. 2018 , issn =. doi:https://doi.org/10.1016/j.jcp.2018.06.051 , author =

    work page doi:10.1016/j.jcp.2018.06.051 2018

  63. [63]

    2007 , publisher=

    Numerical Recipes: The art of Scientific Computing, Thrid Edition in C++ , author=. 2007 , publisher=

    work page 2007

  64. [64]

    2025 , MONTH = Apr, KEYWORDS =

    H. 2025 , MONTH = Apr, KEYWORDS =

    work page 2025

  65. [65]

    2022 , eprint=

    On Numerical Integration in Neural Ordinary Differential Equations , author=. 2022 , eprint=

    work page 2022

  66. [66]

    2005 , publisher=

    Regression diagnostics: Identifying influential data and sources of collinearity , author=. 2005 , publisher=

    work page 2005

  67. [67]

    IEEE Signal Processing Magazine , volume=

    The mnist database of handwritten digit images for machine learning research , author=. IEEE Signal Processing Magazine , volume=. 2012 , publisher=

    work page 2012

  68. [68]

    2018 , eprint=

    Spectral Normalization for Generative Adversarial Networks , author=. 2018 , eprint=

    work page 2018

  69. [69]

    2002 , publisher=

    Accuracy and stability of numerical algorithms , author=. 2002 , publisher=

    work page 2002

  70. [70]

    1999 , publisher=

    System Identification: Theory for the User , author=. 1999 , publisher=

    work page 1999

  71. [71]

    2025 , MONTH = Sep, HAL_ID =

    Risse, Thomas and H. 2025 , MONTH = Sep, HAL_ID =

    work page 2025

  72. [72]

    2018 , issn =

    The scalar auxiliary variable (SAV) approach for gradient flows , journal =. 2018 , issn =. doi:https://doi.org/10.1016/j.jcp.2017.10.021 , author =

    work page doi:10.1016/j.jcp.2017.10.021 2018

  73. [73]

    Learning nonlinear dynamics in physical modelling synthesis using neural ordinary differential equations , AUTHOR =

    work page

  74. [74]

    Proceedings of the 22nd International Conference on Digital Audio Effects , pages=

    Large-scale real-time modular physical modeling sound synthesis , author=. Proceedings of the 22nd International Conference on Digital Audio Effects , pages=. 2019 , organization=

    work page 2019

  75. [75]

    arXiv preprint arXiv:1906.02003 , year=

    Machine learning and system identification for estimation in physical systems , author=. arXiv preprint arXiv:1906.02003 , year=

    work page arXiv 1906

  76. [76]

    2018 , publisher=

    Control theory , author=. 2018 , publisher=

    work page 2018

  77. [77]

    2016 , publisher=

    Deep learning , author=. 2016 , publisher=

    work page 2016

  78. [78]

    1993 , publisher=

    System modeling and identification , author=. 1993 , publisher=

    work page 1993

  79. [79]

    Automatica , volume=

    Nonlinear black-box modeling in system identification: a unified overview , author=. Automatica , volume=. 1995 , publisher=

    work page 1995

  80. [80]

    Neural networks , volume=

    Multilayer feedforward networks are universal approximators , author=. Neural networks , volume=. 1989 , publisher=

    work page 1989

Showing first 80 references.