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REVIEW 3 major objections 6 minor 34 references

DeepF-fNet: a physics-informed neural network for vibration isolation optimization

T0 review · 3 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read DeepF-fNet, a dual-network physics-informed surrogate, estimates locally resonant metamaterial parameters that place a vibration bandgap at a user-specified target frequency in real time.

desk verdict A plausible incremental extension of neural inverse operators to variable-frequency vibration isolation, but the headline speed claim and the SICE4 correction step need serious scrutiny before the paper's conclusions can be trusted. read the letter →

arxiv 2412.21132 v1 pith:FGQG45V6 submitted 2024-12-30 physics.comp-ph cs.LGeess.SP

A. Tollardo , F. Cadini , M. Giglio , L. Lomazzi This is my paper · ORCID
classification physics.comp-phcs.LGeess.SP
keywords physics-informedneuralnetworkDeepONetvibrationisolationlocallyresonantmetamaterialinverseeigenvalueproblembandgaptuningsemi-activecontrolgeneticalgorithm
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

The paper claims that a physics-informed neural network, trained once on finite-element simulations, can invert the bandgap-placement problem for a locally resonant metamaterial: given a target frequency to isolate, it outputs the geometric parameters that put a bandgap there. The authors pair this network, DeepF-fNet, with a prediction-correction scheme called SICE4 that shifts and rescales a stored dispersion curve to approximate the target spectrum before asking the network for parameters. They test the idea on a trichiral honeycomb resonator attached to a steel plate and report parameters close to those found by a genetic algorithm, while running roughly 500,000 times faster. The practical stake is semi-active vibration isolation: a device that can retune itself at control-loop rates instead of requiring minutes or hours of optimization.

What carries the argument

The load-bearing object is the SICE4 prediction-correction update on dispersion curves: a shift by $\Delta$ and a rescaling by Gamma that convert a stored spectrum into an approximate target spectrum without solving the direct eigenvalue problem. This is paired with DeepF-fNet's dual CNN architecture, where the IEPS inverts spectra into structural parameters and the WES enforces physical plausibility by evaluating the wave-equation residual and boundary-condition residual on the predicted mode shapes, making the inverse mapping respect the underlying physics.

What would settle it

A direct check: take the parameters predicted by SICE4 for a target frequency, run a fresh finite-element simulation of that unit cell, and compare the simulated dispersion curves with the corrected spectrum that was fed to the network. If the first bandgap of the simulated cell does not center near the target frequency for a substantial fraction of the tested targets, the shift-and-rescale assumption fails.

Watch

Extended reading notes

Core claim

DeepF-fNet is a dual convolutional network: the inverse eigenvalue problem solver (IEPS) maps a set of dispersion curves to the three unit-cell parameters r, L, and s, while the wave equation solver (WES) takes the discretized geometry defined by those parameters and predicts mode shapes. Training combines a data loss with two physics-informed terms, the harmonic wave-equation residual and the boundary-condition residual, so the inverse mapping is constrained by the governing elastodynamics. The SICE4 deployment algorithm selects from the training dataset the spectrum whose first bandgap is closest to the user's target frequency, shifts all curves except the first by the difference $\Delta$, rescales the first curve by Gamma = min(second curve)/max(first curve), and feeds the corrected spectrum to the IEPS. The paper reports that for a 200 Hz target this pipeline returns parameters essentially equivalent to those found by a genetic algorithm, with a first bandgap centered near the target, while consuming about 0.0157 s of CPU time versus 7690 s for the GA.

Load-bearing premise

The SICE4 correction step assumes that shifting a stored dispersion curve by $\Delta$ and rescaling its first branch by Gamma produces an input spectrum that behaves like a real metamaterial spectrum for the network; if that artificial spectrum lies outside the training distribution, the predicted parameters can be wrong even though the network itself is accurate on genuine spectra.

Editorial extensions

If this is right

  • A trained DeepF-fNet under SICE4 can estimate optimal unit-cell parameters for a time-varying target frequency at control-loop rates; the simulated 2 s, 150 Hz test signal tracks a 250-750-500 Hz sequence.
  • For a single 200 Hz target, SICE4 produces essentially the same first bandgap as a genetic algorithm while using about 500,000 times less CPU time, because training is performed offline.
  • The physics-informed losses allow the inverse mapping to be trained from a relatively small dataset of 2000 finite-element samples and to remain accurate within the parameter range covered by that dataset.
  • The first, lowest bandgap is predicted accurately even though higher eigenfrequencies exhibit spectral bias, so the method is directly usable for low-frequency single-bandgap isolation, while multi-bandgap targets need further development.
  • The framework positions semi-active vibration isolators as a practical replacement for fully active control systems in applications like automotive noise, vibration, and harshness, where rapid retuning is essential.

Reading between the lines

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

  • The shift-and-rescale correction in SICE4 is effectively an affine augmentation of the training manifold; a natural testable extension is to check whether corrected spectra lie close to genuine finite-element spectra in the IEPS latent space, and if not, to add a small number of physically realizable spectra near the target-bandgap manifold to close the distribution shift.
  • Because the method succeeds on the first bandgap, it likely transfers to other periodic structures, such as phononic plates or beams with attached resonators, whose bandgap formation is dominated by the lowest dispersion branches; the same SICE4 logic could be reused if the input spectra are normalized consistently.
  • The reported 500,000x speedup is a single-case comparison against one genetic algorithm implementation without parallelization or warm starting; the practical gain in deployment will be smaller once sensor readout, actuation, and safety margins are included, though the core advantage of offline training with online feed-forward inference remains.
  • The spectral bias at higher eigenfrequencies suggests that multi-bandgap targets will require a higher-fidelity operator, such as a Fourier neural operator, as the authors propose; a focused study on bandgaps above the first would test whether the correction step preserves accuracy there.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper introduces DeepF-fNet, a dual-CNN neural network that maps dispersion curves of a locally resonant metamaterial unit cell to the geometric parameters (r, L, s), and SICE4, a prediction-correction algorithm that transforms a stored dispersion curve to approximate a user-specified target bandgap frequency before querying the network. The intended application is real-time semi-active vibration isolation. The case study trains on 2000 finite-element-computed dispersion spectra, validates the inverse solver (IEPS) by feeding predicted parameters back to the FE solver and comparing the resulting dispersion curves, and compares SICE4 against a genetic algorithm (GA) for a single 200 Hz target. The authors report a speed advantage of up to 5e5 times and comparable first-bandgap accuracy.

Significance. The paper contains a credible validation of the first-bandgap prediction: predicted parameters are re-run through the FE solver and shown to reproduce the target bandgap, which rules out simple overfitting to training labels. The SICE4 heuristic of selecting the nearest stored spectrum and then applying a shift/scale correction is a practical idea for inverse eigenvalue problems in this setting. If the off-manifold generalization of SICE4 were demonstrated across a range of target frequencies, the framework would be a valuable tool for fast structural optimization. However, the current evidence supports only a narrow, single-frequency claim; the broad real-time claims in the abstract and conclusions are not yet backed by the experiments.

major comments (3)
  1. [Section 2.1, Algorithm 1 lines 6-10] The correction step constructs an input spectrum by shifting all branches n>1 by Δ and rescaling branch 1 by Γ = min(new DC_{n=2})/max(DC_{n=1}[ib]). This synthetic spectrum is not a physically realizable dispersion curve of any (r,L,s) triple, because eigenfrequency branches do not transform in this way under geometric changes. The IEPS was trained exclusively on FE-generated spectra from valid parameter triples (Section 3.2), so the corrected input lies on a different manifold. The paper's only FE verification of the full SICE4 pipeline is a single 200 Hz target compared with a single GA run (Table 7, Fig. 10). The time-series simulation in Fig. 9 outputs parameters for 300 targets between 250 and 750 Hz but does not re-run the FE solver to confirm that the resulting bandgaps actually bracket those targets. To support the central claim of real-time optimal parameter estimation over a user-defined frequency range, the authors should FE-verify a representative sample of target frequencies, or otherwise quantify the network's error on artificial SICE4 inputs versus genuine FE spectra.
  2. [Section 3.3, Table 7] The speed comparison is asymmetric. The SICE4 inference time of 0.0157 s excludes the offline cost of generating the 2000-sample FE dataset and training both networks, whereas the reported GA time of 7690 s is the full optimization time. Offline costs can legitimately be amortized in real-time deployment, but the 'up to 500,000 times faster' statement in the abstract and Section 4 should be presented as an inference-time comparison. Moreover, the GA settings (population size, number of generations, objective function definition) are not specified, so the 7690 s figure cannot be assessed or reproduced.
  3. [Section 2.2, Table 5] The PDE loss weight is 10^-14 and the BC loss weight is 10^-12 for both networks, while the data loss weights are 10^8 and 10^12. With these settings the physics terms contribute negligibly to the total loss, so the framework is effectively a purely data-driven surrogate. The paper's claim that 'incorporating physical laws into training reduces data requirements and improves accuracy' (Section 4) is therefore not supported by the experiments. The authors should either show an ablation with meaningful physics weights, or revise the characterization of the method as physics-informed.
minor comments (6)
  1. [Section 3.1] 'Its width (along the y-axis) was optimized' appears to be a typo; the width is along the x-axis according to Eq. (4c) and Fig. 3a.
  2. [Section 3.1, Eq. (6)] The Floquet-Bloch condition is written for a vector w0 but the surrounding text refers to a scalar displacement; please make the notation consistent.
  3. [Section 2.2] The paper describes DeepF-fNet as based on DeepONets, but the architecture in Table 3 is a standard CNN without a branch/trunk structure. Clarify the connection to DeepONets or avoid the term.
  4. [Section 3.3, Fig. 5(e)] The compliance conditions are said to be 'violated for some combinations of parameters,' but it is not specified how these violations are treated during training or what 'empirically observed' means quantitatively; please clarify.
  5. [Algorithm 1] The notation DC, f_bg, and ib is not fully defined; in particular, the index ib runs over dataset samples while n indexes branches. Please define all variables explicitly.
  6. [Section 3.3, Table 7] The GA comparison uses a single run and a single target frequency; please report the GA's population size, generation count, and the objective function, and ideally repeat with several random seeds to provide error bars.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the inverse network is validated against external FE recomputation; SICE4's synthetic-spectrum concern is a generalization issue, not a circular derivation.

full rationale

I walked the claimed derivation chain. The central claim is that SICE4/DeepF-fNet maps a user-specified target frequency to optimal unit-cell parameters. This is an empirical surrogate-inversion claim, not a derivation, and the paper's own evaluations are external to the training labels. The IEPS is trained on FE-generated spectra with labels (r, L, s), and the reported validation recomputes dispersion curves from the predicted parameters using the COMSOL FE solver (Fig. 6 and Tab. 6), which is an independent check of the surrogate. The SICE4 correction step (Algorithm 1, lines 6-10) constructs an input spectrum by shifting and rescaling a database spectrum, and the IEPS then predicts parameters; this is a heuristic input transformation, not a fitted parameter renamed as a prediction. The full-pipeline 200 Hz case is verified by FE recomputation and compared against a genetic algorithm (Tab. 7 and Fig. 10), again an external baseline. No equation in the paper uses the target frequency as a fitted constant, and no output is equivalent to an input by construction. The paper also explicitly discloses limitations: spectral bias at higher eigenfrequencies, dataset-size constraints, and the need for future experimental validation. The skeptic's concerns about out-of-distribution synthetic spectra, the single-frequency FE check, and the very low PDE/BC loss weights are genuine validity and generalization risks, but they are not circularity: there is no self-citation chain, no uniqueness theorem imported from the authors, no ansatz smuggled in via citation, and no fitted parameter predicted as a new result. I therefore find no significant circularity.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central numerical result is a supervised surrogate trained on FE-generated dispersion curves. Its key trust assumptions are the validity of the FE ground truth, the Floquet-Bloch reduction, and the SICE4 correction step that feeds possibly non-physical spectra into the network. No new physical entities such as particles, forces, or dimensions are introduced.

free parameters (4)
  • Loss function weights = w_data_IEPS=10^8; w_data_WES=10^12; w_PDE=10^-14; w_BC=10^-12
    Chosen by hand in Table 5; the PDE and BC weights are orders of magnitude smaller than the data weights, so the physics-informed component is numerically minimal and its claimed benefit is not demonstrated.
  • Network hyperparameters = Learning rates 1e-3 and 1e-2; dropout 0.5; norm clip 1; batch size 40; 40 epochs; CNN filter sizes and layer counts
    Selected without a reported tuning procedure or sensitivity study. These choices affect whether the trained mapping is accurate enough for the claimed speed and accuracy.
  • Eigenfrequency count and wave-number count = N=10, K=40
    Defines the input and output discretization for the IEPS and WES. No resolution study is reported for these sampling counts.
  • Random parameter sampling bounds = r, L, s in [0.04 mm, 4 mm]
    Defines the domain over which the inverse mapping is trained. Optimality claims are restricted to this box and to the compliance constraints of Equation 3.
assumptions (6)
  • standard math Floquet-Bloch theorem applies to the periodic LRM unit cell and reduces the analysis to the first Brillouin zone.
    Invoked in Section 3.1, Equation 6. If the periodic medium is not ideal or has damping, the computed dispersion curves may not represent the physical structure.
  • domain assumption The 2D plane-strain extruded model captures the 3D behavior of the metamaterial.
    Stated in Section 3.1. The out-of-plane dimension is fixed and the model is extruded, but no comparison to a full 3D simulation or experiment is given.
  • domain assumption The effective material properties of the chiral honeycomb in Equation 5 are valid.
    Taken from reference [29]. The inverse problem and the FE dataset generation both rely on these homogenized expressions for density, modulus, and Poisson ratio.
  • domain assumption The COMSOL eigenfrequency study with a Finer physics-controlled mesh and ARPACK solver provides converged ground-truth dispersion curves.
    The dataset and all validation tests depend on this FE solver output. A convergence study is mentioned but no convergence curves or mesh statistics are shown.
  • ad hoc to paper Choosing the training spectrum whose first-bandgap center is closest to the target frequency, then shifting and rescaling it, is a sufficient initialization for finding optimal parameters.
    This is the SICE4 correction rule in Algorithm 1. It is a heuristic that couples the target frequency to a surrogate spectrum and is not derived from the physics or from a formal optimality condition.
  • ad hoc to paper A neural network trained on physically realizable FE spectra will generalize to the artificially shifted and rescaled spectra produced by SICE4.
    The IEPS is trained on genuine spectra, but SICE4 feeds it corrected spectra that may not correspond to any real geometry. This generalization is assumed without an out-of-distribution study.

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Cite this review

Pith. "Pith review of DeepF-fNet: a physics-informed neural network for vibration isolation optimization." pith.science (2026). https://pith.science/paper/FGQG45V6

@misc{pith2026241221132,
  author       = {Pith},
  title        = {Pith review of: DeepF-fNet: a physics-informed neural network for vibration isolation optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGQG45V6}},
  note         = {Machine review of arXiv:2412.21132}
}
read the original abstract

Structural optimization is essential for designing safe, efficient, and durable components with minimal material usage. Traditional methods for vibration control often rely on active systems to mitigate unpredictable vibrations, which may lead to resonance and potential structural failure. However, these methods face significant challenges when addressing the nonlinear inverse eigenvalue problems required for optimizing structures subjected to a wide range of frequencies. As a result, no existing approach has effectively addressed the need for real-time vibration suppression within this context, particularly in high-performance environments such as automotive noise, vibration and harshness, where computational efficiency is crucial. This study introduces DeepF-fNet, a novel neural network framework designed to replace traditional active systems in vibration-based structural optimization. Leveraging DeepONets within the context of physics-informed neural networks, DeepF-fNet integrates both data and the governing physical laws. This enables rapid identification of optimal parameters to suppress critical vibrations at specific frequencies, offering a more efficient and real-time alternative to conventional methods. The proposed framework is validated through a case study involving a locally resonant metamaterial used to isolate structures from user-defined frequency ranges. The results demonstrate that DeepF-fNet outperforms traditional genetic algorithms in terms of computational speed while achieving comparable results, making it a promising tool for vibration-sensitive applications. By replacing active systems with machine learning techniques, DeepF-fNet paves the way for more efficient and cost-effective structural optimization in real-world scenarios.

Figures

Figures reproduced from arXiv: 2412.21132 by the authors.

Figure 1
Figure 1. Example of dispersion curves (blue solid lines), with the highlig [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. DeepF-fNet architecture outputs of both models in a physically coherent manner throughout the train￾ing phase. The comprehensive loss function is composed of three distinct components: a data-driven element, termed the data loss, and two physics￾informed elements, encompassing the partial differential equation (PDE) loss and the boundary condition (BC) loss. Together, these elements contribute to the total loss func… view at source ↗
Figure 3
Figure 3. LRM geometry: (a) unit cell, (b) ALR’s unit cell and (c) perio [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Training and validation losses of the IEPS and WES, compute [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Total IEPS and WES loss functions with their physics-infor [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Graphic validation of the IEPS model with input dispersion cu [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Graphic validation of the WES model for each component of [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Graphic validation of the WES model for each component of [PITH_FULL_IMAGE:figures/full_fig_p021_8.png]
Figure 9
Figure 9. Figure 9: Simulation of application of the SICE4 algorithm to a real wor [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]
Figure 10
Figure 10. Figure 10: Application of the SICE4 and GA algorithms to the single tar [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]

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Reference graph

Works this paper leans on

34 extracted references · 20 canonical work pages

  1. [1]

    Bia/suppress las, Comparison of passive and active reduction of vibrations of mechanical systems, Journal of Achievements in Materials and Man u- facturing Engineering 18 (08 2006)

    K. Bia/suppress las, Comparison of passive and active reduction of vibrations of mechanical systems, Journal of Achievements in Materials and Man u- facturing Engineering 18 (08 2006)

    work page 2006

  2. [2]

    GARDONIO, S

    P. GARDONIO, S. ELLIOTT, Passive and active isolation of struct ural vibration transmission Journal of Sound and Vibration 237 (3) (2000) 483–511. doi:https://doi.org/10.1006/jsvi.2000.3064. URL https://www.sciencedirect.com/science/article/pii/S0022460X00930645

    work page arXiv 2000

  3. [3]

    Y. Liu, T. Waters, M. Brennan, A comparison of semi-active damping control strategies for vibration Journal of Sound and Vibration 280 (1) (2005) 21–39. doi:https://doi.org/10.1016/j.jsv.2003.11.048. URL https://www.sciencedirect.com/science/article/pii/S0022460X04000252

    work page doi:10.1016/j.jsv.2003.11.048 2005

  4. [4]

    Karnopp, M

    D. Karnopp, M. J. Crosby, R. Harwood, Vibration control using semi- active force generators (1974)

    work page 1974

  5. [5]

    Crosby, D

    M. Crosby, D. C. Karnopp, The active damper: a new concept fo r shock and vibration control, Shock and Vibration Bulletin 43 (4) (1973) 119 – 133

    work page 1973

  6. [6]

    M. T. Chu, Inverse eigenvalue problems, SIAM Review 40 (1) (1998) 1–39. arXiv:https://doi.org/10.1137/S0036144596303984, doi:10.1137/S0036144596303984. URL https://doi.org/10.1137/S0036144596303984 24

    doi:10.1137/s0036144596303984 1998

  7. [7]

    G. M. Gladwell, Inverse Problems in Vibration, 2nd Edition, Kluwer Academic Publishers, New York, Boston, Dordrecht, London, Mos cow, 2004

    work page 2004

  8. [8]

    Mishra, R

    S. Mishra, R. Molinaro, Estimates on the generalization error of p hysics informed neural networks (2023). arXiv:2007.01138. URL https://arxiv.org/abs/2007.01138

    arXiv 2023

Show all 34 references
  1. [9]

    Zargham, T

    S. Zargham, T. Ward, R. Ramli, Topology optimization: a review for structural designs under vibra tion problems, Structural and Multidisciplinary Optimization 53 (2016) 1157–1177. doi:https://doi.org/10.1007/s00158-015-1370-5 . URL https://link.springer.com/article/10.1007/s00...

    2016 doi

  2. [10]

    Abdeljaber, O

    O. Abdeljaber, O. Avci, D. J. Inman, Optimization of chiral lattice based metastructures for broadban d vibration suppression using Journal of Sound and Vibration 369 (2016) 50–62. doi:https://doi.org/10.1016/j.jsv.2015.11.048. URL https://www.sciencedirect.com/science/article...

    2016 doi

  3. [11]

    Madeira, H

    J. Madeira, H. Pina, H. Rodrigues, Plate eigenfrequency opti- mization with genetic algorithms and random keys, Vol. 88, 2008. doi:10.4203/ccp.88.73

    2008 doi

  4. [12]

    Hornik, M

    K. Hornik, M. Stinchcombe, H. White, Multilayer feedforward networks are universal approximators, Neural Networks 2 (5) (1989) 359–366. doi:https://doi.org/10.1016/0893-6080(89)90020-8. URL https://www.sciencedirect.com/science/article/pii/0893608089900208

    1989

  5. [13]

    Raissi, P

    M. Raissi, P. Perdikaris, G. Karniadakis, Physics-informed neural networks: A deep learning framework fo r solving forward and inverse Journal of Computational Physics 378 (2019) 686–707. doi:https://doi.org/10.1016/j.jcp.2018.10.045. URL https://www.sciencedirect.com/science/...

    2019 doi

  6. [14]

    Haghighat, M

    E. Haghighat, M. Raissi, A. Moure, H. Gomez, R. Juanes, A physics-informed deep learning framework for inversion and surr ogate modeling in solid mec Computer Methods in Applied Mechanics and Engineering 379 (2021) 25 113741. doi:https://doi.org/10.1016/j.cma.2021.113741. URL ...

    2021

  7. [15]

    A. D. Jagtap, Z. Mao, N. Adams, G. E. Karniadakis, Physics-informed neural networks for inverse problems in supers onic flows, Journal of Computational Physics 466 (2022) 111402. doi:https://doi.org/10.1016/j.jcp.2022.111402. URL https://www.sciencedirect.com/science/article/pi...

    2022

  8. [16]

    S. Hu, M. Liu, S. Zhang, S. Dong, R. Zheng, Physics-informed neural network combined with characteristic-b ased split for solving forward Neurocomputing 573 (2024) 127240. doi:https://doi.org/10.1016/j.neucom.2024.127240 URL https://www.sciencedirect.com/science/article/pii/S0...

    2024

  9. [17]

    L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlinear operators via deeponet based on the universal approximation theorem of op Nature Machine Intelligence 3 (3) (2021) 218–229. doi:10.1038/s42256-021-00302-5 . URL http://dx.doi.org/10.1038/s42256-021-00302-5

    2021 doi

  10. [18]

    J. He, S. Koric, D. Abueidda, A. Najafi, I. Jasiuk, Geom-deeponet: A point-cloud-based deep operator network fo r field predictions on 3d parameterized Computer Methods in Applied Mechanics and Engineering 429 (2024) 117130. doi:https://doi.org/10.1016/j.cma.2024.117130. URL htt...

    2024

  11. [19]

    Molinaro, Y

    R. Molinaro, Y. Yang, B. Engquist, S. Mishra, Neural inverse operators for solving pde inverse problems (2023) . arXiv:2301.11167. URL https://arxiv.org/abs/2301.11167

    2023 arXiv

  12. [20]

    Soleimanian, G

    S. Soleimanian, G. Petrone, F. Franco, S. De Rosa, P. Ko/suppress lakowski, Semi-active vibro-acoustic control of vehicle transmission system s using a metal rubber-based Applied Acoustics 217 (2024) 109861. doi:https://doi.org/10.1016/j.apacoust.2024.109861. URL https://www.s...

    2024

  13. [21]

    URL http://www.comsol.com/products/multiphysics/ 26

    COMSOL-Inc., Comsol multiphysics ® v 6.1 (2024). URL http://www.comsol.com/products/multiphysics/ 26

    2024

  14. [22]

    D. P. Kingma, J. Ba, Adam: A method for stochastic optimization (2017). arXiv:1412.6980. URL https://arxiv.org/abs/1412.6980

    2017 arXiv

  15. [23]

    Abadi, A

    M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Good- fellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaise r, M. Kudlur, J. Levenberg, D. Man´ e, R. Monga, S. Moore, D. Mur- ray, C. Ola...

    2015

  16. [24]

    S. Ioffe, C. Szegedy, Batch normalization: Accelerating deep n etwork training by reducing internal (2015). arXiv:1502.03167. URL https://arxiv.org/abs/1502.03167

    2015 arXiv

  17. [25]

    Pascanu, T

    R. Pascanu, T. Mikolov, Y. Bengio, Understanding the exploding gradient problem, ArXiv abs/1211.506 3 (2012). URL https://api.semanticscholar.org/CorpusID:16074905

    2012

  18. [26]

    Jung, H.-G

    J. Jung, H.-G. Kim, S. Goo, K.-J. Chang, S. Wang, Realisation of a locally resonant metamaterial on the automobile pane l structure to reduce noise Mechanical Systems and Signal Processing 122 (2019) 206–231. doi:https://doi.org/10.1016/j.ymssp.2018.11.050. URL https://www.sci...

    2019 doi

  19. [27]

    X. Yu, J. Zhou, H. Liang, Z. Jiang, L. Wu, Mechanical metamaterials associated with stiffness, rigidity and com pressibility: A brief review Progress in Materials Science 94 (2018) 114–173. doi:https://doi.org/10.1016/j.pmatsci.2017.12.003. URL https://www.sciencedirect.com/sci...

    2018 doi

  20. [28]

    Baravelli, M

    E. Baravelli, M. Ruzzene, Internally resonating lattices for ban dgap generation and low-frequency Journal of Sound and Vibration 332 (25) (2013) 6562–6579. doi:https://doi.org/10.1016/j.jsv.2013.08.014. URL https://www.sciencedirect.com/science/article/pii/S0022460X13006792 27

    2013 doi

  21. [29]

    Mousanezhad, B

    D. Mousanezhad, B. Haghpanah, R. Ghosh, A. M. Hamouda, H. Nayeb-Hashemi, A. Vaziri, Elastic properties of chiral, anti-chiral, and hierarchical honeycom bs: A simple energy-based approac Theoretical and Applied Mechanics Letters 6 (2) (2016) 81–96. doi:https://doi.org/10.1016/...

    2016 doi

  22. [30]

    C. C. Claeys, K. Vergote, P. Sas, W. Desmet, On the potential of tuned resonators to obtain low-frequency vib rational stop bands in periodic Journal of Sound and Vibration 332 (6) (2013) 1418–1436. doi:https://doi.org/10.1016/j.jsv.2012.09.047. URL https://www.sciencedirect.c...

    2013 doi

  23. [31]

    Collet, M

    M. Collet, M. Ouisse, M. Ruzzene, M. Ichchou, Floquet–bloch decomposition for the computation of dispersion of t wo-dimensional periodic, damp International Journal of Solids and Structures 48 (20) (2011) 2837–2848. doi:https://doi.org/10.1016/j.ijsolstr.2011.06.002. URL https...

    2011 doi

  24. [32]

    Sayin, genetic-algorithm-matlab (2012)

    A. Sayin, genetic-algorithm-matlab (2012). URL https://github.com/alpsayin/genetic-algorithm-matlab?tab=readme-ov-file

    2012

  25. [33]

    Z. Li, D. Huang, B. Liu, A. Anandkumar, Fourier neural operator with learned deformations for pdes on ge neral geometries, Journal of Machine Learning Research 24 (388) (2023) 18593–18 618. doi:https://doi.org/10.5555/3648699.3649087. URL https://arxiv.org/abs/2207.05209

    2023

  26. [34]

    DeepF-ONet_Architecture.png

    Z. Li, N. Kovachki, K. Azizzadenesheli, B. Liu, K. Bhattacharya, A. Stuart, A. Anandkumar, Fourier neural operator for parametric partial differential equa tions (2021). arXiv:2010.08895. URL https://arxiv.org/abs/2010.08895 28 This figure "DeepF-ONet_Architecture.png" is avail...

    2021 arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.