Pith. sign in

REVIEW 4 major objections 5 minor 38 references

Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networks

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper derives a frequency-domain acoustic equation for ducts with varying mean flow and temperature, and shows that a physics-informed neural network solves it for both acoustic pressure and particle velocity up to 2000 Hz with…

desk verdict The PINN machinery works, but the governing equation it solves appears to be derived from an isentropic mean flow that the paper's own temperature profile violates—so the 1e-5 matches only show the network solves that equation, not the physical problem. read the letter →

arxiv 2507.22370 v1 pith:RVFEZP6E submitted 2025-07-30 cs.LG cs.SDeess.AS

D. Veerababu , Prasanta K. Ghosh This is my paper
classification cs.LGcs.SDeess.AS MSC 76Q0568T07 PACS 43.20.Ma
keywords physics-informedneuralnetworksductacousticscomplex-valuedacousticfieldtemperaturegradientmeanflowtransferlearningautomaticdifferentiationone-dimensionalwaveequation
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

This paper derives the frequency-domain equation for sound propagation in a one-dimensional duct whose mean temperature and velocity vary along its length, then solves it with a feedforward neural network instead of a classical solver. The network predicts both acoustic pressure and particle velocity up to 2000 Hz, matching the MATLAB boundary-value Runge-Kutta solver bvp4c with maximum relative errors on the order of $10^{-5}$ for both linear and sinusoidal temperature profiles. The work matters because duct acoustics in gas-turbine combustors involves exactly this kind of heterogeneous, mean-flow-carrying medium; a network that reproduces the complex-valued acoustic field without training data is a step toward higher-dimensional physics-constrained formulations. The paper also shows that a separate network trained by transfer learning recovers particle velocity from the momentum equation alone, and that imposing a temperature gradient changes both the spatial growth and the resonance of the acoustic amplitude.

What carries the argument

The load-bearing object is the derived second-order governing equation for complex acoustic pressure, whose coefficients zeta1, zeta2, and zeta3 are given in Eqs. (28)-(30). These coefficients encode the mean-flow Mach number M, its gradient dM/dx, the logarithmic density gradient alpha, and the second logarithmic derivative beta, all computed from an inlet condition and a prescribed axial temperature profile under a first-order Maclaurin approximation of the steady momentum equation. Two pieces of machinery turn this ODE into a trainable loss: the trial-solution method, which enforces the boundary conditions exactly by construction, and a real/imaginary split of the complex equation that makes the loss function real-valued. The network itself is a seven-layer feedforward net with sine activations and 90 neurons per hidden layer; automatic differentiation supplies the first and second derivatives needed to evaluate the residual, and a transfer-learned second network recovers particle velocity from the momentum equation.

What would settle it

Run the same neural-network formulation against an exact analytical solution for a duct with a steep temperature gradient at higher Mach number, where |M $\alpha$| is no longer small compared to k, and compare both the network and bvp4c against the exact solution; if the approximation is invalid, the deviation from the exact solution will exceed the reported O($10^{-5}$) even though the network and bvp4c agree with each other.

Watch

Extended reading notes

Core claim

The paper claims that the acoustic field in a one-dimensional uniform duct with axially varying mean flow and temperature obeys a single second-order linear ordinary differential equation with complex coefficients, Eq. (27), and that a feedforward neural network can solve this equation to high accuracy without any simulation or experimental data. The authors split the complex ODE into two real-valued equations, embed the boundary conditions directly into a trial solution, and minimize the residual loss at 10,000 collocation points using the L-BFGS optimizer. Against the bvp4c solution of the same equation, the network predicts acoustic pressure with relative errors between $10^{-7}$ and $10^{-5}$ across 500-2000 Hz for both temperature profiles; particle velocity, whose magnitude is several orders smaller, is recovered from the pressure field with similar accuracy. They further claim that the temperature gradient increases acoustic amplitude along the duct and shifts resonance frequencies, and that the formulation demonstrates the utility of automatic differentiation and transfer learning for acoustic applications.

Load-bearing premise

The derived governing equation, Eq. (27), is taken to be the true physics of the duct: specifically, the small-gradient expansion |M $\alpha$| << k in Eq. (17), the neglect of Mach-number terms beyond $M^{2}$ in Eq. (19), and the first-order Maclaurin approximation of the steady momentum equation in Eq. (33) must all hold for the regime being modeled; if they fail, the neural network could match bvp4c perfectly while both solve the wrong equation.

Editorial extensions

If this is right

  • One trained network predicts both acoustic pressure and particle velocity even though their magnitudes differ by several orders, so quantities like acoustic intensity, impedance, and transfer functions can be computed directly from the network.
  • Because the network is a differentiable function of position, automatic differentiation can extract other acoustic state variables without re-solving the governing equation.
  • A transfer-learned second network recovers particle velocity from the momentum equation alone, on a coarser collocation grid, and can be stored and recalled without retaining the pressure field.
  • Imposing a temperature gradient changes the acoustic field qualitatively: peak amplitude grows along the duct and resonance frequencies shift, consistent with exact solutions in the literature.
  • The trial-solution formulation extends naturally to higher dimensions or more complicated physics because it only requires a residual evaluation and boundary-embedded trial functions.

Reading between the lines

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

  • Because the reported errors are measured against the same derived equation solved by bvp4c, the paper demonstrates solver agreement rather than physical-model validity; an exact analytical solution or an experiment would be needed to test the equation itself.
  • The small-gradient assumption |M alpha| << k bounds the valid regime; a testable extension is to increase Mach number or temperature-gradient steepness until the errors degrade, mapping the method's operational envelope.
  • The trial-solution structure ties each network to one domain and one set of boundary conditions, a limitation the authors state; conditioning the network on frequency as an additional input would let a single model sweep the acoustic spectrum without retraining.
  • The success on two temperature profiles suggests the network could serve as a fast surrogate inside combustor thermoacoustic instability solvers, where repeated acoustic-field evaluations at varying temperature states are needed.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper derives a second-order ordinary differential equation, Eq. (27), for the complex acoustic pressure in a one-dimensional duct carrying a mean flow with an axial temperature gradient, under assumptions of inviscid perfect-gas flow and isentropic acoustics. The equation is solved with a feedforward neural network: a trial solution enforces the boundary conditions, the complex ODE is split into real and imaginary parts, and a residual-based loss is minimized with L-BFGS. Results are reported for two temperature profiles (linear and sinusoidal) and four frequencies (500?2000 Hz), with the neural-network solution compared against the MATLAB bvp4c solver; relative errors are on the order of 10^-5 to 10^-7 for acoustic pressure and particle velocity. The paper also describes a transfer-learning variant for particle velocity and a uniform-flow limit check.

Significance. If the governing equation were correct, the paper would make a useful contribution by demonstrating a physics-informed neural-network approach for a complex-valued duct-acoustics problem with non-uniform mean flow and temperature, and the reported O(10^-5) errors would be a strong numerical result. The trial-solution construction for enforcing boundary conditions, the real/imaginary splitting of a complex ODE, and the use of automatic differentiation for particle velocity are all sensible methodological elements, and the uniform-flow reduction in Eq. (62) is a good consistency check. However, the central physical derivation is flawed: the mean-state relation used to derive the governing equation is inconsistent with the mean-flow model solved in the paper. Because all numerical results are solutions of this same equation, the significance of the paper is not currently established. The transfer-learning claim also lacks the promised quantitative demonstration.

major comments (4)
  1. [Section 2, Eq. (7)] The relation dρ/dx = (ρ/(γp)) dp/dx is derived in Appendix A from the assumption p/ρ^γ = constant for the mean state, but the mean state computed in Section 2 via Eqs. (33)-(38) does not satisfy that assumption. For the linear profile with T0=1600 K, TL=800 K, p0=1e5 Pa, and M0=0.2, p/ρ^γ changes by a factor of roughly 2.7 between x=0 and x=1. Consequently Eq. (7) is false: at x=1 the exact density gradient from p=ρRT is approximately 0.44 kg/m^4, whereas the right-hand side of Eq. (7) is approximately 0.009 kg/m^4, a discrepancy of about a factor of 50. More fundamentally, substituting Eq. (14) into Eq. (7) gives α = M^2 α, which forces α=0 for M≠1; the two equations are mutually inconsistent. Since Eq. (7) is used to obtain Eq. (8), the coefficients ζ1-ζ3 in Eq. (27) do not represent the linearized acoustics of the stated mean flow. In particular, the u' ∂ρ/∂x term in Eq. (4) is under-represented by roughly a factor of 1/M^2 ≈ 25 for M=0.2. This is a load-bearing error because every reported result is a solution of Eq. (27).
  2. [Section 4.1, Tables 2-5] The validation is circular with respect to the physical model. The 'true solution' is the bvp4c solution of Eq. (27), which is the very equation whose residual defines the neural-network loss. The O(10^-5) errors in Tables 2-5 therefore demonstrate only that the neural network solves Eq. (27) accurately; they do not provide evidence that Eq. (27) is the correct equation for the acoustic field. The uniform-flow reduction in Section 4.3 is a necessary consistency check but cannot detect errors that vanish when α=0 and dM/dx=0. Please validate the nonuniform results against an independent solution of the linearized Euler equations (for example, direct numerical integration of Eqs. (4)-(5) without the isentropic mean-state assumption) or against an exact analytical solution for a temperature gradient.
  3. [Section 4.1, Tables 2-5] All reported relative errors are from a single training run, with no error bars, no statistics over multiple initializations, no convergence curves, and no statement of the number of L-BFGS iterations or stopping criterion. The central quantitative claim of O(10^-5) maximum relative error is therefore not established as robust. Since the reported errors span 3e-7 to 8e-5, a few additional runs with different random seeds and a report of mean±standard deviation would be needed to support the accuracy claim.
  4. [Section 4.2, Eqs. (59)-(61)] The transfer-learning variant for particle velocity is described in detail, and the text states that 'both methods yield similar results,' but no quantitative comparison, figure, or table is provided for this method. Since the abstract and the conclusion list transfer learning as a demonstrated contribution, this claim is currently unsupported. Please include the promised quantitative comparison or remove the transfer-learning claim from the abstract and conclusions.
minor comments (5)
  1. [Section 2, Eqs. (8)-(9)] The time-harmonic convention is inconsistent: the paper states ρ', p', u' ∝ e^{-jωt}, but Eqs. (8) and (9) appear to have been derived with the opposite sign convention (e^{+jωt}). Please make the sign convention consistent throughout, or explicitly state the convention used in the derivation.
  2. [Appendix B.4, Eq. (B.11)] Equation (B.11) contains a typo: the first term on the right-hand side should be (1/c) du/dx rather than (1/c) dM/dx.
  3. [Conclusion, point (1)] The statement that a single neural network architecture predicts both acoustic pressure and particle velocity is misleading: the particle velocity reported in Figs. 7 and 8 is evaluated algebraically from the predicted pressure using Eq. (57), not predicted by the same network. The transfer-learning variant uses a separate network with its own parameters.
  4. [Section 4.3] The text states that the uniform-flow case was validated with a maximum relative error of 0.01% but explicitly excludes the results 'to avoid redundancy.' Either include this comparison in a figure or table, or remove the numerical value, since an unreported validation result cannot be assessed.
  5. [Section 2, Eq. (33)] The description of Eq. (33) as a 'first-order approximation of the Maclaurin series' is imprecise. With the steady mass-conservation relation ρu = ρ0u0, the integrated momentum equation p + ρ0u0u = constant is exact, not an approximation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the neural network is trained on the residual of the derived ODE, not on the bvp4c solution, so the reported agreement is a genuine numerical verification of the solver rather than a fit to the benchmark.

full rationale

The central comparison is a numerical-convergence check, not a constructional identity. Section 4.1 states that the 'true solution' is obtained from 'the traditional Runge-Kutta solver bvp4c in MATLAB', while the training loss in Eqs. (48)-(50) is built only from the residual of Eq. (27) via the trial solution in Eqs. (51)-(52); the bvp4c solution is never used as a training target. The particle-velocity comparison is also non-circular: the predicted velocity is obtained from the network pressure through Eq. (58), and the 'true' velocity from the bvp4c pressure through the same algebraic relation Eq. (57), so the two quantities are not equal by construction but inherit the agreement of the underlying pressure fields. The paper does not invoke a uniqueness theorem, an ansatz smuggled through the authors' own prior work, or a fitted parameter renamed as a prediction; the only self-citations (refs. 3 and 4) are motivational speech-recognition examples and are not load-bearing. One physical-modeling concern, flagged by the skeptic, is real but not circular: Eq. (7), dρ/dx = (ρ/γp) dp/dx, is derived from the isentropic relation p/ρ^γ = constant, whereas the mean-flow profiles computed from Eqs. (36)-(38) are not isentropic, making the coefficients of Eq. (27) potentially inconsistent with the paper's own mean state. This is a correctness and validation-interpretation risk, because both the network and bvp4c solve the same derived ODE, but it does not reduce the prediction to its inputs by construction. No specific circular step can therefore be quoted.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central numerical experiment depends only on standard math, the approximate low-Mach duct equation, and hand-chosen network hyperparameters. No new physical entities are introduced, and the only fitted quantities are the neural network weights themselves, which are the solution representation rather than a model of the physics. The unexamined choices are the network hyperparameters and the validation against bvp4c.

free parameters (1)
  • Neural network architecture and training hyperparameters = 7 hidden layers, 90 neurons each, sine activation, N=10000, L-BFGS
    Chosen by hand without a sensitivity study or search; the reported accuracy depends on these settings and on the single random initialization used, so the central claim carries this unexamined choice.
assumptions (5)
  • domain assumption The sound propagation is isentropic and the medium is a perfect gas, so p/rho^gamma is constant and c^2 = gamma R T.
    Used in Section 2 and Appendix A to linearize the continuity and momentum equations and to express density perturbations in terms of pressure perturbations.
  • domain assumption The mean-flow gradient is small enough that |M alpha| << k, allowing the expansion 1/(1 + j M alpha/k) ~ 1 - j M alpha/k (Eq. 17).
    This approximation enters the elimination step that produces Eq. (19) and ultimately the coefficients of Eq. (27); if it fails, the neural network solves an inaccurate governing equation.
  • domain assumption Terms of Mach number order beyond M^2 can be neglected when deriving Eq. (19).
    This truncation is applied without a stated error bound; it is a modeling choice inherited from the low-Mach regime and is not tested against exact solutions.
  • domain assumption The steady mean flow can be computed from a first-order Maclaurin approximation of the momentum equation, Eq. (33), yielding a quadratic for u(x).
    The quadratic root selection and subsequent p and rho profiles depend on this low-order expansion, which may bias the mean-flow field for large temperature gradients.
  • domain assumption bvp4c provides a sufficiently accurate 'true' solution for validation.
    The comparisons in Tables 2-5 are against bvp4c solving the same ODE, not against an exact analytical or experimental field; both methods share any modeling error in Eq. (27).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networks." pith.science (2026). https://pith.science/paper/RVFEZP6E

@misc{pith2026250722370,
  author       = {Pith},
  title        = {Pith review of: Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RVFEZP6E}},
  note         = {Machine review of arXiv:2507.22370}
}
read the original abstract

Neural networks constrained by the physical laws emerged as an alternate numerical tool. In this paper, the governing equation that represents the propagation of sound inside a one-dimensional duct carrying a heterogeneous medium is derived. The problem is converted into an unconstrained optimization problem and solved using neural networks. Both the acoustic state variables: acoustic pressure and particle velocity are predicted and validated with the traditional Runge-Kutta solver. The effect of the temperature gradient on the acoustic field is studied. Utilization of machine learning techniques such as transfer learning and automatic differentiation for acoustic applications is demonstrated.

Figures

Figures reproduced from arXiv: 2507.22370 by the authors.

Figure 1
Figure 1. Schematic diagram of a feedforward neural network. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Schematic diagram of a feedforward neural network for the complex-valued acoustic pressure. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Schematic diagram of a uniform duct with boundary conditions. [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Temperature profiles: Linear temperature profile, Sinusoidal temperature profile. formulation (predicted solution) against that obtained from the traditional Runge-Kutta solver bvp4c in MATLAB (true solution) for the linear temperature profile. It can be ob￾served that…
Figure 5
Figure 5. Figure 5: Acoustic pressure with linear temperature profile: [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Acoustic pressure with sinusoidal temperature profile: [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Particle velocity with linear temperature profile: [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Particle velocity with sinusoidal temperature profile: [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Effect of the temperature gradient on the acoustic pressure: [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

38 extracted references · 36 canonical work pages

  1. [1]

    P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton University Press, New Jersey, 1986), 1 edition

    work page 1986

  2. [2]

    Munjal, Acoustics of Ducts and Mufflers (John Wiley and Sons Ltd., West Sussex, UK, 2014), 2nd edition

    M. Munjal, Acoustics of Ducts and Mufflers (John Wiley and Sons Ltd., West Sussex, UK, 2014), 2nd edition

    work page 2014

  3. [3]

    Singh, A

    A. Singh, A. Rao, R. Vaideeswaran, C. Yarra and P. K. Ghosh, A study on native American english speech recognition by Indian listeners with varying word familiarity level, in 2021 24th Conf. Oriental COCOSDA (2021), pp. 13–18

    work page 2021

  4. [4]

    Srinivasan, A

    G. Srinivasan, A. Illa and P. K. Ghosh, A study on robustness of articulatory features for automatic speech recognition of neutral and whispered speech, in ICASSP 2019 (2019), pp. 5936–5940

    work page 2019

  5. [5]

    Xydis, N

    A. Xydis, N. Perraudin, R. Rust, K. Heutschi, G. Casas, O. R. Grognuz, K. Eggenschwiler, M. Kohler and F. Perez-Cruz, Gir dataset: A geometry and real impulse response dataset for machine learning research in acoustics, Applied Acoustics 208 (2023) 109333

    work page 2023

  6. [6]

    Yeh and Y.-S

    C.-Y. Yeh and Y.-S. Tsay, Using machine learning to predict indoor acoustic indicators of multi- functional activity centers, Applied Sciences 11 (2021) 5641

    work page 2021

  7. [7]

    Green and D

    M. Green and D. Murphy, Environmental sound monitoring using machine learning on mobile devices, Applied Acoustics 159 (2020) 107041

    work page 2020

  8. [8]

    Y.-J. Cha, A. Mostafavi and S. S. Benipal, Dnoisenet: Deep learning-based feedback active noise control in various noisy environments, Engineering Applications of Artificial Intelligence 121 (2023) 105971

    work page 2023

Show all 38 references
  1. [9]

    Zhang and D

    H. Zhang and D. Wang, Deep anc: A deep learning approach to active noise control, Neural Networks 141 (2021) 1–10

    2021

  2. [10]

    Rafiee, F

    J. Rafiee, F. Arvani, A. Harifi and M. Sadeghi, Intelligent condition monitoring of a gearbox using artificial neural network, Mech. Syst. Signal Process. 21 (2007) 1746–1754

    2007

  3. [11]

    Bangalore, S

    P. Bangalore, S. Letzgus, D. Karlsson and M. Patriksson, An artificial neural network-based con- dition monitoring method for wind turbines, with application to the monitoring of the gearbox, Wind Energy 20 (2017) 1421–1438

    2017

  4. [12]

    M. J. Bianco, P. Gerstoft, J. Traer, E. Ozanich, M. A. Roch, S. Gannot and C.-A. Deledalle, Machine learning in acoustics: Theory and applications, The Journal of the Acoustical Society of America 146 (2019) 3590–3628. September 3, 2025 8:33 ws-jtca 1-D Acoustic Field Predicti...

    2019

  5. [13]

    Cuomo, V

    S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi and F. Piccialli, Scientific machine learning through physics–informed neural networks: Where we are and what’s next, J. Sci. Comput. 92 (2022) 88

    2022

  6. [14]

    Raissi, P

    M. Raissi, P. Perdikaris and G. E. Karniadakis, Physics-informed neural networks: A deep learn- ing framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378 (2019) 686–707

    2019

  7. [15]

    Radu and C

    A. Radu and C. A. Duque, Neural network approaches for solving Schr¨ odinger equation in arbitrary quantum wells, Sci. Rep. 12 (2022) 2535

    2022

  8. [16]

    Zhou, Parallel physics-informed neural networks method with regularization strategies for the forward-inverse problems of the variable coefficient modified KdV equation, J

    H. Zhou, Parallel physics-informed neural networks method with regularization strategies for the forward-inverse problems of the variable coefficient modified KdV equation, J. Syst. Sci. Complexity 37 (2024) 511–544

    2024

  9. [17]

    M. S. Mathias, W. P. de Almeida, J. F. Coelho, L. P. de Freitas, F. M. Moreno, C. F. Netto, F. G. Cozman, A. H. Reali Costa, E. A. Tannuri, E. S. Gomi and M. Dottori, Augmenting a physics-informed neural network for the 2D Burgers equation by addition of solution data points, ...

    2022

  10. [18]

    Basir and I

    S. Basir and I. Senocak, Physics and equality constrained artificial neural networks: Application to forward and inverse problems with multi-fidelity data fusion, J. Comput. Phys. 463 (2022) 111301

    2022

  11. [19]

    Maddu, D

    S. Maddu, D. Sturm, C. L. M¨ uller and I. F. Sbalzarini, Inverse Dirichlet weighting enables reli- able training of physics informed neural networks, Mach. Learn. Sci. Technol. 3 (2022) 015026

    2022

  12. [20]

    X. Jin, S. Cai, H. Li and G. E. Karniadakis, NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations, J. Comput. Phys. 426 (2021) 109951

    2021

  13. [21]

    Oldenburg, F

    J. Oldenburg, F. Borowski, A. ¨Oner, K. Schmitz and M. Stiehm, Geometry aware physics in- formed neural network surrogate for solving Navier–Stokes equation (GAPINN), Adv. Model. Simul. Eng. Sci. 9 (2022) 1–15

    2022

  14. [22]

    Alkhalifah, C

    T. Alkhalifah, C. Song, U. bin Waheed and Q. Hao, Wavefield solutions from machine learned functions constrained by the Helmholtz equation, Artif. Intell. Geosci. 2 (2021) 11–19

    2021

  15. [23]

    T. Cui, Z. Wang and X. Xiang, An efficient neural network method with plane wave activation functions for solving Helmholtz equation, Comput. Math. Appl. 111 (2022) 34–49

    2022

  16. [24]

    M. A. Heckl, S. M. Gopinathan and A. Surendran, A unified framework for acoustic instabilities based on the tailored green’s function, Journal of Sound and Vibration 541 (2022) 117279

    2022

  17. [25]

    Li and A

    J. Li and A. S. Morgans, The one-dimensional acoustic field in a duct with arbitrary mean axial temperature gradient and mean flow, J. Sound Vib. 400 (2017) 248–269

    2017

  18. [26]

    Karthik, B

    B. Karthik, B. M. Kumar and R. Sujith, Exact solutions to one-dimensional acoustic fields with temperature gradient and mean flow, The Journal of the Acoustical Society of America 108 (2000) 38–43

    2000

  19. [27]

    Munjal and M

    M. Munjal and M. Prasad, On plane-wave propagation in a uniform pipe in the presence of a mean flow and a temperature gradient, The Journal of the Acoustical Society of America 80 (1986) 1501–1506

    1986

  20. [28]

    Cummings, Sound generation and transmission in flow ducts with axial temperature gradi- ents, Journal of Sound and Vibration 57 (1978) 261–279

    A. Cummings, Sound generation and transmission in flow ducts with axial temperature gradi- ents, Journal of Sound and Vibration 57 (1978) 261–279

    1978

  21. [29]

    W. H. Beyer, CRC Standard Mathematical Tables (CRC Press, Boca Raton, 1987), 28th edition

    1987

  22. [30]

    Hornik, M

    K. Hornik, M. Stinchcombe and H. White, Multilayer feedforward networks are universal ap- proximators, Neural Netw. 2 (1989) 359–366

    1989

  23. [31]

    D. V. Dung, N. D. Song, P. S. Palar and L. R. Zuhal, On the choice of activation functions in physics-informed neural network for solving incompressible fluid flows, in AIAA SCITECH 2023 Forum (National Harbor, Maryland, USA, 2023), p. 1803

    2023

  24. [32]

    I. E. Lagaris, A. Likas and D. I. Fotiadis, Artificial neural networks for solving ordinary and September 3, 2025 8:33 ws-jtca 20 D. Veerababu & Prasanta K. Ghosh partial differential equations, IEEE Trans. Neural Netw. 9 (1998) 987–1000

    1998

  25. [33]

    K. He, X. Zhang, S. Ren and J. Sun, Delving deep into rectifiers: Surpassing human-level per- formance on imagenet classification, in Proceedings of the IEEE International Conference on Computer Vision (2015), pp. 1026–1034

    2015

  26. [34]

    C. Xu, B. T. Cao, Y. Yuan and G. Meschke, Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios, Comput. Methods Appl. Mech. Eng. 405 (2023) 115852

    2023

  27. [35]

    Y. Gao, K. C. Cheung and M. K. Ng, SVD-PINNs: Transfer learning of physics-informed neural networks via singular value decomposition, in 2022 IEEE Symp. Series Comput. Intell. (Singa- pore, 2022), pp. 1443–1450

    2022

  28. [36]

    Pellegrin, B

    R. Pellegrin, B. Bullwinkel, M. Mattheakis and P. Protopapas, Transfer learning with physics- informed neural networks for efficient simulation of branched flows, arXiv:2211.00214 (2022) 1–5

    2022 arXiv

  29. [37]

    Desai, M

    S. Desai, M. Mattheakis, H. Joy, P. Protopapas and S. Roberts, One-shot transfer learning of physics-informed neural networks, arXiv:2110.11286v2 (2022) 1–14

    2022 arXiv

  30. [38]

    Sujith, G

    R. Sujith, G. Waldherr and B. Zinn, An exact solution for one-dimensional acoustic fields in ducts with an axial temperature gradient, Journal of Sound and Vibration 184 (1995) 389–402. Appendix A. Derivation of p − ρ relations for an isentropic process Since the sound propaga...

    1995

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.