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Neural network based approach for solving problems in plane wave duct acoustics

D. Veerababu, Prasanta K. Ghosh

TL;DR

The paper tackles the challenge of solving frequency-domain duct acoustics with neural networks by adopting a boundary-condition–satisfying trial solution, thereby mitigating vanishing-gradient issues at high frequencies. It extends this physics-informed neural network framework to uniform, gradually varying, and narrow-tube ducts, including complex-valued pressures via real-imag decomposition, and demonstrates strong agreement with analytical solutions. A transfer-learning strategy enables estimating particle velocity from existing pressure fields, both with and without mean flow, and the authors analyze sensitivity to activation functions and collocation-point density. The results point to a robust, meshless solver capable of handling visco-thermal and convective effects, with implications for scalable acoustics simulations in higher dimensions.

Abstract

Neural networks have emerged as a tool for solving differential equations in many branches of engineering and science. But their progress in frequency domain acoustics is limited by the vanishing gradient problem that occurs at higher frequencies. This paper discusses a formulation that can address this issue. The problem of solving the governing differential equation along with the boundary conditions is posed as an unconstrained optimization problem. The acoustic field is approximated to the output of a neural network which is constructed in such a way that it always satisfies the boundary conditions. The applicability of the formulation is demonstrated on popular problems in plane wave acoustic theory. The predicted solution from the neural network formulation is compared with those obtained from the analytical solution. A good agreement is observed between the two solutions. The method of transfer learning to calculate the particle velocity from the existing acoustic pressure field is demonstrated with and without mean flow effects. The sensitivity of the training process to the choice of the activation function and the number of collocation points is studied.

Neural network based approach for solving problems in plane wave duct acoustics

TL;DR

The paper tackles the challenge of solving frequency-domain duct acoustics with neural networks by adopting a boundary-condition–satisfying trial solution, thereby mitigating vanishing-gradient issues at high frequencies. It extends this physics-informed neural network framework to uniform, gradually varying, and narrow-tube ducts, including complex-valued pressures via real-imag decomposition, and demonstrates strong agreement with analytical solutions. A transfer-learning strategy enables estimating particle velocity from existing pressure fields, both with and without mean flow, and the authors analyze sensitivity to activation functions and collocation-point density. The results point to a robust, meshless solver capable of handling visco-thermal and convective effects, with implications for scalable acoustics simulations in higher dimensions.

Abstract

Neural networks have emerged as a tool for solving differential equations in many branches of engineering and science. But their progress in frequency domain acoustics is limited by the vanishing gradient problem that occurs at higher frequencies. This paper discusses a formulation that can address this issue. The problem of solving the governing differential equation along with the boundary conditions is posed as an unconstrained optimization problem. The acoustic field is approximated to the output of a neural network which is constructed in such a way that it always satisfies the boundary conditions. The applicability of the formulation is demonstrated on popular problems in plane wave acoustic theory. The predicted solution from the neural network formulation is compared with those obtained from the analytical solution. A good agreement is observed between the two solutions. The method of transfer learning to calculate the particle velocity from the existing acoustic pressure field is demonstrated with and without mean flow effects. The sensitivity of the training process to the choice of the activation function and the number of collocation points is studied.
Paper Structure (19 sections, 58 equations, 15 figures, 6 tables)

This paper contains 19 sections, 58 equations, 15 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Physics-based neural network formulation
  3. Uniform duct
  4. Duct with gradually varying cross-sectional area
  5. Duct with narrow-tube effects
  6. Results and Discussion
  7. Acoustic pressure: Uniform duct
  8. Acoustic pressure: Duct with gradually varying cross-sectional area
  9. Acoustic pressure: Duct with narrow-tube effects
  10. Particle velocity estimation
  11. Effect of activation function on the training process
  12. Effect of number of data points on the training process
  13. Conclusion
  14. Illustration of bias towards one of the loss functions while training a neural network
  15. Analytical solution for the acoustic pressure distribution in a uniform duct
  16. ...and 4 more sections

Figures (15)

  • Figure 1: Schematic diagram of a feedforward neural network.
  • Figure 2: Schematic diagram of a feedforward neural network to predict complex-valued acoustic pressure.
  • Figure 3: Schematic diagram of a uniform duct with boundary conditions.
  • Figure 4: Acoustic pressure in a uniform duct:true solution, predicted solution.
  • Figure 5: Schematic diagram of a gradually varying rectangular duct.
  • ...and 10 more figures