Physics-Informed Neural Networks for Plane Wave Acoustics
By D. Veerababu, Prasanta K. Ghosh, Indian Institute of Science, Bangalore, India
Introduction
Physics-informed neural networks (PINNs) emerged as an alternate numerical tool to solve differential equations that govern complicated physics. The researchers successfully solved Burger’s equation, Schrodinger wave equation, Navier-Stokes equations, Poisson equation etc., using PINNs. Attempts have also been made to solve the Helmholtz equation. However, the success rate of solving it is marginal at higher frequencies due to the vanishing gradient problem that occurs during the neural network training process.
This article presents a methodology that bypasses the vanishing gradient problem and predicts the acoustic field in a 1-D uniform duct with axial temperature gradient. This work is useful to study thermoacoustic instabilities that have practical significance in the aerospace and gas turbine industries.
Governing Equation
The 1-D acoustic pressure in a uniform duct having a stationary medium and the axial temperature gradient can be found by solving the following differential equation
subjected to the boundary conditions P(0) = Po and P(L) = Pl. Here, P is the acoustic pressure, T is the temperature in Kelvin, 𝜔 is the angular frequency, 𝛾 is the specific heat ratio, and R is the universal gas constant.
PINNs Formulation
In PINNs, the required acoustic field P is approximated to the output of a feedforward neural network P^ as shown in Figure 1, and the parameters of the network 𝜃 are obtained by solving the optimization problem:
where 𝜆 is the Lagrange multiplier, Ld and Lb are the loss functions associated with the differential equation and the boundary conditions, respectively. Finding appropriate 𝜆 that works for different frequencies is challenging even with automatic 𝜆 update algorithms. Hence, a trial neural network that eliminates Lb from the optimization problem is constructed as follows:
Figure 1: Schematic diagram of a feedforward neural network
and the loss function is calculated as
Here, N is the number of collocation points, and x^i is the i-th collocation point.
Figure 2: Schematic diagram of a uniform duct considered for the study
Results and Discussion
Figure 2 shows a schematic diagram of the duct considered for the study with prescribed boundary values and temperatures. Figure 3 shows the acoustic pressure inside the duct at different frequencies. Here, the temperature is assumed to be varying linearly along the axial direction, and the properties of air are assumed to be 𝛾=1.4 and R=287 J/Kg.K. The architecture of the network used is as follows: m= 6, n= 90, N= 10000, activation function is sine, and optimizer is L-BFGS. The relative errors (RE) indicate that the PINNs formulation is able to predict the acoustic field from the governing equation and the boundary conditions like traditional numerical tools (FEM, BEM and CFD).
Figure 3: Acoustic pressure inside the duct with linearly varying temperature: Red- Predicted solution; Blue- True solution (Analytical solution)
Summary:
- Trial solution method eliminates the boundary loss from the optimization process.
- The method can be merged with the other data-driven methods.
- Neural networks can serve as a mesh-less acoustic solver in the near future.
- The work is useful in biomedical, speech, automotive and aerospace sectors.
Other relevant articles from the authors:
1. D. Veerababu and Prasanta K. Ghosh, “Prediction of acoustic field in 1-D uniform duct with varying mean flow and temperature using neural networks”, Journal of Theoretical and Computational Acoustics (2025). https://doi.org/10.1142/S2591728524400036
2. D. Veerababu and Prasanta K. Ghosh, “Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks”, Journal of Sound and Vibration, (2025). https://doi.org/10.1016/j.jsv.2025.119022
3. D. Veerababu and Prasanta K. Ghosh, “Neural network based approach for solving problems in plane wave duct acoustics”, Journal of Sound and Vibration, 585, 118476, (2024). https://doi.org/10.1016/j.jsv.2024.118476
4. D. Veerababu, Namra Quasim, and Prasanta K. Ghosh, “Estimation of Acoustic Field in a Uniform Duct with Mean Flow using Neural Networks”, International Journal of Acoustics and Vibration, 29(4), 391-399, (2024). https://doi.org/10.20855/ijav.2024.29.42062
Code repositories from the authors:
1) https://github.com/d-veerababu/1d-helmholtz-pinn-solver.git
2) https://github.com/d-veerababu/1d-acoustics-meanflow-pinns.git
3) https://github.com/d-veerababu/1d-acoustics-tempgradients-pinn.git
4) https://github.com/d-veerababu/1d-acoustics-meanflow-tempgradients-pinn.git
5) https://github.com/d-veerababu/2d-acoustics-circle-pinn.git
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