AI Predicts Acoustic Boundaries with Precision

By Muhammad OsamaReviewed by Susha Cheriyedath, M.Sc.Jun 12 2024

In a recent paper published in the journal Mechanical Systems and Signal Processing, researchers presented an innovative data-driven approach to predict acoustic boundary conditions by employing physics-informed neural networks (PINNs). They demonstrated that PINNs could accurately learn the sound pressure field and implicitly solve the inverse problem of characterizing acoustic boundary admittance from noisy data.

Background

Acoustic boundary conditions are essential for accurately simulating interior acoustics by defining the interaction between sound waves and domain boundaries. However, real-time measurement of boundary admittance poses significant challenges, often requiring computationally expensive inverse methods that rely on validated forward models.

Machine learning techniques, especially deep learning methods using NNs, offer promising directions for data-driven modeling and scientific computation. PINNs integrate prior physical knowledge by incorporating the residual of the partial differential equation (PDE) into their loss functions.

About the Research

In this article, the authors focused on solving the Helmholtz equation in the frequency domain, which governs the behavior of acoustic waves within a specified domain. This second-order linear PDE depends on variables such as sound pressure, wave number, and boundary conditions, which can be of Dirichlet, Neumann, or Robin type, depending on the boundary's characteristics. The researchers specifically addressed Robin-type boundary conditions, which relate the normal derivative of sound pressure to acoustic admittance, a complex measure representing the ratio of normal particle velocity to sound pressure at the boundary.

To address this, the study proposed a PINN that took spatial coordinates as inputs and outputs both the real and imaginary components of sound pressure. The PINN was trained on noisy sound pressure data obtained from finite element simulations or experimental measurements. Its loss function comprised two terms: a data-driven term measuring the mean squared error between the PINN's prediction and the training data, and a physics-based term quantifying the mean squared residual of the Helmholtz equation. This latter term was evaluated at collocation points randomly sampled within the domain. The PINN was trained using the Adam optimization algorithm with a learning rate decay strategy.

Method Validation

The introduced method's efficacy was tested through two numerical examples and one experimental validation. The first numerical example involved a two-dimensional model of an acoustic duct with a harmonic velocity excitation on one end and a normalized admittance boundary condition on the opposite end. The second example utilized a two-dimensional model of a sound-radiating circle, with a harmonic velocity excitation applied to the circular surface and two distinct normalized admittances on the rectangular boundaries.

For experimental validation, an acoustic impedance tube measurement was conducted, measuring sound pressure at four designated locations along the tube to estimate the acoustic admittance of the sample at the tube's end.

Research Findings

The outcomes showed that the PINN accurately learned the sound pressure field and implicitly solved the inverse problem by providing a precise estimate of the underlying boundary admittance, even with spatially varying boundary conditions. The PINN's predictions for sound pressure and boundary admittance were in excellent agreement with the reference solutions, achieving root mean square errors (RMSEs) on the order of 10-3 Pa for sound pressure and 10-4 for boundary admittance.

Additionally, this approach outperformed a purely data-driven supervised NN, which did not include the physical loss term, by more than an order of magnitude in prediction accuracy. The method was further validated using experimental data from an acoustic impedance tube measurement, demonstrating its capability to handle realistic scenarios with measurement noise and uncertainties.

Applications

The new technique holds significant potential across various fields of computational acoustics, including room acoustics, noise control, and acoustic metamaterials. It can estimate the acoustic boundary conditions of complex structures or materials, such as porous media, perforated plates, or membranes, based on sound pressure measurements.

Moreover, it can be used to design acoustic devices or systems with specific boundary conditions, such as absorbers, reflectors, or filters, by training a PINN with a target admittance profile. Additionally, the method can be adapted to other types of boundary conditions, such as Dirichlet or mixed boundary conditions, and extended to other physical equations, such as the wave equation or the Navier-Stokes equations.

Conclusion

In summary, the novel approach proved effective for constructing a data-driven model from noisy sound pressure data and estimating unknown acoustic boundary conditions. It did not require prior assumptions about the forward model or boundary conditions and could handle spatially varying boundary conditions without prior parameterization.

Moreover, it demonstrated robustness and generalization ability when applied to experimental data. The authors suggested that PINNs could be a powerful tool for solving inverse problems in acoustics and other fields of computational physics. For future work, they recommended applying the method to three-dimensional problems, incorporating multiple frequencies or time-domain data, and exploring other types of inverse problems, such as source localization or material parameter identification.