Convergence of physics-informed neural networks modeling time-harmonic wave fields

TL;DR

This paper investigates the convergence behavior of physics-informed neural networks (PINNs) in modeling 3D low-frequency room acoustics, emphasizing the importance of training data density and boundary conditions for accurate wave field predictions.

Contribution

It provides a detailed analysis of PINN convergence in 3D acoustic wave modeling, highlighting the necessary training points per wavelength and the effects of boundary conditions and absorption modeling.

Findings

At least six training points per wavelength are needed for accurate predictions.

Convergence depends on boundary condition modeling and source definitions.

Complex speed of sound models help account for absorption effects.

Abstract

Studying physics-informed neural networks (PINNs) for modeling partial differential equations to solve the acoustic wave field has produced promising results for simple geometries in two-dimensional domains. One option is to compute the time-harmonic wave field using the Helmholtz equation. Compared to existing numerical models, the physics-informed neural networks forward problem has to overcome several topics related to the convergence of the optimization toward the "true" solution. The topics reach from considering the physical dimensionality (from 2D to 3D), the modeling of realistic sources (from a self-similar source to a realistic confined point source), the modeling of sound-hard (Neumann) boundary conditions, and the modeling of the full wave field by considering the complex solution quantities. Within this contribution, we study 3D room acoustic cases at low frequency, varying the source definition and the number of boundary condition sets and using a complex speed of sound model to account for some degree of absorption. We assess the convergence behavior by looking at the loss landscape of the PINN architecture, the $L^2$ error compared to a finite element reference simulation for each network architecture and configuration. The convergence studies showed that at least six training points per wavelength are necessary for accurate training and subsequent predictions of the PINN. The developments are part of an initiative aiming to model the low-frequency behavior of room acoustics, including absorbers.