# Physics-Informed Neural Networks for Modal Wave Field Predictions in 3D Room Acoustics
## 1 Introduction  
Today, there are persisting challenges in room acoustics simulations, with errors mainly arising from uncertain input parameters of used building material, resulting in a high uncertainty in the used boundary conditions [1,2]. In particular, at low-frequency waves, the material parameters are off [3,4], hindering more accurate simulation results. These uncertain simulation input data underline the importance of using algorithms that allow for the on-the-fly minimization of the model discrepancy with reality. As a selected algorithm, physics-informed neural networks (PINNs) can predict the wave field and infer the material coefficients or boundary condition parameters with the use of data [3]. However, the computation of the time-harmonic acoustic wave field in large 3D geometries represents a challenge for current wave-based numerical and machine-learned methods [4,5]. The solutions to the Helmholtz equation in 3D are essential for understanding low-frequency room acoustic scenarios. The Finite Element Methods (FEM) analysis provides accurate results [4], but the extension to inverse parameters estimation is tedious compared to PINNs. PINNs show promising capabilities for that purpose, such that the interest in tailoring these algorithms to the accuracy of FEM has become an interesting topic. So far, PINNs have been used to solve the Helmholtz equation [6, 7, 8] only in 1D and 2D domains for acoustic simulations [3,9, 10, 11, 12, 13, 14, 15], ultra-sound application [16], electromagnetic waveguides [17, 18], and aeroacoustics [19]. It was already observed in [11] that the training deteriorates with increasing wave number due to the frequency-principle [20,21] and deteriorates when changing from the Dirichlet boundary condition to a Neumann boundary condition. A weak point regarding the acoustics predictions of previous studies using PINNs is that the solution of the inhomogeneous Helmholtz equation is similar to the widespread exciting source of the partial differential equation. A cosine function in space describes the wave field, and a scaled and unrealistic cosine function models the source covering the whole room. However, typical sources in acoustics are confined [22,23], resulting in a substantial increase in the prediction discrepancies from a 2.5% to 92% relative error [5]. These confined sources excite a spatial periodic wave field. This characteristic is known as the dispersion relation of the acoustic wave. Compared to previous studies focusing on unrealistic smooth excitation [9,11], we will analyze the network's generalization capabilities towards realistic excitation scenarios caused by a (loudspeaker-like) confined source. A structured hyperparameter study provides intuition about the network key parameter changes (concerning the results of [11]), the activation functions, and various backends, including PyTorch [24], TensorFlow 1 and 2 [25], and JAX [26]. All results are compared to results from a validated FEM analysis using openCFS [4]. Rerunning the training of the PINN reported in [5] on a Tesla T4 GPU with Tensorflow 1, the training time of the 3D PINN took about 84 min compared to a solver time of the FEM analysis of 16 min on a standard notebook CPU. In [11], the randomly distributed point-wise \(l^{2}\) error measure was used as a loss metric. It is highly questionable whether a random distribution of evaluation points computes a reliable error measure for assessing numerical convergence. The accuracy is evaluated in a best-practice manner by comparing the results on a structured evaluation grid with the FEM results and computing an \(L^{2}\) error norm. As a follow-up to the hyperparameter study, additional data points are added by residual-based adaptive refinement. In addition to the standard neural network architectures, the locally adaptive activation functions with slope recovery [27] and the multi-scale Fourier feature network [28] are investigated for potentially improved neural network performance.