# Physics-Informed Neural Networks for Modal Wave Field Predictions in 3D Room Acoustics
## 4 Hypotheses  
The core part of this article is the investigation of the influence on the prediction accuracy through narrowing the spread of the excitation source by decreasing the sharpness parameter \(s\). In this preliminary section, the generalization capabilities of the standard neural network for confined sources are investigated. The source function \(g(\mathbf{x})\) is modeled more by a gradually confined source in the center of the room modeled by Equation (3) and illustrated in Figure 5a. The sharpness of the source can be tuned by the parameter \(s\) in the exponential part of Equation (3). In Figure 5a, we see a range of \(s\)-parameters ranging from a distributed source for \(s\rightarrow\infty\) to a relatively confined source \(s=0.02\) m. When sharpening the excitation \(g(\mathbf{x})\), the neural network must model additional spatial modes and feature a multi-scale problem. As pointed out [28] and indicated by the modal description of a point source [52], it may be difficult for a standard fully-connected neural network to produce accurate results of the acoustic pressure. Figure 5b shows the results provided by the FEM analysis on the selected center line \(y=z=0.5\) m. The FEM analysis results show that independently of the source function, the acoustic pressure is primarily dominated by the wave number connected to the time-harmonic oscillation of \(0.5\) m.  
Figure 4: Schematic representation of a PINN with multi-scale Fourier feature architecture.  
The maximum pressure amplitude is \(\max(p)=\{1.01,1.02,1.11,0.903,0.505,0.123\}\) Pa for \(s=\{1,0.5,0.2,0.1,0.05,0.02\}\) m, respectively. The lowest level of \(s=0.02\) was selected to respect the frequency considerations in Section 2.4 for accurate FEM solutions of the source concerning the FEM mesh. The standard network with hyperparameters \(\theta_{\text{NN}}=\{L-1=3,W=180,w_{\text{NBC}}=5,\) sin, PyTorch\(\}\) (motivated by [5; 28]), and after 200,000 iterations of training with a learning rate of \(10^{-3}\) previously resulted in large relative errors [5]. The recalculation of the PINNs with respect to the converged FEM solution presented in Table 1 shows that the standard network with hyperparameters \(\theta_{\text{NN}}=\{L-1=3,W=180,\)\(w_{\text{NBC}}=5,\) sin, PyTorch\(\}\) and after 200,000 iterations of training with a learning rate of \(10^{-3}\) resulted in an error, with an increase in the relative error as the source sharpened \(s\to 0\) (see Table 2).  
Therefore, the following hypotheses (H1-H4) were formed to investigate this behavior and improve prediction accuracy.  
**H1.**: _Tuning the hyperparameters positively influences the generalization and prediction accuracy._  
**H2.**: _A layer-wise local adaptive activation function positively influences the generalization and prediction accuracy._  
**H3.**: _Adding data points to the training dataset by loss-residual-based adaptive refinement positively influences the generalization and prediction accuracy._  
**H4.**: _A multi-scale Fourier feature network positively influences the generalization and prediction accuracy._  
H1-H4 will be tested by the FEM reference solution.  
\begin{table}
\begin{tabular}{c c c c c} \hline \(s\) in m & 0.2 & 0.1 & 0.05 & 0.02 \\ \(e_{\text{rel}}\) in \% & 4.52 & 5.72 & 7.05 & no convergence \\ \hline \end{tabular}
\end{table}
Table 2: Summary of the relative errors of the standard PINN for various source sharpness parameters \(s\) compared to the FEM reference solution.  
Figure 5: Higher order oscillation and spatial dependence of (**a**) the forcing and (**b**) the pressure results as a function of the source sharpness parameter \(s\). (**a**) Forcing depending on the \(x\) coordinate at a line with \(y=z=0.5\) m. (**b**) FEM analysis pressure results depending on the \(x\) coordinate at a line with \(y=z=0.5\) m.