# Physics-Informed Neural Networks for Modal Wave Field Predictions in 3D Room Acoustics
## 5 Results
### Multi-Scale Fourier Feature Networks  
The prior knowledge of spatial scales in the system is strongly driven by the behavior of the wave and the emissions coinciding with the wavelength \(\lambda\). Furthermore, it is expected that based on the sharpening of the force, the spatial frequency content gets more dispersed over the wave number and can lead to approximation difficulties using a modal field [52]. This was formally shown in Section 2.2. We selected an architecture with the hyperparameters \(\theta_{\text{NN}}=\{L-1=2,W=80,w_{\text{NBC}}=5,\text{sin},\text{TensorFlow 1}\}\) and a learning rate of \(10^{-3}\), and after 200,000 iterations, the errors were evaluated. The Fourier features were selected from a normal distribution with a standard deviation of \(\hat{s}_{1}=0.5\) m and \(\hat{s}_{1}=1.0\) m, which is in the range of the wavelength based on the selected wave number. In the case of \(s\rightarrow\infty\), the relative error (22) with 1.4% was in the range of the standard feed-forward network with the same number of trainable parameters. This was also observed for the case \(s=1\) m with an error of 1.93%. We observed no convergence for all the repeated test runs for \(s<1\) m, which was unexpected. In a second test, the activation function of the network neurons was altered to tanh to avoid the possible poor performance of a  
Figure 8: The added points based on adaptive refinement for \(s=0.2\) m are highlighted in (**a**) in red, whereas the initial data points are the tiny black points. (**b**) shows the results of the acoustic pressure on the line \(y=z=0.5\) m compared to the FEM reference solution. Figure 7: The added points based on adaptive refinement for \(s=1\) m are highlighted in (**a**) in red, whereas the initial data points are the tiny black points. (**b**) shows the results of the acoustic pressure on the line \(y=z=0.5\) m compared to the FEM reference solution. combination of Fourier feature inputs and sinus activation functions. Similarly, the relative error measure was 1.4%, 2.3%, and no convergence for \(s\rightarrow\infty\), for \(s=1\) m, and \(s=0.2\) m, respectively. The simulation time was evenly distributed for all tests, around 2500 s.  
Overall, the multi-scale Fourier feature networks performed similarly to the standard model, and the means of H4 did not improve prediction accuracy. A possible explanation is based on the exact linearization of the input-output relation due to the wave's dispersion relation [29]. In the precise form, the cosine and sine transformations for the \(s\)-dependent FEM results are illustrated in Figure 9.  
The cosine transformation in Figure 9a analytically fulfills the boundary condition of sound-hard walls and yields approximately linear slopes. These linear and nearly linear behaviors can be predicted easily by the neural network. However, for a slightly shifted wavelength (by 10%), one can identify that the cosine and sine both transform into ellipses, making it comparably hard to solve the wave equation and estimate the solution. As soon as there is a deviation in the wave number, inherently present in the multi-scale network by the random draw of the Fourier features, it is fundamentally hard for the network architecture to find the solution of the wave equation. These facts motivate us to form hypothesis 5 (H5) based on the given angular frequency \(\omega\), wave number \(k\), and wavelength \(\lambda\). **H5. **_Deterministic Fourier features as the input [54; 55] that satisfy the dispersion relation of the wave will improve the prediction accuracy and the performance of the PINN._  
Figure 9: Input feature generation depending on the wavelength and the results for various source sharpness parameter \(s\).