# Physics-Informed Neural Networks for Modal Wave Field Predictions in 3D Room Acoustics
## 5 Results
### Input Feature Generation  
The ordinary input \(x\) and additionally the transformed input are fed into the neural network. We select \(\lambda=0.5\) m so that the underlying application's acoustic wave number frequency relation is respected. The transform is motivated by Green's function of the waves in a box. It can be modeled by a complete orthonormal solution basis as a product of solutions for each Cartesian coordinate  
\[\psi(\mathbf{x})=\prod_{x_{i}\in x,y,z}\cos\left(\frac{2\pi}{\lambda_{i}}\,x_{i} \right). \tag{23}\]  
It is important to note that an additive ansatz would fail and not reduce the complexity of the PINN problem. We selected an architecture with hyperparameters \(\theta_{\text{NN}}=\{L-1=2\), \(W=10,w_{\text{NBC}}=5\), sin, TensorFlow 1\(\}\) and a learning rate of \(10^{-3}\), and, after 20,000 iterations, the errors were evaluated. In doing so, the following errors were reached: \(2\times 10^{-4}\%\), \(0.20\%\), \(17\%\), and \(45\%\), \(s\rightarrow\infty\), \(s=1\) m, \(s=0.2\) m, and \(s=0.02\) m, respectively. This significantly improves the other network structures by ingraining expert knowledge. For \(s\rightarrow\infty\), the errors are in the range of the FEM method, and training took a few seconds, which is even faster than the FEM solution. The execution time is in the range reported in [11] for the 2D simulations without fixed boundary conditions (Quadro RTX 8000 Nvidia (Santa Clara, CA, USA)). We obtained the convergence of the amplitudes in the case of \(s\rightarrow\infty\) and \(s=1\) m. Discrepancies in the amplitudes and shapes for the cases of \(s=0.2\) m and \(s=0.02\) m are given. In Figure 10, the pressure prediction of the PINN compared to the analytic or FEM results is illustrated for \(s\rightarrow\infty\), \(s=1\) m, \(s=0.2\) m, and \(s=0.02\) m. In the case of \(s=1\) m, the pressure agrees well with the FEM results. For \(s=0.2\) m, additional features with wave numbers of \(0.5k_{0}\), \(1.5k_{0}\), \(2k_{0}\), and \(2.5k_{0}\) are necessary to obtain convergence as illustrated in Figure 10c. The graph of the pressure starts to deviate at the domain's boundaries from the FEM solution. In the case of \(s=0.02\) m, the pressure signal does not adapt to the reference FEM solution.  
As a result of the fast training speed (below a minute and 20 times faster than the FEM simulation) in the feature-engineered neural network architecture, an automated hyperparameter optimization using a Gaussian processes-based Bayesian optimization [11] was performed in scikit-learn 1.5.2. Five hundred automated hyperparameter model iterations were performed. The learning rate, the network depth, the network width, the activation function, and the boundary condition loss weight were investigated. In doing so for \(s=1\) m, the relative error was reduced from 0.2% to 0.1938%. In contrast to the claimed strong improvements by automated hyperparameter optimization in the loss reduction [11], only a slight improvement in the accuracy of the model was obtained (see Table 8). The fastest model took 50 s to train and had an error of 0.204%. Regarding hypothesis H5, it was confirmed that the feature-engineered network outperformed the other architectures presented in that article for \(s>1\).  
\begin{table}
\begin{tabular}{c c c c c} \hline  & **Overall Best** & **Best Sin** & **Best Sigmoid** & **Fastest** \\ \hline \(e_{\text{rel}}\) in \% & 0.1938 & 0.195 & 0.21 & 0.204 \\ Training time in s & 621 & 404 & 89 & 50 \\ Learning rate & 0.0009 & 0.00077 & 0.025 & 0.001 \\ \(L-1\) & 9 & 3 & 10 & 2 \\ \(W\) & 80 & 80 & 12 & 35 \\ \(\sigma\) & tanh & sin & sigmoid & sin \\ \(w_{\text{NBC}}\) & 1 & 0.1 & 4.7 & 5 \\ \hline \end{tabular}
\end{table}
Table 8: Summary of the relative errors and network parameters of the automated hyperparameter optimization with \(s=1\).