# Physics-Informed Neural Networks for Modal Wave Field Predictions in 3D Room Acoustics
## 5 Results
### Explorative Hyperparameter Study for \(s=1\) m  
Compared to the previous studies conducted for \(s\to\infty\), the study investigates the networks' capability of calculating the results for a slightly distorted field based on \(s=1\) m. In this case, the amplitudes of the solution are marginally higher due to reflections at the sound-hard walls than for the simple example \(s\to\infty\). A relative error less than the relative \(l^{2}\) deviations \(e_{\rm rel}<3.14\%\) from the case \(s\to\infty\) indicates that the network has learned a meaningful description of the field. The number of layers for a configuration improved the accuracy, as it can be seen in Table 3 and for the first cohort (separated by rules in the table). As a training speed accuracy compromise, the networks with \(L-1=2\) and \(L-1=3\) were selected for further investigations. This trend was observable for all hyperparameter tests provided in Table 3. In the second and third cohorts, a combination of switching the optimizer during the training was tested, and the effect of this on performance was examined. Interestingly, when using \(w_{\rm NBC}=50\), the PDE and the BC loss were initially of the same level and caused an excellent prediction when combining it with the LBFGS optimizer to further iterate on the network trainable parameters (orange row in Table 3). In the final two cohorts, the boundary condition weights were tested. It was found that a relative weight increase from 5 to 500 improves the accuracy. As indicated by [53], an increasing relative boundary condition weight can positively influence the optimization by a more regular loss landscape. In a subsequent study, the network width was tested, as was its influence on the accuracy (Table 4). As expected, narrow networks performed comparably poor. In some occasions for a layer width of 40 neurons, no convergence to the FEM results was obtained. In conclusion, the layer width between 120 and 240 gave acceptable results for this case. Compared to the network's depth, the width shows less sensitivity in terms of accuracy for a neuron range of 120-240 neurons. The accuracy was in the bounds for all tested backends (Table 5), with a somewhat superior performance of TensorFlow 1 leading to an error below 0.12% and the highest training speed of about 20 min (being in the range of the FEM simulation). Regarding the accuracy, the PINN using JAX as a backend led to a low relative L\({}^{2}\) error of 0.34%. This low relative error is comparable to the refined FEM simulation. Finally, the PINN was trained on an alternative GPU (NVIDIA A100-PCIE-40 GB (NVIDIA,Santa Clara, CA, USA)) with a newer tensor core technology, reducing the training time by a factor of more than 2.5. The execution times are in the range reported previously [11]. Following up on the layer number and width study (Table 6), the activation functions were tested \(\sigma=\{\sin\), ELU, GELU, ReLU, SELU, sigmoid, SiLU, swish, and tanh\(\}\). The solution oscillates around zero, yielding positive and negative values. The functions ELU, ReLU, SELU, and sigmoid performed poorly potentially due to their predominantly positive branch. It was confirmed in the tests that none of the networks using those activation functions reached the converged results with an error of less than 50%. The sinus function performed best; the GELU, swish, tanh, and SiLU activation reached similar errors, which were already quite large. In conclusion to H1, the hyperparameter optimization improved the accuracy from initially 0.45% to 0.019%. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{L-1}\) & \(\mathbf{W}\) & \(\mathbf{w_{\text{NBC}}}\) & \(\sigma\) & \(\mathbf{\mathcal{B}}\) & **Training Sequence** & \(\mathbf{e_{\text{rel}}}\) **in \%** \\ \hline  
\hline \end{tabular}
\end{table} 
Table 3: Error \(e_{\text{rel}}\) in percentages for the number of layers and PDE/BC balancing weight hyperparameter of \(\theta_{\text{NN}}\) compared to FEM results.