# Physics-Informed Neural Networks for Modal Wave Field Predictions in 3D Room Acoustics
## 5 Results
### Adaptive Refinement of Training Set  
The pre-trained standard network architecture with hyperparameters \(\theta_{\text{NN}}=\{L-1=3\), \(W=180,w_{\text{NBC}}=5,\sin\), PyTorch\(\}\), and a learning rate of \(10^{-3}\), after 200,000 iterations, is used for the adaptive fine-tuning of the model. The idea that adaptive refinement may improve the results emerged by looking at Figure 6a and the overlaid black dots, which seemed to be distributed nicely but were relatively rare compared to a FEM mesh. The adaptive refinement procedure works in the following steps: in the first data refinement step, \(j=1,\dots,10\), the domain was sampled uniformly for a new dataset \(\mathcal{T}_{\text{samp,H,j}}\). At these data points, the residual of the loss (15) was evaluated, and the arbitrary choice of 150 data points with the most significant residual were added to \(\mathcal{T}_{\text{train,O,j}}=\mathcal{T}_{\text{train,O}}\cup(\mathcal{T}_{ \text{samp,O,j}}\mid\max_{i=1\dots 150}(\mathcal{L}(x_{i})))\,\forall x_{i}\in \mathcal{T}_{\text{samp,O,n}}\). After the uniform refinement step, another 25 data refinement steps \(j=11,\dots,35\) with random data point sampling were added to the training \begin{table}
\begin{tabular}{c c c c c c c} \hline \(L-1\) & \(W\) & \(w_{\text{NBC}}\) & \(\sigma\) & \(\mathcal{B}\) & **Training Sequence** & \(e_{\text{rel}}\) **in \%** \\ \hline
2 & 180 & 5 & sin & P & 90,000 Adams & 0.77 \\
2 & 180 & 5 & ELU & P & 90,000 Adams & nc \\
2 & 180 & 5 & GELU & P & 90,000 Adams & 1.56 \\
2 & 180 & 5 & ReLU & P & 90,000 Adams & nc \\
2 & 180 & 5 & SELU & P & 90,000 Adams & nc \\
2 & 180 & 5 & sigmoid & P & 90,000 Adams & nc \\
2 & 180 & 5 & SiLU & P & 90,000 Adams & 9.77 \\
2 & 180 & 5 & swish & P & 90,000 Adams & 6.09 \\
2 & 180 & 5 & tanh & P & 90,000 Adams & 2.53 \\ \hline \end{tabular}
\end{table} 
Table 6: Error \(e_{\text{rel}}\) in percentages for the activation function as a hyperparameter variation compared to FEM results. Yellow-marked entries describe the best-performing parameter set in the direction of variation in the line search. \begin{table}
\begin{tabular}{c c c c c c} \hline \(L-1\) & \(W\) & \(w_{\text{NBC}}\) & \(\sigma\) & \(\mathcal{B}\) & **Training Sequence** & \(e_{\text{rel}}\) **in \%** \\ \hline
2 & 180 & 5 & sin & P & 90,000 Adams & 0.77 \\
2 & 180 & 5 & LAAF-n 1 & T1 & 90,000 Adams & 0.086 \\
2 & 180 & 5 & LAAF-n 2 & T1 & 90,000 Adams & 0.370 \\
2 & 180 & 5 & LAAF-n 4 & T1 & 90,000 Adams & 0.120 \\
2 & 180 & 5 & LAAF-n 8 & T1 & 90,000 Adams & 0.096 \\ \hline \end{tabular}
\end{table} 
Table 7: Error \(e_{\text{rel}}\) in percentages for the layer-wise locally adaptive activation function network compared to FEM results. Yellow-marked entries describe the best-performing parameter set in the direction of variation in the line search. data points. In each refinement cycle, 1000 iterations of the optimizer were carried out, with early stopping activated. The relative error results (see Equation (22)) for the case \(s\to\infty\) present a reduction in the error from 0.35% to 0.33% in the first refinement stage and then another reduction to 0.28% after the whole procedure of adaptive refinement. This reduction of 0.07% points is an improvement of the network's prediction accuracy by 20%. Figure 5(a) illustrates the pressure field predicted by the PINN in the evaluation plane (\(z=0.5\) m) with a colormap representing the pressure magnitude. Overlaid black dots highlight the training points close to the evaluation plane. Overlaid red dots highlight the training points that the adaptive refinement procedure added and indicate regions where the training error was high before the refinement. Figure 5(b) shows the pressure field predicted by the PINN compared to the analytic solution on the evaluation line (\(y=z=0.5\) m). Notably, the graphs do coincide qualitatively. In the case of \(s=1\) m, the relative error results in (22) were reduced from 0.45% to 0.21% in the first refinement stage and then further decrease to 0.18% after the whole procedure of adaptive refinement.