## 3 Experimental Setup and Methodology
### Neuronal Network Training  
In order to enable extrapolation, the neural network must be trained. For this purpose, a training data set was created from 60,000 synthetically generated sound pressure fields. These sound pressure fields were modeled after a tomographic measurement of a Kundt's tube at an angular access of 180\({}^{\circ}\)(see Figure 5). The measured model is a hexagonal array of Helmholtz resonators with a circular resonator aperture. The size of the resonator holes (\(r_{x}\); \(r_{y}\)), the number of resonators \(n_{holes}\), the distance between the resonators (\(x_{dist}\); \(y_{dist}\)), the amplitude above the resonators (\(a_{hole}\)) and the orientation of the start scan angle (\(\alpha\)) were varied. This is superimposed by a static sound pressure throughout the cylinder (offset). In the region of the apertures there is a Gaussian reduction of the sound pressure. The parameter range of the randomly varied quantities is listed in Table 1. By randomly choosing the above parameters, an asymmetric pattern is guaranteed for each image. Only in case of such asymmetric modeling an angular scan of 180\({}^{\circ}\) is necessary. Extrapolation of the neural network by repetition and mirroring of already existing areas of the input data can thus be excluded.  
For training, 80% of the data set was used. The remaining data were used as a validation data set. A total of 24 epochs were trained to converge the loss function to a acceptable remaining loss and thus, the extrapolation results can not be substantially improved. Additionally, the problem of overfitting is minimal. The overfitting and thus the difference of the loss function between training data set and validation data set after 24 epochs was less than 0.3%.  
The training results are shown in Figure 8. Good reconstruction results have been obtained. Mean squared error (MSE) and structural similarity (SSIM) [37] were used as comparison values. The MSE is calculated according to the following formula:  
\[MSE=\frac{1}{n_{x}\cdot n_{y}}\sqrt{\sum_{x=-50}^{50\text{ mm}}\sum_{y=-50}^{50 \text{ mm}}(Y(x,y)-\hat{Y}(x,y))^{2}} \tag{9}\]  
where the coordinates \(x\) and \(y\) are the lateral and axial positions of a reconstructed plane.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Offset in Pa** & \(r_{x}\) **in mm** & \(r_{y}\) **in mm** & \(a_{hole}\) **in Pa** & \(n_{holes}\) & \(x_{dist}\) **in mm** & \(y_{dist}\) **in mm** & \(\alpha\) **in \({}^{\circ}\)** \\ \hline training set & [0.5; 30] & [0.4; 1] & [0.4; 1] & [0.1; 30] & [1; 130] & [5; 15] & [3; 10] & [0; 180] \\ example set & 2.8 & 1 & 1 & 0.8 & 35 & 6 & 4 & 12 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters used for sound field generation.  
Figure 8: Visualization of training process with 60,000 data sets (48,000 training data set; 24,000 validation data set) over 24 epochs. As validation parameters mean squared error (MSE) and structural similarity (SSIM) are shown.  
Starting with the 2 nd epoch, an MSE lower than 0.1% is reached. In epoch 7, there is a sudden increase of the error to 0.27% in the validation data set. This can be explained by the NADAM optimization algorithm used. The algorithm does not run into a local minimum, but tries to reach the global minimum. Thus, due to abrupt changes in the optimized weights, the error may increase abruptly [38]. After 24 epochs, an error of the MSE of less than 0.005% was reached. The impact of an overfitting effect is negligible after 24 epochs. The same result can be seen for SSIM, which reaches a value of 99.4% after 24 epochs of training. The training took 11 h on a high performance workstation with an Nvidia TITAN RTX GPU. After training the extrapolation process took in summary 35 s for one extrapolation process. This time is mainly limited by the loading and saving process of the data.