## 2 Materials and Methods
### Inverse Parameter Identification of the Acoustic Model Parameters
Between these sets, a model \(f\), mapping the four geometry parameters \(\mathbf{X}\) to the five model parameters \(\mathbf{Y}\), is built:  
\[\mathbf{Y}= f(\mathbf{X}) \tag{1}\] \[f:\mathbb{R}^{4}\rightarrow\mathbb{R}^{5}\quad.\]  
The model \(f\) is built within this contribution using an artificial neural network (ANN), using the keras/tensorflow library [53]. A \"classic\" structure as a densely connected feedforward neural network with three hidden layers is used. All layers use the Rectified Linear Unit (ReLU) activation function. The input data are scaled to the unit interval before being presented to the network. The final structure of the network is found in a trial-and-error approach. Since the resulting network is very similar to the one presented in [36], the reader is referred to that study for more details on the network. The network has a total of 1,111,721 parameters and is trained using the Adam algorithm and the \"mean squared error\" error measure. The results are reported using the coefficient of determination \(R^{2}\). The model can be seen schematically in Figure 3.  
The available dataset is split into three parts, namely the training set, the validation set and the test set. The test dataset amounts to 20 % of the data, and is not used for the training and model generation. The remaining 80 % of the data are used in a threefold cross-validation. Thus, the model is trained using two-thirds of the data (the training dataset) and is tested against the remaining one-third (validation dataset). This training procedure is repeated using different model structures. The best results have been obtained with the model shown in Figure 3.  
Hereafter, a short overview of the model performance is given. Since no general changes are made and only slight accuracy improvements are achieved compared to the preceding study, the results of the performance evaluation are shown in this section. The training score of this model is \(R^{2}=0.83\pm 0.12\), whereas the reported uncertainty accounts for twice the standard deviation (\(\approx\)95 % confidence interval) and is computed over the three cross-validation runs. The model is finally evaluated using the (yet unseen) test dataset, resulting in a test score of \(R^{2}=0.93\). The increase in the test score relative to the training score here results from an increase in available training data, as the model is retrained using the entire training and validation dataset before computing the test score. The ability of the model to predict the JCA model parameters from the geometry values is shown using correlation plots in Figure 4. In the plots, the prediction of the ANN with the geometry values from the test dataset as input is shown in the vertical axis. The horizontal axis shows the correct model parameter that corresponds to the geometry parameter taken by the ANN. It can be seen that the correlation is good for the flow resistivity and acceptable for the remaining parameters. Figure 3: The artificial neural network used within this contribution. The size of the hidden layers is not to scale.