## 3 Artificial Neural Network Model and Training Results  
It was thought that it would be difficult to train an artificial neural network when the number of scatterers was varied, so two learning models were developed for each case of two and three scatterers. For learning, we apply \(N=16\) incident \(\boldsymbol{\theta}_{n}\) and observation \(\boldsymbol{\theta}_{n}\) directions for full-view data such that  
\[\boldsymbol{\theta}_{n}=\bigg{(}\cos\frac{2(n-1)\pi}{N},\sin\frac{2(n-1)\pi}{ N}\bigg{)},\quad n=1,2,\cdots,N=16.\]Throughout this paper, we assume that there exist 2 or 3 scatterers with the same radii \(\alpha_{m}=0.01\) and permittivities \(\varepsilon_{m}=5\). For a limited-view situation, we apply first \(N/2=8\) incident \(\mathbf{\theta}_{n}\) and observation directions \(\mathbf{\vartheta}_{n}\). With this setting, the far-field pattern data \(\Delta S_{j,l}=u_{\infty}(\mathbf{\vartheta}_{j},\mathbf{\theta}_{l})\) of \(\mathbb{K}\) at wavenumber \(k=2\pi/0.4\) was generated by solving the Foldy-Lax formulation (see [49], for instance). Therefore, \(16\times 16\) values of the MSR matrix \(\mathbb{M}\) of full-view data are partitioned into four \(8\times 8\) data groups as shown in Figure 2, and then data preprocessing is performed to convert group 1 data into input data.  
Then, three training data sets were created using the data of groups 2, 3, and 4 as the correct label, and then three artificial neural networks were trained using each data. That is, in the first artificial neural network, the data of groups 1 and 2 are training data sets, in the second artificial neural network, the data of groups 1 and 3 are training data sets, and in the third artificial neural network, the data of groups 1 and 4 are used as training data sets, which were applied to each artificial neural network as a training data set.  
In order to find a suitable learning model to solve the problem of constructing full view data from limited view data, the number of layers and the number of nodes of hidden layers were adjusted in CNN and ANN models, and experiments were conducted using various activation functions. In the CNN model. a suitable model could not be found. However, when the ReLU function was used as the activation function in the ANN model and the number of hidden layers was 3, a learning model with a small mean square error could be found. In order to solve the overfitting problem, which is the major problem of ANN, we proceeded the learning. Even though the mean square error of the training data decreased, the mean square error of the test data did not show increase or decrease. In this case, we proceeded the learning process trying another model or increased the amount of training data. In addition, in order to reduce the complexity of the model, the number of hidden layers and the frequency of learning were adjusted to be as small as possible. In order to increase the learning rate and apply the ReLU function, the data cleansing of \(\Delta S_{j,l}S\) resulted in \(A_{j,l}\). Then, the data output through the artificial neural network, \(B_{j,l}\), was converted to \(C-j,l\) by performing data preprocessing, and the final output was made.  
The artificial neural network model was constructed to generate three \(1\times 64\) data using limited-view \(8\times 8\) data as shown in Figure 3, and the artificial neural network consists of an input layer and three hidden layers and an output layer. Activation function used ReLU, and optimization algorithm used Adam, and \(8\times 8\) data were preprocessed into \(1\times 64\) data, to be an input into the artificial neural network. and inputting three different artificial neural networks, and made them learned [50; 51; 52; 53; 54]. By processing three \(1\times 64\) output data drawn from each artificial neural network and using \(8\times 8\) data used for input, \(16\times 16\) data became a final output.  
Figure 2: Partitioning of the MSR matrix.