## 2 DUT and Modelling Methodologies
The ANNs are multi-layer perceptron (MLP) networks, which consist of layers of neurons: an input layer, an output layer, and one or more hidden layers. The frequency and the temperature are the two input neurons, whereas the real and imaginary parts of the considered parameter are the two output neurons. The number of hidden layers and their size (i.e., the number of neurons in each hidden layer) were determined during the training. The adopted training algorithm was the Levenberg-Marquardt (L-M) method, which is a modification of the most frequently used optimization backpropagation algorithm [62]. MATLAB software environment was used for training and testing the ANNs. As illustrated in Table 1, the trained ANNs have two hidden layers, each with a different number of neurons. It should be underlined that two separate one-output ANNs were used for modeling the real and imaginary parts of \(S_{12}\) because of its more complex behavior that was not possible to reproduce accurately by using a single ANN for both real and imaginary parts. The training data are all the available measurements, with the exception of those at 145 \({}^{\circ}\)C, which are used for assessing the model generalization ability. Figure 2: Topology of the small-signal equivalent-circuit model for the tested GaN HEMT. Figure 4 illustrates the GRU-based temperature-dependent model adopted for modeling the tested GaN-based HEMT. GRU is served as an efficient method for modeling as its exceeding capability of solving issues such as vanishing and exploding gradients. Compared with techniques such as long short-term memory (LSTM) networks, GRU produces fewer parameters in the model training process, thus making the modeling process more efficient. The proposed GRU model is built and trained using Tensorflow, which is an open-source software library designed for developing and deploying state-of-the-art machine learning algorithms. In the test, each \(S\)-parameter is modeled with a separate GRU model. As shown in Figure 4, each GRU model has two GRU building blocks, and each GRU blocks share the same structure, with one GRU layer and one dense layer. Figure 5 depicts the internal architecture of a basic unit of the GRU networks. A GRU has two layers with sigmoid functions, corresponding to the so-called reset gate and update gate and a tan-sigmoid layer. Each GRU model consists of a GRU layer and a dense layer. Table 2 shows the detailed layer size information for both the GRU layer and the dense layer for the GRU model. After optimizing weight and bias terms inside the model, two trained models, each for the real part and imaginary part of measured data, are combined to fit and predict the small-signal behavior of the DUT. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **ANN Structure** \\ \hline \(S_{11}\) & 2-4-4-2 \\ \hline \(S_{21}\) & 2-4-4-2 \\ \hline \(\text{Re}\ (S_{12})\) & 2-5-5-1 \\ \hline \(\text{Im}\ (S_{12})\) & 2-5-4-1 \\ \hline \(S_{22}\) & 2-5-5-2 \\ \hline \end{tabular}
\end{table} 
Table 1: Structure of the trained artificial neural networks: the number of neurons for the input, first hidden, second hidden, and output layers. Figure 3: Illustration of the model based on using artificial neural networks for the tested GaN HEMT: (**a**) \(S_{11}\), \(S_{21}\), \(S_{22}\), and (**b**) \(S_{12}\).