## 3 Inclusion of Temperature in the DNN Model-Based Method
### Training with Validation  
The feed-forward DNN is trained using the popular backpropagation method [25; 26; 27], in order to align the outputs (\(I_{fc,cal}\)) as closely as possible with the measured values (\(I_{fc,meas}\)). Backpropagation is an iterative gradient descent optimization technique applied to the network's weight space, enabling the weights to be updated in each iteration and to progressively minimize the cost function (CF).  
At the first stage of each epoch, forward propagation is performed, where the inputs (\(U,P,Q,\) and \(\theta\)) are fed to the DNN and the output value is calculated based on those inputs and with the initialization values of the weight, as stated in Equation (3).  
\[I_{fc,cal}=F\cdot\left[\sum_{k=1}^{K}w_{F,\ 2k}h_{2k}\left(\sum_{j=1}^{J}w_{2k, \ 1h_{1j}}(w_{1j,U}\cdot U+w_{1j,P}\cdot P+w_{1j,Q}\cdot Q+w_{1j,\theta}\cdot \theta+w_{j,0})\right)\ +w_{k,0}\right)+w_{F,0}\right] \tag{3}\]  
where \(wF,2k,\)\(w2k,1\), and \(w1j,U,\)\(w1j,P,\)\(w1j,Q,\)\(w1j,\theta\) are the synaptic weights from the neurons of the second hidden layer to the output neuron, from the neurons of the first hidden layer to the second hidden layer, and from the neurons of the input layer to the ones of the first hidden layer, respectively.  
To summarize Equation (3), it is better to write the weights vector as in Equation (4):  
\[W_{t}=\left[w_{F,\ 2k}\ w_{2k,\ 1j}\ w_{1j,\ U}\ w_{1j,\ P}\ w_{1j,\ Q}\ w_{1j,\ \theta}\ \right] \tag{4}\]  
with \(j\) from \(1\) to \(J\), and \(k\) from \(1\) to \(K\).  
At the second stage, the algorithm calculates the value adopted by the CF and subsequently evaluates its gradient by propagating the errors from the output layer back to the input layer through the chain rule. Through stochastic gradient descent, accordingly, moving towards the negative direction of the gradient, the iteration is concluded by updating the weights and biases.  
The progressive adjustment of the network is performed by minimizing the mean square error (MSE) between the predicted and the measured exciter field currents, serving as the CF. The cost function calculation is shown in (5).  
\[CF=\frac{1}{N}\sum_{n=1}^{N}\left(I_{fc,\ cal}-I_{fc,\ mea}\right)^{2} \tag{5}\]  
The adaptive moment estimation (ADAM) algorithm, which employs the exponential moving average of both the gradient or first momentum (\(\
abla\)CF) and the squared gradient or second momentum (\(\
abla^{2}\)CF), is used for gradient descent optimization. The diagram of the employed backpropagation method is provided in Figure 6.  
Also, the following equations allow the weights vector at epoch \(t\) to be determined from the vectors at the previous epoch (\(t-1\)):  
\[W_{t}=W_{t-1}-l\cdot\frac{\hat{m_{t}}}{\sqrt{\hat{v_{t}}+\varepsilon}} \tag{6}\]  
\[\hat{m_{t}}=m_{t}\cdot\frac{1}{1-\beta_{m}} \tag{7}\]  
\[m_{t}=\beta_{m}\cdot m_{t-1}+(1-\beta_{m})\cdot\
abla CF \tag{8}\]  
\[\hat{v_{t}}=v_{After training, hyperparameter tuning is performed to achieve the best-fitting network. During this validation phase, various neural networks were evaluated to determine the optimal number of hidden layers and nodes in each layer. The network that reduced the maximum absolute error (MAE) to 0.1% was selected.