## 2 Forward Model
### Surrogate Model  
In order to solve the coupled-field problem in a fast way, a surrogate model is implemented. Specifically, for predicting the temperature on the disk profile and knowing the amplitudes and frequencies of the current inductors, three methods are investigated: Support Vector Regression (SVR), Gaussian Process Regression (GPR), and a fully connected Neural Network (NN). The SVR predicts the outputs as:  
\[f(x^{*})=\sum_{i=1}^{n}{(\alpha_{i}-\alpha_{i}^{*})K(x_{i},x^{*})}+b \tag{6}\]  
where \(\alpha_{i},\alpha_{i}^{*}\) are the variables determined during training, \(K(x_{i},x^{*})\) is the kernel function, \(b\) is the bias term, which will be optimized during training, and \(x_{i}\) is a support vector (subset of the training vector points that are on or outside the epsilon margin). In the paper, the cubic kernel is utilized:  
\[K(x_{i},x^{*})=\left(x_{i}^{T}x^{*}\right)^{3} \tag{7}\]  
where \(x^{*}\) is the input and \(x_{i}\) is a training point. The epsilon-insensitive loss function is used:  
\[L(y,\hat{y})= \begin{cases}0&\quad if\ |y-\hat{y}|\leq\epsilon\\ |y-\hat{y}|-\epsilon&\quad\text{otherwise}\end{cases}  \tag{8}\]  
where \(y\) is the true value, \(\hat{y}=f(x)\) is the predicted value, and \(\epsilon\) is the tolerance margin and the input features were standardized to zero mean and unit variance before training. The training procedure solves a dual optimization problem to find \(\alpha_{i},\alpha_{i}^{*}\)[30]. The SVR algorithm used in this study is available in Matlab 2024b [31]. In this study, GPR was implemented with a constant basis function (\(\beta_{0}\)) and an isotropic exponential kernel function, which defines the covariance:  
\[K(x,xt)=\sigma_{f}^{2}e^{(-\frac{|x-xt|}{l})} \tag{9}\]  
where \(\sigma_{f}^{2}\) is the signal variance and \(l\) is the length scale and the same length scale is used for all input dimensions. The exponential kernel is commonly used in GPR because it is computationally efficient. GPR's objective function is an optimization process which focuses on optimizing the hyperparameters (\(\sigma_{f}^{2},l\), noise variance \(\sigma_{n}^{2}\)) and the basis function (\(\beta_{0}\)) by maximizing the log marginal likelihood [32]. Using training data \(\{X,y\}\), for a new point \(x^{*}\), the predicted average and variance are:  
\[\mu(x^{*})=\beta_{0}+k^{T}\Big{[}K+\sigma_{n}^{2}I\Big{]}^{-1}(y-\beta_{0}) \tag{10}\]  
\[\sigma^{2}(x^{*})=K(x^{*},x^{*})-k^{T}\Big{[}K+\sigma_{n}^{2}I\Big{]}^{-1}k \tag{11}\]  
where the \(k\) is the covariance vector between \(x^{*}\) and the training points, \(K\) is the covariance matrix of the training data, and \(K(x^{*},x^{*})\) is the covariance of \(x^{*}\) with itself. The GPR algorithm used in this study is available in Matlab [33]. The neural network that is used in this study for the forward model is a fully connected deep neural network (DNN) with three hidden layers with 100 neurons in each one. In all layers, the ReLU (Rectified Linear Unit) activation function is used:  
\[ReLU(x)=\max(0,x), \tag{12}\]The loss function is the well-known mean squared error (MSE):  
\[MSE=\frac{1}{n}{\sum_{i=1}^{n}}(y_{i}-\hat{y}_{i})^{2}, \tag{13}\]  
where \(y_{i}\) is the actual value and \(\hat{y}_{i}\) is the predicted value. LBFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) is chosen for the solver, which utilizes an estimation of the Hessian matrix. This solver is selected because it has some of the advantages of Newton-like methods while it has a much lower computational cost [34]. The maximum number of iterations for training is set to 1000. A database, namely DB1, based on the FE model, is built for training purposes. It is composed of \(N_{s}=5832\) samples. Each sample contains two vectors, one with the current amplitudes and frequencies and another with the relevant \(N_{p}=250\) temperature values along the \(\alpha\) line (see Figure 1). All-in-one \(N_{s}\) couples of vectors compose the database.