## 4 Results
### Second Approach: Deep Neural Network Approach  
The optimal design problem was solved using a fully connected DNN with 6 layers with 80, 40, 30, 20, 10, and 8 neurons per layer from the first to the last layer, respectively. The following statement is assumed: the input of the NN is the prescribed temperature profile and the outputs are the current and frequency of the second coil. The NN is trained utilizing the database DB2. Specifically, DB2 contains those solutions with temperature profiles around 1100 \({}^{\circ}\)C with \(\pm 5\%\) range, considering the first 25 mm of the graphite disk. In fact, the main effect of the first coil is on the first 25 mm, and it guarantees that for all  
Figure 6: Starting and arrival population of the multi-objective optimization.  
Figure 7: Temperature profiles of 3 optimal individuals P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\) (see Figure 6).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Points** & **v\({}_{1}\)** & **v\({}_{2}\) (\({}^{\circ}\)C)** & \(f_{1}\) (kHz)** & \(I_{1}\) (A) & \(f_{2}\) (kHz) & \(I_{2}\) (A) \\ \hline P\({}_{1}\) & 74 & 238.95 & 100 & 305 & 1 & 1137 \\ P\({}_{2}\) & 200 & 10.02 & 20 & 100 & 1.04 & 400 \\ P\({}_{3}\) & 97 & 64.16 & 20 & 288 & 1.01 & 1134 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Optimal points of the Pareto front.  
DB2 samples (of which there are 998), the first coil acts in the selected range. In DB2, the average \(f_{1 od}\) of the frequency f\({}_{1}\) of the samples is considered and it is equal to 38.5 kHz.  
The DNN is trained using 70% of samples for training, 15% for validation, and 15% for testing. The RMSE and MAPE errors, calculated with values normalized in the range \([-1,1]\), for the validation and test sets, are 418 and 9.72%, respectively. In Figure 8, the predicted versus actual values of the validation and test sets are plotted.  
By solving the optimal design problem with the prescribed temperature profile being homogenous and equal to 1100 \({}^{\circ}\)C, the optimal values of \(f_{2}\) and \(I_{2}\) are identified: \(f_{2 od}\) = 1.38 kHz and \(I_{2 od}\) = 1.02 kA.  
Once the optimal design problem has been solved, the optimal control problem is handled.  
Knowing the optimal values of \(f_{1}\) (being the average of the frequencies in DB2) and \(f_{2}\) (obtained by solving the optimal design problem), the database DB3 is prepared. Specifically, new samples are generated utilizing the forward surrogate model GPR, considering \(f_{1 od}\) = 38.5 kHz, \(f_{2 od}\) = 1.38 kHz, and the current varying between 100 and 400 A for the first coil and between 400 and 2000 A for the second one. This way, the new database DB3 with 2665 samples is created.  
The optimal control problem is solved using a fully connected NN with 6 layers with 80, 40, 30, 20, 10, and 8 neurons per layer from the first to the last layer, respectively. The following statement is assumed: the input of the NN is the prescribed temperature profile and the outputs are the current intensity of the two coils.  
The DNN is trained using 70% of samples for training, 15% for validation, and 15% for testing. The RMSE and MAPE errors, calculated with values normalized in the range \([-1,1]\), for the validation and test sets, are 1.95 and 0.21%, respectively. In Figure 9, the predicted versus actual values of the validation and test sets are plotted.  
By solving the optimal control problem with the prescribed temperature profile being homogenous and equal to 1100 \({}^{\circ}\)C, the optimal values of \(I_{1}\) and \(I_