## 5 Discussion  
The three surrogate models trained for solving the coupled field problem are able to provide a rather accurate temperature profile along a line of the graphite disk, specifically, the \(\alpha\) line, with known source terms i.e., magnitude and frequency of the current of each of the two inductors. Considering the evaluation parameters in Table 2, they are in agreement for each surrogate model in terms of accuracy: SVR shows the worst performance, with its R-squared being the lowest among the three methods and its RMSE and MAPE being the highest. Overall, the performance of SVR is the worst for our case study. The NN model shows values of R-squared, RMSE, and MAPE in the middle of the three methods. The best method is GPR because its R-squared is the closest to 1 among the three methods and its RMSE and MAPE errors are the lowest. This performance can be also assessed from the example shown in Figure 5, where the temperature profile for one case, approximated by means of the three methods, is shown and compared to the one calculated by means of the Finite Element method. The worst approximation is given by the SVR model, while NN and GPR better approximate the actual profile. However, the temperature profile given by the NN is less smooth than the one approximated by GPR, at a few points. Considering the training time, the GPR model is the most expensive in terms of computational burden, while the NN model is the lightest. Figure 10: Optimal temperature profile obtained using the DNN approach [\(f_{1\text{d}}\) = 38.5 kHz, \(I_{1\text{d}}\) = 255 A,\(f_{2\text{cc}}\) = 1.38 kHz, \(I_{2\text{cc}}\) = 996 A]. Figure 9: Optimal control DNN actual values vs. predicted values (test and validation sets). However, based on the accuracy, the GPR model is the best one, therefore, it was used as the surrogate model in the solution of the inverse problem, applied with both of the two approaches. The multi-objective approach found a Pareto front, depicted in Figure 6, which can be divided into two parts. The right part of the front, characterized by low \(v_{2}\) values, is representative of homogeneous temperature profiles, with a small difference between the maximum and lowest values along the profile. However, these profiles have many points outside the gray zone (\(\pm\)1% of 1100 \({}^{\circ}\)C, see Figure 7), which means high values for the objective function \(v_{1}\). An example of this behavior is shown by point P\({}_{2}\) in Figure 7: a rather flat temperature profile, far from the target temperature. Conversely, the left part of the front is characterized by low values of \(v_{1}\), which means a small number of points of the temperature profile outside the gray zone, but high \(v_{2}\) values, i.e., a high discrepancy between the minimum and maximum values of the temperature profile. The temperature profile of point P\({}_{1}\), belonging to this part of the front, is shown in Figure 7: it is not a flat profile but many of its points are close to the target temperature. Finally, point P\({}_{3}\) is a trade-off between P\({}_{1}\) and P\({}_{2}\), being in the middle, also called the knee-point, of the front. P\({}_{3}\) shows a rather homogenous profile, as depicted in Figure 7, and it is close to the target temperature. Looking at Table 3, where the design variables of the three solutions P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\) are shown, it can be noted that the two solutions with a rather homogenous profile, P\({}_{2}\) and P\({}_{3}\), show a low value of frequency \(f_{1}\) (inner coil). While they have a significantly different current \(I_{1}\) (higher for P\({}_{3}\), which has higher temperatures close to the disk center, specifically close to the target temperature). Considering the outer coil, points P\({}_{1}\) and P\({}_{3}\), which have temperature profiles close to the target temperature, have a higher current \(I_{2}\) with respect to point P\({}_{2}\), which has a temperature profile far from the target temperature.