Optimal Multi-Physics Synthesis of a Dual-Frequency Power Inductor Using Deep Neural Networks and Gaussian Process Regression  
[PERSON]  
1 Department of Electrical Computer and Biomedical Engineering, University of Pavia, 27100 Pavia, Italy; [EMAIL_ADDRESS] (P.D.B. ); [EMAIL_ADDRESS] (A.G.)1 Department of Industrial Engineering, University of Padova, 35131 Padova, Italy; [EMAIL_ADDRESS] (F.D. ); [EMAIL_ADDRESS] (M.F. )2 Department of Theoretical and Applied Sciences, University of Insubria, 21100 Varese, Italy; [EMAIL_ADDRESS]; Tel. : +39-0382-98578512  
[PERSON]  
1 Department of Electrical Computer and Biomedical Engineering, University of Pavia, 27100 Pavia, Italy; [EMAIL_ADDRESS] (P.D.B. ); [EMAIL_ADDRESS] (A.G.)1 Department of Industrial Engineering, University of Padova, 35131 Padova, Italy; [EMAIL_ADDRESS] (F.D. ); [EMAIL_ADDRESS] (M.F. )2 Department of Theoretical and Applied Sciences, University of Insubria, 21100 Varese, Italy; [EMAIL_ADDRESS] (F.  
rapid heating and precise temperature control, allowing uniform heating in the whole workpiece, which is essential in processes such as hardening, annealing, and brazing in metalworking. Furthermore, induction heating systems are clean and safe because they do not involve open flames or toxic byproducts, enhancing workplace safety and meeting stringent industry standards for emissions and pollutants. However, effective induction heating requires precise calibration of parameters, including frequency and power. A wrong setup of the inductors or badly designed inductors deteriorates the efficiency of the process. Hence, one of the main challenges in induction heating is the design of inductors and their parameter tuning. To this end, automated optimal designs and control of inductors have been proposed in the literature in the last few decades. Moreover, numerical methods such as Finite Element Analysis have replaced analytical formulations, increasing the accuracy of the solution [6; 7]. Specifically, the field problem requires the solution of coupled electromagnetic and thermal problems to simulate the temperature in the workpiece generated by the power supplied by an inductor. In complicated geometry, this task requires a big effort in terms of computation time and hardware resources since an electromagnetic problem is coupled to a thermal problem [8; 9]. These features made this problem an interesting and well-known benchmark, solved in different areas of computational electromagnetism in a comparative way [10]. With the advent of innovative algorithms based, e.g., on Neural Networks (NNs), it is possible to build reliable surrogate models for predicting the temperature distribution in a workpiece during induction heating processes [11; 12]. Actually, surrogate models help very much in the design of inductors, because a typical optimization routine, using, e.g., genetic algorithms, requires the solution of a large number of parametric geometries to find a set of improved solutions [13; 14; 15]. For example, when a multi-objective optimization is considered, the Pareto approach coupled with FEA is very time-expensive; therefore, in this area, where the field problem involves both electromagnetic and temperature fields, giving rise to a coupled problem, it is convenient to use a surrogate model to reduce the number of required FEA analyses [13; 16; 17; 18]. In terms of applications, in this paper, an induction heating system for epitaxial growth of silicon is considered, as described in two patents ([PERSON]; [PERSON]; [PERSON] Deposition Reactor with Inductors and Electromagnetic Shields 2021, EP3870734A1; [PERSON]; [PERSON]; [PERSON] [PERSON] Inductively Heatable Susceptor and Epitaxial Deposition Reactor 2017, WO2017137872A1) and a study [19]. The proposed geometry is a pancake inductor that uniformly heats a graphite disk. The aim of the device design is to obtain a prescribed temperature profile, which should be as uniform as possible along the disk surface and should reach a given temperature.