{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,17]],"date-time":"2026-04-17T15:28:41Z","timestamp":1776439721143,"version":"3.51.2"},"reference-count":35,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T00:00:00Z","timestamp":1735776000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>In this paper, a multi-physics case study belonging to the class of induction heating problem is considered. Finite Element Analysis is used to evaluate the temperature along a line on a graphite disk heated by two power inductors. In order to build a surrogate field model of the device, i.e., to compute the temperature profile on the disk, given the amplitudes and frequencies of the supply currents, three methods have been used (Support Vector Regression (SVR), fully connected Neural Network (NN) and Gaussian Process Regression (GPR)). In turn, to solve the inverse problem, i.e., to identify frequencies and currents of the two coils, given a prescribed temperature profile, two approaches have been implemented. The former is an optimization approach based on a multi-objective formulation, solved by means of the NSGA-II algorithm; the latter is a two-step procedure, based on fully connected Deep Neural Networks (DNNs), solving an optimal design problem first and, subsequently, an optimal control problem.<\/jats:p>","DOI":"10.3390\/a18010010","type":"journal-article","created":{"date-parts":[[2025,1,2]],"date-time":"2025-01-02T06:05:10Z","timestamp":1735797910000},"page":"10","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal Multi-Physics Synthesis of a Dual-Frequency Power Inductor Using Deep Neural Networks and Gaussian Process Regression"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5293-1809","authenticated-orcid":false,"given":"Paolo","family":"Di Barba","sequence":"first","affiliation":[{"name":"Department of Electrical Computer and Biomedical Engineering, University of Pavia, 27100 Pavia, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-2419-5466","authenticated-orcid":false,"given":"Arash","family":"Ghafoorinejad","sequence":"additional","affiliation":[{"name":"Department of Electrical Computer and Biomedical Engineering, University of Pavia, 27100 Pavia, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3300-0296","authenticated-orcid":false,"given":"Maria Evelina","family":"Mognaschi","sequence":"additional","affiliation":[{"name":"Department of Electrical Computer and Biomedical Engineering, University of Pavia, 27100 Pavia, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fabrizio","family":"Dughiero","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, University of Padova, 35131 Padova, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2565-2983","authenticated-orcid":false,"given":"Michele","family":"Forzan","sequence":"additional","affiliation":[{"name":"Department of Industrial Engineering, University of Padova, 35131 Padova, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5297-0576","authenticated-orcid":false,"given":"Elisabetta","family":"Sieni","sequence":"additional","affiliation":[{"name":"Department of Theoretical and Applied Sciences, University of Insubria, 21100 Varese, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,1,2]]},"reference":[{"key":"ref_1","first-page":"11","article-title":"Efficient Heating by Electromagnetic Sources in Metallurgical Processes: Recent Applications and Development Trends","volume":"86","author":"Baake","year":"2010","journal-title":"Przegl\u0105d Elektrotech."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Rudnev, V., Loveless, D., and Cook, R. (2017). Handbook of Induction Heating. Manufacturing Engineering and Materials Processing, CRC Press. [2nd ed.].","DOI":"10.1201\/9781315117485"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Rudnev, V., and Totten, G.E. (2014). Induction Heating of Selective Regions. Induction Heating and Heat Treatment, ASM International.","DOI":"10.31399\/asm.hb.v04c.9781627081672"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hetnarski, R.B. (2014). Induction Heating. Encyclopedia of Thermal Stresses, Springer.","DOI":"10.1007\/978-94-007-2739-7"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Rapoport, E., and Pleshivtseva, Y. (2007). Optimal Control of Induction Heating Processes, CRC\/Taylor & Francis. Mechanical Engineering.","DOI":"10.1201\/9781420019490"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Brazhnik, D.S., and Bolotin, K.E. (2020, January 27\u201330). Different Approaches to Taking Joule Heat into Induction Heating of Graphite Crucible. Proceedings of the 2020 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), St. Petersburg and Moscow, Russia.","DOI":"10.1109\/EIConRus49466.2020.9039247"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"012065","DOI":"10.1088\/1757-899X\/643\/1\/012065","article-title":"Numerical Simulation of the Induction Heating Process of a Disk Profile","volume":"643","author":"Mannanov","year":"2019","journal-title":"IOP Conf. Ser. Mater. Sci. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"818","DOI":"10.1016\/j.apm.2022.07.009","article-title":"Coupled Electromagnetic-Thermal Solution Strategy for Induction Heating of Ferromagnetic Materials","volume":"111","author":"Fisk","year":"2022","journal-title":"Appl. Math. Model."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2770","DOI":"10.1016\/j.apm.2015.10.006","article-title":"Approximate Analytical Solution for Induction Heating of Solid Cylinders","volume":"40","author":"Jankowski","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1108\/03321640310452213","article-title":"Optimal Shape Design of Devices and Systems for Induction-Heating: Methodologies and Applications","volume":"22","author":"Dughiero","year":"2003","journal-title":"COMPEL Int. J. Comput. Math. Electr. Electron. Eng."},{"key":"ref_11","first-page":"1","article-title":"Training Sample Selection Strategy Applied to CNN in Magneto-Thermal Coupled Analysis","volume":"57","author":"Gong","year":"2021","journal-title":"IEEE Trans. Magn."},{"key":"ref_12","first-page":"1","article-title":"A Numerical Twin Model for the Coupled Field Analysis of TEAM Workshop Problem 36","volume":"59","author":"Mognaschi","year":"2023","journal-title":"IEEE Trans. Magn."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2074","DOI":"10.1016\/j.apm.2012.04.058","article-title":"Numerical Optimisation for Induction Heat Treatment Processes","volume":"37","author":"Naar","year":"2013","journal-title":"Appl. Math. Model."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1109\/TMAG.2011.2175712","article-title":"Numerical Model of Induction Shrink Fits in Monolithic Formulation","volume":"48","author":"Karban","year":"2012","journal-title":"IEEE Trans. Magn."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"305","DOI":"10.1108\/03321640510571327","article-title":"The Optimisation of Induction Heating System Based on Multiquadric Function Approximation","volume":"24","author":"Zgraja","year":"2005","journal-title":"COMPEL Int. J. Comput. Math. Electr. Electron. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1109\/TMAG.2013.2284024","article-title":"Shared-Memory Parallelism and Low-Rank Approximation Techniques Applied to Direct Solvers in FEM Simulation","volume":"50","author":"Amestoy","year":"2014","journal-title":"IEEE Trans. Magn."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"413","DOI":"10.3233\/JAE-2008-994","article-title":"Finite Element Analysis of Coupled Electromagnetic and Thermal Fields within a Practical Induction Heating Cooker","volume":"28","author":"Chen","year":"2008","journal-title":"Int. J. Appl. Electromagn. Mech."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3064","DOI":"10.1109\/20.34370","article-title":"Simulation of Induction Machine Operation Using Complex Magnetodynamic Finite Elements","volume":"25","author":"Vassent","year":"1989","journal-title":"IEEE Trans. Magn."},{"key":"ref_19","unstructured":"Forzan, M., Maccalli, G., Valente, G., and Crippa, D. (2006, January 8\u20139). Design of an Innovative Heating Process System for the Epitaxial Growth of Silicon Carbide Layers Wafer. Proceedings of the MMP-Modelling for Material Processing, Riga, Latvia."},{"key":"ref_20","first-page":"139","article-title":"New Solutions to a Multi-Objective Benchmark Problem of Induction Heating: An Application of Computational Biogeography and Evolutionary Algorithms","volume":"67","author":"Dughiero","year":"2018","journal-title":"Arch. Electr. Eng."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"182","DOI":"10.1109\/4235.996017","article-title":"A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II","volume":"6","author":"Deb","year":"2002","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Deb, K. (2004). Multi-Objective Optimization Using Evolutionary Algorithms, Wiley.","DOI":"10.1142\/9789812702838_0003"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1109\/4235.797969","article-title":"Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach","volume":"3","author":"Zitzler","year":"1999","journal-title":"IEEE Trans. Evol. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1162\/evco.1994.2.3.221","article-title":"Multiobjective Optimization Using Nondominated Sorting in Genetic Algorithms","volume":"2","author":"Srinivas","year":"1994","journal-title":"Evol. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Meunier, G. (2008). The Finite Element Method for Electromagnetic Modeling, Wiley.","DOI":"10.1002\/9780470611173"},{"key":"ref_26","unstructured":"Binns, K.J., Lawrenson, P.J., and Trowbridge, C.W. (1992). The Analytical and Numerical Solution of Electric and Magnetic Fields, Wiley."},{"key":"ref_27","unstructured":"Carslaw, H. (1986). Conduction of Heat in Solids, Oxford University Press. [2nd ed.]."},{"key":"ref_28","unstructured":"COMSOL (2024, November 11). Multiphysics Software for Optimizing Designs. Available online: https:\/\/www.comsol.com\/."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.finel.2016.07.012","article-title":"Simulation of Multi-Frequency-Induction-Hardening Including Phase Transitions and Mechanical Effects","volume":"121","author":"Liu","year":"2016","journal-title":"Finite Elem. Anal. Des."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1023\/B:STCO.0000035301.49549.88","article-title":"A Tutorial on Support Vector Regression","volume":"14","author":"Smola","year":"2004","journal-title":"Stat. Comput."},{"key":"ref_31","unstructured":"(2024, December 24). Train Kernel Approximation Model Using Regression Learner App. Available online: https:\/\/it.mathworks.com\/help\/stats\/train-kernel-approximation-using-regression-learner-app.html."},{"key":"ref_32","unstructured":"Rasmussen, C.E., and Williams, C.K.I. (2008). Adaptive Computation and Machine Learning. Gaussian Processes for Machine Learning, MIT Press. 3. print."},{"key":"ref_33","unstructured":"(2024, December 24). Regression Learner. Available online: https:\/\/it.mathworks.com\/help\/stats\/regressionlearner-app.html."},{"key":"ref_34","unstructured":"Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep Learning, The MIT Press."},{"key":"ref_35","unstructured":"(2024, December 24). NSGA-II: A Multi-Objective Optimization Algorithm. Available online: https:\/\/it.mathworks.com\/matlabcentral\/fileexchange\/10429-nsga-ii-a-multi-objective-optimization-algorithm."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/1\/10\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T15:23:27Z","timestamp":1759850607000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/1\/10"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,1,2]]},"references-count":35,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["a18010010"],"URL":"https:\/\/doi.org\/10.3390\/a18010010","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,1,2]]}}}