{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T02:15:40Z","timestamp":1773368140645,"version":"3.50.1"},"reference-count":72,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T00:00:00Z","timestamp":1678406400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Micro-electro-mechanical-systems are complex structures, often involving nonlinearites of geometric and multiphysics nature, that are used as sensors and actuators in countless applications. Starting from full-order representations, we apply deep learning techniques to generate accurate, efficient, and real-time reduced order models to be used for the simulation and optimization of higher-level complex systems. We extensively test the reliability of the proposed procedures on micromirrors, arches, and gyroscopes, as well as displaying intricate dynamical evolutions such as internal resonances. In particular, we discuss the accuracy of the deep learning technique and its ability to replicate and converge to the invariant manifolds predicted using the recently developed direct parametrization approach that allows the extraction of the nonlinear normal modes of large finite element models. Finally, by addressing an electromechanical gyroscope, we show that the non-intrusive deep learning approach generalizes easily to complex multiphysics problems.<\/jats:p>","DOI":"10.3390\/s23063001","type":"journal-article","created":{"date-parts":[[2023,3,10]],"date-time":"2023-03-10T03:35:42Z","timestamp":1678419342000},"page":"3001","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Reduced Order Modeling of Nonlinear Vibrating Multiphysics Microstructures with Deep Learning-Based Approaches"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1914-9123","authenticated-orcid":false,"given":"Giorgio","family":"Gobat","sequence":"first","affiliation":[{"name":"Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8599-6588","authenticated-orcid":false,"given":"Stefania","family":"Fresca","sequence":"additional","affiliation":[{"name":"MOX\u2014Department of Mathematics, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8277-2802","authenticated-orcid":false,"given":"Andrea","family":"Manzoni","sequence":"additional","affiliation":[{"name":"MOX\u2014Department of Mathematics, Politecnico di Milano, P.za Leonardo da Vinci 32, 20133 Milano, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7100-1321","authenticated-orcid":false,"given":"Attilio","family":"Frangi","sequence":"additional","affiliation":[{"name":"Department of Civil and Environmental Engineering, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1007\/s00466-020-01902-5","article-title":"Non-intrusive reduced order modeling for the dynamics of geometrically nonlinear flat structures using three-dimensional finite elements","volume":"66","author":"Vizzaccaro","year":"2020","journal-title":"Comput. 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