{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T03:33:13Z","timestamp":1762918393154,"version":"3.40.5"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,9,7]],"date-time":"2024-09-07T00:00:00Z","timestamp":1725667200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,9,7]],"date-time":"2024-09-07T00:00:00Z","timestamp":1725667200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100005073","name":"Agency for Defense Development","doi-asserted-by":"publisher","award":["UD230016JD","UD230016JD","UD230016JD"],"award-info":[{"award-number":["UD230016JD","UD230016JD","UD230016JD"]}],"id":[{"id":"10.13039\/501100005073","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002551","name":"Seoul National University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100002551","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2025,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Finite element (FE) analysis is one of the most accurate methods for predicting electromagnetic field scatter; however, it presents a significant computational overhead. In this study, we propose a data-driven parametric model-order reduction \u00a0(pMOR)\u00a0framework to predict the scattered electromagnetic field of FE analysis. The surface impedance of a coated component is selected as parameter of analysis. A physics-aware (PA) neural network incorporated within a least-squares hierarchical-variational autoencoder (LSH-VAE) is selected for the data-driven pMOR method. The proposed PA-LSH-VAE framework directly accesses the scattered electromagnetic field represented by a large number of degrees of freedom (DOFs). Furthermore, it captures the behavior along with the variation of the complex-valued multi-parameters. A parallel computing approach is used to generate the training data efficiently. The PA-LSH-VAE framework is designed to handle over 2 million DOFs, providing satisfactory accuracy and exhibiting a second-order speed-up factor.<\/jats:p>","DOI":"10.1007\/s00366-024-02056-1","type":"journal-article","created":{"date-parts":[[2024,9,7]],"date-time":"2024-09-07T11:01:56Z","timestamp":1725706916000},"page":"785-799","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Physics-aware neural network-based parametric model-order reduction of the electromagnetic analysis for a coated component"],"prefix":"10.1007","volume":"41","author":[{"given":"SiHun","family":"Lee","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Seung-Hoon","family":"Kang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sangmin","family":"Lee","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"SangJoon","family":"Shin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,9,7]]},"reference":[{"key":"2056_CR1","doi-asserted-by":"publisher","first-page":"1235","DOI":"10.1007\/s42405-021-00372-0","volume":"22","author":"D-H Son","year":"2021","unstructured":"Son D-H, Hyun J-M, Chaki S, Park CH, Lee J-R (2021) Evaluation of mechanical\/electromagnetic preformation of single-sided active frequency selective surface for stealth radomes. Int J Aeronaut Space Sci 22:1235\u20131242. https:\/\/doi.org\/10.1007\/s42405-021-00372-0","journal-title":"Int J Aeronaut Space Sci"},{"key":"2056_CR2","volume-title":"The finite element method in electromagnetics","author":"J-M Jin","year":"2014","unstructured":"Jin J-M (2014) The finite element method in electromagnetics, 3rd edn. Wiley-IEEE Press, Hoboken","edition":"3"},{"key":"2056_CR3","unstructured":"ANSYS Inc. (2021) HFSS help; release 2021R1. ANSYS Inc., Canonsburg"},{"issue":"12","key":"2056_CR4","doi-asserted-by":"publisher","first-page":"889","DOI":"10.5139\/JKSAS.2022.50.12.889","volume":"50","author":"S-H Kang","year":"2022","unstructured":"Kang S-H, Song D-H, Choi JW, Shin SJ (2022) Parallel computation on the three-dimensional electromagnetic field by the graph partitioning and multi-frontal method. J Korean Soc Aeronaut Space Sci 50(12):889\u2013898. https:\/\/doi.org\/10.5139\/JKSAS.2022.50.12.889. (in Korean)","journal-title":"J Korean Soc Aeronaut Space Sci"},{"issue":"10","key":"2056_CR5","doi-asserted-by":"publisher","first-page":"3000","DOI":"10.1109\/TAP.2006.882191","volume":"54","author":"Y-J Li","year":"2006","unstructured":"Li Y-J, Jin J-M (2006) A vector dual-primal finite element tearing and interconnecting method for solving 3-D large-scale electromagnetic problems. IEEE Trans Antennas Propag 54(10):3000\u20133009. https:\/\/doi.org\/10.1109\/TAP.2006.882191","journal-title":"IEEE Trans Antennas Propag"},{"issue":"10","key":"2056_CR6","doi-asserted-by":"publisher","first-page":"2803","DOI":"10.1109\/TAP.2007.905954","volume":"55","author":"Y-J Li","year":"2007","unstructured":"Li Y-J, Jin J-M (2007) A new dual-primal domain decomposition approach for finite element simulation of 3-D large-scale electromagnetic problems. IEEE Trans Antennas Propag 55(10):2803\u20132810. https:\/\/doi.org\/10.1109\/TAP.2007.905954","journal-title":"IEEE Trans Antennas Propag"},{"issue":"3","key":"2056_CR7","doi-asserted-by":"publisher","first-page":"2193","DOI":"10.1137\/080728536","volume":"31","author":"V Dolean","year":"2009","unstructured":"Dolean V, Gander MJ, Gerardo-Giorda L (2009) Optimized Schwarz methods for Maxwell\u2019s equations. SIAM J Sci Comput 31(3):2193\u20132213. https:\/\/doi.org\/10.1137\/080728536","journal-title":"SIAM J Sci Comput"},{"key":"2056_CR8","doi-asserted-by":"publisher","first-page":"920","DOI":"10.1016\/j.jcp.2014.06.040","volume":"274","author":"M-F Xue","year":"2014","unstructured":"Xue M-F, Jin J-M (2014) A preconditioned dual-primal finite element tearing and interconnecting method for solving three-dimensional time-harmonic Maxwell\u2019s equations. J Comput Phys 274:920\u2013935. https:\/\/doi.org\/10.1016\/j.jcp.2014.06.040","journal-title":"J Comput Phys"},{"issue":"6","key":"2056_CR9","doi-asserted-by":"publisher","first-page":"2604","DOI":"10.1109\/TAP.2015.2417977","volume":"63","author":"I Voznyuk","year":"2015","unstructured":"Voznyuk I, Tortel H, Litman A (2015) 3-D electromagnetic scattering computation in free-space with the FETI-FDP2 method. IEEE Trans Antennas Propag 63(6):2604\u20132613. https:\/\/doi.org\/10.1109\/TAP.2015.2417977","journal-title":"IEEE Trans Antennas Propag"},{"issue":"4","key":"2056_CR10","doi-asserted-by":"publisher","first-page":"1886","DOI":"10.1109\/TAP.2017.2670541","volume":"65","author":"F-X Roux","year":"2017","unstructured":"Roux F-X, Barka A (2017) Block Krylov recycling algorithms for FETI-2LM applied to 3-D electromagnetic wave scattering and radiation. IEEE Trans Antennas Propag 65(4):1886\u20131895. https:\/\/doi.org\/10.1109\/TAP.2017.2670541","journal-title":"IEEE Trans Antennas Propag"},{"key":"2056_CR11","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1016\/j.cma.2019.06.018","volume":"355","author":"D Xiao","year":"2019","unstructured":"Xiao D (2019) Error estimation of the parametric non-intrusive reduced order model using machine learning. Comput Methods Appl Mech Eng 355:513\u2013534. https:\/\/doi.org\/10.1016\/j.cma.2019.06.018","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"3","key":"2056_CR12","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1002\/nme.5624","volume":"113","author":"A Moosavi","year":"2018","unstructured":"Moosavi A, \u015etef\u0103nescu R, Sandu A (2018) Multivariate predictions of local reduced-order-model errors and dimensions. Int J Numer Methods Eng 113(3):512\u2013533. https:\/\/doi.org\/10.1002\/nme.5624","journal-title":"Int J Numer Methods Eng"},{"key":"2056_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113999","volume":"384","author":"S Lee","year":"2021","unstructured":"Lee S, Jang K, Cho H, Kim H, Shin SJ (2021) Parametric non-intrusive model order reduction for flow-fields using unsupervised machine learning. Comput Methods Appl Mech Eng 384:113999. https:\/\/doi.org\/10.1016\/j.cma.2021.113999","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2056_CR14","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/s00366-023-01782-2","volume":"40","author":"S Lee","year":"2024","unstructured":"Lee S, Jang K, Lee S, Cho H, Shin SJ (2024) Parametric model order reduction by machine learning for fluid\u2013structure interaction analysis. Eng Comput 40:45\u201360. https:\/\/doi.org\/10.1007\/s00366-023-01782-2","journal-title":"Eng Comput"},{"key":"2056_CR15","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/j.jcp.2018.02.037","volume":"363","author":"JS Hesthaven","year":"2018","unstructured":"Hesthaven JS, Ubbiali S (2018) Non-intrusive reduced order modeling of nonlinear problems using neural networks. J Comput Phys 363:55\u201378. https:\/\/doi.org\/10.1016\/j.jcp.2018.02.037","journal-title":"J Comput Phys"},{"key":"2056_CR16","unstructured":"Gonzalez FJ, Balajewicz M (2018) Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems. arXiv:1808.01346"},{"key":"2056_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.advwatres.2021.104098","volume":"160","author":"T Kadeethum","year":"2022","unstructured":"Kadeethum T, Ballarin F, Choi Y, O\u2019Malley D, Yoon H, Bouklas N (2022) Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques. Adv Water Resour 160:104098. https:\/\/doi.org\/10.1016\/j.advwatres.2021.104098","journal-title":"Adv Water Resour"},{"key":"2056_CR18","doi-asserted-by":"publisher","first-page":"20654","DOI":"10.1038\/s41598-022-24545-3","volume":"12","author":"T Kadeethum","year":"2022","unstructured":"Kadeethum T, Ballarin F, O\u2019Malley D, Choi Y, Bouklas N, Yoon H (2022) Reduced order modeling for flow and transport problems with Barlow Twins self-supervised learning. Sci Rep 12:20654. https:\/\/doi.org\/10.1038\/s41598-022-24545-3","journal-title":"Sci Rep"},{"key":"2056_CR19","doi-asserted-by":"publisher","first-page":"2165","DOI":"10.1007\/s11071-022-07733-8","volume":"110","author":"H Kim","year":"2022","unstructured":"Kim H, Cheon S, Jeong I, Cho H, Kim H (2022) Enhanced model reduction method via combined supervised and unsupervised learning for real-time solution of nonlinear structural dynamics. Nonlinear Dyn 110:2165\u20132195. https:\/\/doi.org\/10.1007\/s11071-022-07733-8","journal-title":"Nonlinear Dyn"},{"issue":"1","key":"2056_CR20","doi-asserted-by":"publisher","DOI":"10.1063\/1.5067313","volume":"9","author":"N Omata","year":"2019","unstructured":"Omata N, Shirayama S (2019) A novel method of low-dimensional representation for temporal behavior of flow fields using deep autoencoder. AIP Adv 9(1):015006. https:\/\/doi.org\/10.1063\/1.5067313","journal-title":"AIP Adv"},{"key":"2056_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.108973","volume":"404","author":"K Lee","year":"2020","unstructured":"Lee K, Carlberg KT (2020) Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J Comput Phys 404:108973. https:\/\/doi.org\/10.1016\/j.jcp.2019.108973","journal-title":"J Comput Phys"},{"key":"2056_CR22","unstructured":"Kingma DP, Welling M (2013) Auto-encoding variational bayes. arXiv:1312.6114"},{"key":"2056_CR23","unstructured":"S\u00f8nderby CK, Raiko T, Maal\u00f8e L, S\u00f8nderby SK, Winther O (2016) Ladder variational autoencoders. In: Proceedings of the 30th international conference on neural information processing systems. NIPS\u201916. Curran Associates Inc., Red Hook, NY, USA, pp 3745\u20133753"},{"key":"2056_CR24","unstructured":"Vahdat A, Kautz J (2020) NVAE: a deep hierarchical variational autoencoder. In: Proceedings of the 34th international conference on neural information processing systems. NIPS \u201920. Curran Associates Inc., Red Hook, NY, USA. pp 19667\u201319679"},{"key":"2056_CR25","doi-asserted-by":"publisher","first-page":"2385","DOI":"10.1007\/s00366-023-01916-6","volume":"40","author":"S Lee","year":"2024","unstructured":"Lee S, Lee S, Jang K, Cho H, Shin SJ (2024) Data-driven nonlinear parametric model order reduction framework using deep hierarchical variational autoencoder. Eng Comput. https:\/\/doi.org\/10.1007\/s00366-023-01916-6","journal-title":"Eng Comput"},{"key":"2056_CR26","unstructured":"Lee S (2024) Development of a deep hierarchical varitaional autoencoder for efficient analysis including flude-structure interaction. PhD thesis, Seoul National University, Seoul"},{"issue":"4","key":"2056_CR27","doi-asserted-by":"publisher","first-page":"779","DOI":"10.1109\/LAWP.2022.3224983","volume":"22","author":"HH Zhang","year":"2023","unstructured":"Zhang HH, Yao HM, Jiang L, Ng M (2023) Fast full-wave electromagnetic forward solver based on deep conditional convolutional autoencoders. IEEE Antennas Wirel Propag Lett 22(4):779\u2013783. https:\/\/doi.org\/10.1109\/LAWP.2022.3224983","journal-title":"IEEE Antennas Wirel Propag Lett"},{"key":"2056_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2023.115271","volume":"431","author":"X-F He","year":"2023","unstructured":"He X-F, Li L, Lanteri S, Li K (2023) Model order reduction for parameterized electromagnetic problems using matrix decomposition and deep neural networks. J Comput Appl Math 431:115271. https:\/\/doi.org\/10.1016\/j.cam.2023.115271","journal-title":"J Comput Appl Math"},{"issue":"6","key":"2056_CR29","doi-asserted-by":"publisher","first-page":"1782","DOI":"10.1109\/TEMC.2023.3316916","volume":"65","author":"R Choupanzadeh","year":"2023","unstructured":"Choupanzadeh R, Zadehgol A (2023) A deep neural network modeling methodology for efficient EMC assessment of shielding enclosures using MECA-generated RCS training data. IEEE Trans Electromagn Comput 65(6):1782\u20131792. https:\/\/doi.org\/10.1109\/TEMC.2023.3316916","journal-title":"IEEE Trans Electromagn Comput"},{"issue":"1","key":"2056_CR30","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1109\/TAES.2022.3182303","volume":"59","author":"L Ye","year":"2023","unstructured":"Ye L, Hu S, Yan T, Xie Y (2023) GAF representation of millimeter wave drone RCS and drone classification method based on deep fusion network using ResNet. IEEE Trans Aerosp Electron Syst 59(1):336\u2013346. https:\/\/doi.org\/10.1109\/TAES.2022.3182303","journal-title":"IEEE Trans Aerosp Electron Syst"},{"issue":"8","key":"2056_CR31","doi-asserted-by":"publisher","DOI":"10.1063\/5.0053979","volume":"33","author":"J Wang","year":"2021","unstructured":"Wang J, He C, Li R, Chen H, Zhai C, Zhang M (2021) Flow field prediction of supercritical airfoils via variational autoencoder based deep learning framework. Phys Fluids 33(8):086108","journal-title":"Phys Fluids"},{"issue":"10","key":"2056_CR32","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1002\/mop.4650021010","volume":"2","author":"JP Webb","year":"1989","unstructured":"Webb JP, Kanellopoulos VN (1989) Absorbing boundary conditions for the finite element solution of the vector wave equation. Microwave Opt Technol Lett 2(10):370\u2013372. https:\/\/doi.org\/10.1002\/mop.4650021010","journal-title":"Microwave Opt Technol Lett"},{"issue":"2","key":"2056_CR33","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1109\/8.214614","volume":"41","author":"A Chatterjee","year":"1993","unstructured":"Chatterjee A, Jin JM, Volakis JL (1993) Edge-based finite elements and vector ABCs applied to 3-D scattering. IEEE Trans Antennas Propag 41(2):221\u2013226. https:\/\/doi.org\/10.1109\/8.214614","journal-title":"IEEE Trans Antennas Propag"},{"key":"2056_CR34","doi-asserted-by":"publisher","DOI":"10.1109\/9780470544655","volume-title":"Finite element method electromagnetics: antennas, microwave circuits, and scattering applications","author":"JL Volakis","year":"1998","unstructured":"Volakis JL, Chatterjee A, Kempel LC (1998) Finite element method electromagnetics: antennas, microwave circuits, and scattering applications. IEEE Press, Piscataway"},{"key":"2056_CR35","unstructured":"Kingma DP, Salimans T, Jozefowicz R, Chen X, Sutskever I, Welling M (2016) Improved variational inference with inverse autoregressive flow. In: Proceedings of the 30th international conference on neural information processing systems. NIPS\u201916. Curran Associates Inc., Red Hook, NY, USA. pp 4743\u20134751"},{"issue":"1","key":"2056_CR36","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1137\/S1064827595287997","volume":"20","author":"G Karypis","year":"1998","unstructured":"Karypis G, Kumar V (1998) A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J Sci Comput 20(1):359\u2013392. https:\/\/doi.org\/10.1137\/S1064827595287997","journal-title":"SIAM J Sci Comput"},{"issue":"1","key":"2056_CR37","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1145\/3242094","volume":"45","author":"PR Amestoy","year":"2019","unstructured":"Amestoy PR, Buttari A, L\u2019Excellent J-Y, Mary T (2019) Performance and scalability of the block low-rank multifrontal factorization on multicore architectures. ACM Trans Math Softw 45(1):2. https:\/\/doi.org\/10.1145\/3242094","journal-title":"ACM Trans Math Softw"},{"issue":"1","key":"2056_CR38","doi-asserted-by":"publisher","first-page":"73","DOI":"10.1109\/TAP.1987.1143972","volume":"35","author":"J Richmond","year":"1987","unstructured":"Richmond J (1987) Scattering by a ferrite-coated conducting sphere. IEEE Trans Antennas Propag 35(1):73\u201379. https:\/\/doi.org\/10.1109\/TAP.1987.1143972","journal-title":"IEEE Trans Antennas Propag"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-024-02056-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-024-02056-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-024-02056-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T22:09:37Z","timestamp":1747087777000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-024-02056-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,7]]},"references-count":38,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2025,4]]}},"alternative-id":["2056"],"URL":"https:\/\/doi.org\/10.1007\/s00366-024-02056-1","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"type":"print","value":"0177-0667"},{"type":"electronic","value":"1435-5663"}],"subject":[],"published":{"date-parts":[[2024,9,7]]},"assertion":[{"value":"19 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 August 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 September 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}