{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T06:14:59Z","timestamp":1783750499600,"version":"3.55.0"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T00:00:00Z","timestamp":1783728000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T00:00:00Z","timestamp":1783728000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Engineering with Computers"],"published-print":{"date-parts":[[2026,8]]},"DOI":"10.1007\/s00366-026-02382-6","type":"journal-article","created":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T06:05:49Z","timestamp":1783749949000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A boundary integral-based neural operator for mesh deformation"],"prefix":"10.1007","volume":"42","author":[{"given":"Zhengyu","family":"Wu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jun","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei","family":"Wang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,11]]},"reference":[{"issue":"8","key":"2382_CR1","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1109\/LMWC.2019.2927113","volume":"29","author":"F Feng","year":"2019","unstructured":"Feng F, Zhang J, Zhang W, Zhao Z, Jin J, Zhang Q-J (2019) Coarse-and fine-mesh space mapping for em optimization incorporating mesh deformation. IEEE Microwave Wirel Compon Lett 29(8):510\u2013512","journal-title":"IEEE Microwave Wirel Compon Lett"},{"key":"2382_CR2","doi-asserted-by":"crossref","unstructured":"Wei W, Vouvakis MN (2011) Fast geometric parameter sweep of fem models via a nonlinear bt-pod model reduction. In: 2011 IEEE international symposium on antennas and propagation (APSURSI), pp 2472\u20132475. IEEE","DOI":"10.1109\/APS.2011.5997024"},{"issue":"5","key":"2382_CR3","doi-asserted-by":"publisher","first-page":"2973","DOI":"10.1109\/TMTT.2023.3330766","volume":"72","author":"W Liu","year":"2023","unstructured":"Liu W, Zhang Q-J, Feng F, Wang X, Xue J, Zhang J, Ma K (2023) Electromagnetic parametric modeling using mor-based neuro-impedance matrix transfer functions for two-port microwave components. IEEE Trans Microw Theory Tech 72(5):2973\u20132989","journal-title":"IEEE Trans Microw Theory Tech"},{"issue":"7","key":"2382_CR4","doi-asserted-by":"publisher","first-page":"3065","DOI":"10.1109\/TMTT.2023.3239363","volume":"71","author":"M Baranowski","year":"2023","unstructured":"Baranowski M, Balewski \u0141, Lamecki A, Mrozowski M, Galdeano J (2023) The design of cavity resonators and microwave filters applying shape deformation techniques. IEEE Trans Microw Theory Tech 71(7):3065\u20133074","journal-title":"IEEE Trans Microw Theory Tech"},{"key":"2382_CR5","doi-asserted-by":"crossref","unstructured":"Guo L, Feng F, Liu W, Ma K, Zhang Q-J (2024) An optimization method of five-order sisl microstrip bandpass filter using mesh deformation. In: 2024 international applied computational electromagnetics society symposium (ACES-China), pp 1\u20133. IEEE","DOI":"10.1109\/ACES-China62474.2024.10699478"},{"key":"2382_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2021.113492","volume":"392","author":"M Morelli","year":"2021","unstructured":"Morelli M, Bellosta T, Guardone A (2021) Efficient radial basis function mesh deformation methods for aircraft icing. J Comput Appl Math 392:113492","journal-title":"J Comput Appl Math"},{"issue":"1","key":"2382_CR7","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/s11831-012-9070-4","volume":"19","author":"K Takizawa","year":"2012","unstructured":"Takizawa K, Tezduyar TE (2012) Computational methods for parachute fluid-structure interactions. Arch Comput Methods Eng 19(1):125\u2013169","journal-title":"Arch Comput Methods Eng"},{"issue":"12","key":"2382_CR8","doi-asserted-by":"publisher","first-page":"817","DOI":"10.1002\/(SICI)1096-9853(199712)21:12<817::AID-NAG902>3.0.CO;2-D","volume":"21","author":"GA Fenton","year":"1997","unstructured":"Fenton GA, Griffiths DV (1997) A mesh deformation algorithm for free surface problems. Int J Numer Anal Meth Geomech 21(12):817\u2013824","journal-title":"Int J Numer Anal Meth Geomech"},{"key":"2382_CR9","doi-asserted-by":"crossref","unstructured":"Wang W, Vouvakis MN (2012) Mesh morphing strategies for robust geometric parameter model reduction. In: Proceedings of the 2012 IEEE international symposium on antennas and propagation, pp 1\u20132. IEEE","DOI":"10.1109\/APS.2012.6349139"},{"key":"2382_CR10","doi-asserted-by":"crossref","unstructured":"Dwight RP (2006) Robust mesh deformation using the linear elasticity equations. In: Computational fluid dynamics 2006: proceedings of the fourth international conference on computational fluid dynamics, ICCFD, Ghent, Belgium, 10-14, pp 401\u2013406. Springer, 2009","DOI":"10.1007\/978-3-540-92779-2_62"},{"issue":"11\u201314","key":"2382_CR11","doi-asserted-by":"publisher","first-page":"784","DOI":"10.1016\/j.compstruc.2007.01.013","volume":"85","author":"A De Boer","year":"2007","unstructured":"De Boer A, Van der Schoot MS, Bijl H (2007) Mesh deformation based on radial basis function interpolation. Comput Struct 85(11\u201314):784\u2013795","journal-title":"Comput Struct"},{"key":"2382_CR12","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/j.advengsoft.2018.11.011","volume":"128","author":"F Gagliardi","year":"2019","unstructured":"Gagliardi F, Giannakoglou KC (2019) A two-step radial basis function-based cfd mesh displacement tool. Adv Eng Softw 128:86\u201397","journal-title":"Adv Eng Softw"},{"key":"2382_CR13","doi-asserted-by":"publisher","first-page":"997","DOI":"10.1016\/j.jcp.2016.05.036","volume":"321","author":"T Gillebaart","year":"2016","unstructured":"Gillebaart T, Blom DS, Van Zuijlen AH, Bijl H (2016) Adaptive radial basis function mesh deformation using data reduction. J Comput Phys 321:997\u20131025","journal-title":"J Comput Phys"},{"key":"2382_CR14","doi-asserted-by":"crossref","unstructured":"Witteveen J, Bijl H (2009) Explicit mesh deformation using inverse distance weighting interpolation. In: 19th AIAA computational fluid dynamics, pp 3996","DOI":"10.2514\/6.2009-3996"},{"key":"2382_CR15","doi-asserted-by":"crossref","unstructured":"Sorgente T, Biasotti S, Manzini G, Spagnuolo M (2023) A survey of indicators for mesh quality assessment. In: Computer graphics forum, vol 42, pp 461\u2013483. Wiley Online Library","DOI":"10.1111\/cgf.14779"},{"issue":"4","key":"2382_CR16","doi-asserted-by":"publisher","first-page":"863","DOI":"10.1007\/s10543-010-0283-3","volume":"50","author":"SM Shontz","year":"2010","unstructured":"Shontz SM, Vavasis SA (2010) Analysis of and workarounds for element reversal for a finite element-based algorithm for warping triangular and tetrahedral meshes. BIT Numer Math 50(4):863\u2013884","journal-title":"BIT Numer Math"},{"issue":"7","key":"2382_CR17","doi-asserted-by":"publisher","first-page":"787","DOI":"10.1007\/s00371-008-0260-x","volume":"24","author":"G Rong","year":"2008","unstructured":"Rong G, Cao Y, Guo X (2008) Spectral mesh deformation. Vis Comput 24(7):787\u2013796","journal-title":"Vis Comput"},{"issue":"11","key":"2382_CR18","doi-asserted-by":"publisher","first-page":"3400","DOI":"10.1109\/TMTT.2016.2605672","volume":"64","author":"A Lamecki","year":"2016","unstructured":"Lamecki A (2016) A mesh deformation technique based on solid mechanics for parametric analysis of high-frequency devices with 3-d fem. IEEE Trans Microw Theory Tech 64(11):3400\u20133408","journal-title":"IEEE Trans Microw Theory Tech"},{"issue":"3","key":"2382_CR19","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1007\/s10915-022-01939-z","volume":"92","author":"S Cuomo","year":"2022","unstructured":"Cuomo S, Cola VSD, Giampaolo F, Rozza G, Raissi M, Piccialli F (2022) Scientific machine learning through physics-informed neural networks: where we are and what\u2019s next. J Sci Comput 92(3):88","journal-title":"J Sci Comput"},{"key":"2382_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.engappai.2023.106660","volume":"125","author":"A Aygun","year":"2023","unstructured":"Aygun A, Maulik R, Karakus A (2023) Physics-informed neural networks for mesh deformation with exact boundary enforcement. Eng Appl Artif Intell 125:106660","journal-title":"Eng Appl Artif Intell"},{"issue":"8","key":"2382_CR21","doi-asserted-by":"publisher","DOI":"10.1063\/5.0286075","volume":"37","author":"X Liu","year":"2025","unstructured":"Liu X, Guan Z, Wang H, Tan P, Wang X (2025) Physics-informed surrogate model for mesh deformation with hard constraints. Phys Fluids 37(8):087224","journal-title":"Phys Fluids"},{"key":"2382_CR22","unstructured":"Anandkumar A, Azizzadenesheli K, Bhattacharya K, Kovachki N, Li Z, Liu B, Stuart A (2020) Neural operator: graph kernel network for partial differential equations. In: ICLR 2020 workshop on integration of deep neural models and differential equations"},{"key":"2382_CR23","unstructured":"Li Z, Kovachki N, Azizzadenesheli K, Liu B, Bhattacharya K, Stuart A, Anandkumar A (2020) Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895"},{"key":"2382_CR24","unstructured":"Tripura T, Chakraborty S (2022) Wavelet neural operator: a neural operator for parametric partial differential equations. arXiv:2205.02191"},{"key":"2382_CR25","unstructured":"Raonic B, Molinaro R, Rohner T, Mishra S, de\u00a0Bezenac E (2023) Convolutional neural operators. In: ICLR 2023 workshop on physics for machine learning"},{"issue":"89","key":"2382_CR26","first-page":"1","volume":"24","author":"N Kovachki","year":"2023","unstructured":"Kovachki N, Li Z, Liu B, Azizzadenesheli K, Bhattacharya K, Stuart A, Anandkumar A (2023) Neural operator: learning maps between function spaces with applications to pdes. J Mach Learn Res 24(89):1\u201397","journal-title":"J Mach Learn Res"},{"key":"2382_CR27","unstructured":"Kashi A, Daw A, Meena MG, Lu H (2024) Learning the boundary-to-domain mapping using lifting product fourier neural operators for partial differential equations. ICML 2024 AI for Science Workshop"},{"key":"2382_CR28","unstructured":"L\u00f6tzsch W, Ohler S, Otterbach J (2022) Learning the solution operator of boundary value problems using graph neural networks. In: ICML 2022 2nd AI for science workshop"},{"key":"2382_CR29","unstructured":"Lu L, Jin P, Karniadakis GM (2019) Deeponet: learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv preprint arXiv:1910.03193"},{"key":"2382_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.117130","volume":"429","author":"J He","year":"2024","unstructured":"He J, Koric S, Abueidda D, Najafi A, Jasiuk I (2024) Geom-deeponet: a point-cloud-based deep operator network for field predictions on 3d parameterized geometries. Comput Methods Appl Mech Eng 429:117130","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2382_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114778","volume":"393","author":"L Lu","year":"2022","unstructured":"Lu L, Meng X, Cai S, Mao Z, Goswami S, Zhang Z, Karniadakis GE (2022) A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data. Comput Methods Appl Mech Eng 393:114778","journal-title":"Comput Methods Appl Mech Eng"},{"key":"2382_CR32","unstructured":"Lin G, Hu P, Chen F, Chen X, Chen J, Wang J, Shi Z (2021) Binet: learning to solve partial differential equations with boundary integral networks. arXiv preprint arXiv:2110.00352"},{"issue":"1","key":"2382_CR33","first-page":"103","volume":"11","author":"G Lin","year":"2023","unstructured":"Lin G, Chen F, Pipi H, Chen X, Chen J, Wang J, Shi Z (2023) Bi-greennet: learning green\u2019s functions by boundary integral network. Commun Math Stat 11(1):103\u2013129","journal-title":"Commun Math Stat"},{"issue":"3","key":"2382_CR34","doi-asserted-by":"publisher","first-page":"475","DOI":"10.1162\/neco_a_01647","volume":"36","author":"Z Fang","year":"2024","unstructured":"Fang Z, Wang S, Perdikaris P (2024) Learning only on boundaries: a physics-informed neural operator for solving parametric partial differential equations in complex geometries. Neural Comput 36(3):475\u2013498","journal-title":"Neural Comput"},{"issue":"2","key":"2382_CR35","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1007\/s00466-020-01950-x","volume":"67","author":"A Shamanskiy","year":"2021","unstructured":"Shamanskiy A, Simeon B (2021) Mesh moving techniques in fluid-structure interaction: robustness, accumulated distortion and computational efficiency. Comput Mech 67(2):583\u2013600","journal-title":"Comput Mech"},{"key":"2382_CR36","unstructured":"Carlos A, Brebbia J, Dominguez, (1994) Boundary elements: an introductory course. WIT press"},{"key":"2382_CR37","doi-asserted-by":"crossref","unstructured":"Maz\u2019ya V, Movchan A, Nieves M, et\u00a0al (2013) Green\u2019s kernels and meso-scale approximations in perforated domains, volume 2077. Springer","DOI":"10.1007\/978-3-319-00357-3"},{"key":"2382_CR38","unstructured":"Rahaman N, Baratin A, Arpit D, Draxler F, Lin M, Hamprecht F, Bengio Y, Courville A (2019) On the spectral bias of neural networks. In: International conference on machine learning, pp 5301\u20135310. PMLR"},{"issue":"5","key":"2382_CR39","doi-asserted-by":"publisher","first-page":"1746","DOI":"10.4208\/cicp.OA-2020-0085","volume":"28","author":"Z-QJ Xu","year":"2020","unstructured":"Xu Z-QJ, Zhang Y, Luo T, Xiao Y, Ma Z (2020) Frequency principle: Fourier analysis sheds light on deep neural networks. Commun Comput Phys 28(5):1746\u20131767","journal-title":"Commun Comput Phys"},{"key":"2382_CR40","unstructured":"Knupp P (1999) Matrix norms and the condition number: a general framework to improve mesh quality via node-movement (Technical report)"},{"key":"2382_CR41","doi-asserted-by":"crossref","unstructured":"He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for image recognition. In: Proceedings of the IEEE conference on computer vision and pattern recognition, pp 770\u2013778","DOI":"10.1109\/CVPR.2016.90"},{"issue":"5","key":"2382_CR42","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.2514\/1.2255","volume":"41","author":"PG Luana Huang","year":"2004","unstructured":"Luana Huang PG, Huang RPLB, Hauser T (2004) Numerical study of blowing and suction control mechanism on naca0012 airfoil. J Aircr 41(5):1005\u20131013","journal-title":"J Aircr"},{"key":"2382_CR43","doi-asserted-by":"crossref","unstructured":"Sieger D, Menzel S, Botsch M (2013) High quality mesh morphing using triharmonic radial basis functions. In: Proceedings of the 21st international meshing roundtable, pp 1\u201315. Springer","DOI":"10.1007\/978-3-642-33573-0_1"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-026-02382-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-026-02382-6","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-026-02382-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,11]],"date-time":"2026-07-11T06:05:53Z","timestamp":1783749953000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-026-02382-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,11]]},"references-count":43,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,8]]}},"alternative-id":["2382"],"URL":"https:\/\/doi.org\/10.1007\/s00366-026-02382-6","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,11]]},"assertion":[{"value":"15 April 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 June 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 July 2026","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 no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"143"}}