{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,10]],"date-time":"2026-10-10T12:12:31Z","timestamp":1791634351839,"version":"4.3.4"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T00:00:00Z","timestamp":1694044800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T00:00:00Z","timestamp":1694044800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"crossref","award":["2021R1A2B5B03002410"],"award-info":[{"award-number":["2021R1A2B5B03002410"]}],"id":[{"id":"10.13039\/501100003725","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":[[2024,6]]},"DOI":"10.1007\/s00366-023-01871-2","type":"journal-article","created":{"date-parts":[[2023,9,7]],"date-time":"2023-09-07T09:02:06Z","timestamp":1694077326000},"page":"1717-1739","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":27,"title":["Automatically imposing boundary conditions for boundary value problems by unified physics-informed neural network"],"prefix":"10.1007","volume":"40","author":[{"given":"Khang A.","family":"Luong","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thang","family":"Le-Duc","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5056-829X","authenticated-orcid":false,"given":"Jaehong","family":"Lee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,9,7]]},"reference":[{"key":"1871_CR1","doi-asserted-by":"crossref","unstructured":"Park JJ, Florence P, Straub J, Newcombe R, Lovegrove S (2019) Deepsdf: learning continuous signed distance functions for shape representation. In: Proceedings of the IEEE\/CVF conference on computer vision and pattern recognition, pp 165\u2013174","DOI":"10.1109\/CVPR.2019.00025"},{"key":"1871_CR2","doi-asserted-by":"publisher","first-page":"3697","DOI":"10.1109\/JSTARS.2022.3168799","volume":"15","author":"J Ni","year":"2022","unstructured":"Ni J, Xiang D, Lin Z, L\u00f3pez-Mart\u00ednez C, Hu W, Zhang F (2022) Dnn-based polsar image classification on noisy labels. IEEE J Sel Top Appl Earth Obs Remote Sens 15:3697\u20133713","journal-title":"IEEE J Sel Top Appl Earth Obs Remote Sens"},{"issue":"6","key":"1871_CR3","doi-asserted-by":"publisher","first-page":"82","DOI":"10.1109\/MSP.2012.2205597","volume":"29","author":"G Hinton","year":"2012","unstructured":"Hinton G, Deng L, Yu D, Dahl GE, Mohamed A-R, Jaitly N, Senior A, Vanhoucke V, Nguyen P, Sainath TN et al (2012) Deep neural networks for acoustic modeling in speech recognition: the shared views of four research groups. IEEE Signal Process Mag 29(6):82\u201397","journal-title":"IEEE Signal Process Mag"},{"key":"1871_CR4","doi-asserted-by":"publisher","DOI":"10.1016\/j.finel.2022.103893","volume":"216","author":"HT Mai","year":"2023","unstructured":"Mai HT, Truong TT, Kang J, Mai DD, Lee J (2023) A robust physics-informed neural network approach for predicting structural instability. Finite Elem Anal Des 216:103893","journal-title":"Finite Elem Anal Des"},{"issue":"5","key":"1871_CR5","doi-asserted-by":"publisher","first-page":"1901266","DOI":"10.1002\/adem.201901266","volume":"22","author":"JK Wilt","year":"2020","unstructured":"Wilt JK, Yang C, Gu GX (2020) Accelerating auxetic metamaterial design with deep learning. Adv Eng Mater 22(5):1901266","journal-title":"Adv Eng Mater"},{"issue":"8","key":"1871_CR6","doi-asserted-by":"publisher","first-page":"11618","DOI":"10.1364\/OE.384875","volume":"28","author":"Y Chen","year":"2020","unstructured":"Chen Y, Lu L, Karniadakis GE, Dal Negro L (2020) Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Opt Express 28(8):11618\u201311633","journal-title":"Opt Express"},{"key":"1871_CR7","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112732","volume":"361","author":"L Sun","year":"2020","unstructured":"Sun L, Gao H, Pan S, Wang J-X (2020) Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Comput Methods Appl Mech Eng 361:112732","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1871_CR8","unstructured":"Mai HT, Lieu QX, Kang J, Lee J (2022) A novel deep unsupervised learning-based framework for optimization of truss structures. Eng Comput 1\u201324"},{"key":"1871_CR9","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","volume":"378","author":"M Raissi","year":"2019","unstructured":"Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686\u2013707","journal-title":"J Comput Phys"},{"key":"1871_CR10","unstructured":"Guo H, Zhuang X, Rabczuk T (2021) A deep collocation method for the bending analysis of Kirchhoff plate. arXiv:2102.02617"},{"key":"1871_CR11","doi-asserted-by":"crossref","unstructured":"Chen J, Du R, Wu K (2020) A comparison study of deep Galerkin method and deep Ritz method for elliptic problems with different boundary conditions. arXiv:2005.04554","DOI":"10.4208\/cmr.2020-0051"},{"key":"1871_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.109675","volume":"419","author":"J Huang","year":"2020","unstructured":"Huang J, Wang H, Yang H (2020) Int-deep: a deep learning initialized iterative method for nonlinear problems. J Comput Phys 419:109675","journal-title":"J Comput Phys"},{"key":"1871_CR13","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113933","volume":"383","author":"W Li","year":"2021","unstructured":"Li W, Bazant MZ, Zhu J (2021) A physics-guided neural network framework for elastic plates: comparison of governing equations-based and energy-based approaches. Comput Methods Appl Mech Eng 383:113933","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"1","key":"1871_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40304-018-0127-z","volume":"6","author":"B Yu","year":"2018","unstructured":"Yu B et al (2018) The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems. Commun Math Stat 6(1):1\u201312","journal-title":"Commun Math Stat"},{"key":"1871_CR15","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2019.112790","volume":"362","author":"E Samaniego","year":"2020","unstructured":"Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh VM, Guo H, Hamdia K, Zhuang X, Rabczuk T (2020) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Comput Methods Appl Mech Eng 362:112790","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1871_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/j.euromechsol.2019.103874","volume":"80","author":"VM Nguyen-Thanh","year":"2020","unstructured":"Nguyen-Thanh VM, Zhuang X, Rabczuk T (2020) A deep energy method for finite deformation hyperelasticity. Eur J Mech A\/Solids 80:103874","journal-title":"Eur J Mech A\/Solids"},{"key":"1871_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114096","volume":"386","author":"VM Nguyen-Thanh","year":"2021","unstructured":"Nguyen-Thanh VM, Anitescu C, Alajlan N, Rabczuk T, Zhuang X (2021) Parametric deep energy approach for elasticity accounting for strain gradient effects. Comput Methods Appl Mech Eng 386:114096","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1871_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.euromechsol.2021.104225","volume":"87","author":"X Zhuang","year":"2021","unstructured":"Zhuang X, Guo H, Alajlan N, Zhu H, Rabczuk T (2021) Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. Eur J Mech A\/Solids 87:104225","journal-title":"Eur J Mech A\/Solids"},{"issue":"5","key":"1871_CR19","doi-asserted-by":"publisher","first-page":"4569","DOI":"10.1007\/s00366-022-01661-2","volume":"38","author":"A Chakraborty","year":"2022","unstructured":"Chakraborty A, Anitescu C, Zhuang X, Rabczuk T (2022) Domain adaptation based transfer learning approach for solving pdes on complex geometries. Eng Comput 38(5):4569\u20134588","journal-title":"Eng Comput"},{"key":"1871_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115757","volume":"403","author":"A Li","year":"2023","unstructured":"Li A, Zhang YJ (2023) Isogeometric analysis-based physics-informed graph neural network for studying traffic jam in neurons. Comput Methods Appl Mech Eng 403:115757","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1871_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2020.110085","volume":"428","author":"H Sheng","year":"2021","unstructured":"Sheng H, Yang C (2021) Pfnn: a penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries. J Comput Phys 428:110085","journal-title":"J Comput Phys"},{"issue":"6","key":"1871_CR22","doi-asserted-by":"publisher","DOI":"10.1115\/1.4046892","volume":"20","author":"V Dwivedi","year":"2020","unstructured":"Dwivedi V, Srinivasan B (2020) Solution of biharmonic equation in complicated geometries with physics informed extreme learning machine. J Comput Inf Sci Eng 20(6):061004","journal-title":"J Comput Inf Sci Eng"},{"key":"1871_CR23","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115839","volume":"405","author":"J Gasick","year":"2023","unstructured":"Gasick J, Qian X (2023) Isogeometric neural networks: a new deep learning approach for solving parameterized partial differential equations. Comput Methods Appl Mech Eng 405:115839","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1871_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114502","volume":"390","author":"H Gao","year":"2022","unstructured":"Gao H, Zahr MJ, Wang J-X (2022) Physics-informed graph neural Galerkin networks: a unified framework for solving pde-governed forward and inverse problems. Comput Methods Appl Mech Eng 390:114502","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1871_CR25","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114333","volume":"389","author":"N Sukumar","year":"2022","unstructured":"Sukumar N, Srivastava A (2022) Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks. Comput Methods Appl Mech Eng 389:114333","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1871_CR26","volume-title":"Approximate calculation of integrals","author":"H Krylov","year":"2006","unstructured":"Krylov H (2006) Approximate calculation of integrals. Macmillan, New York"},{"key":"1871_CR27","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1007\/978-0-387-21738-3","volume-title":"Introduction to numerical analysis, Texts in applied mathematics","author":"J Stoer","year":"2002","unstructured":"Stoer J, Bulirsch R (2002) Introduction to numerical analysis, Texts in applied mathematics, vol 12. Springer, New York, p 30"},{"key":"1871_CR28","volume-title":"Energy principles and variational methods in applied mechanics","author":"JN Reddy","year":"2017","unstructured":"Reddy JN (2017) Energy principles and variational methods in applied mechanics. Wiley, New York"},{"key":"1871_CR29","unstructured":"Liu Z, Yang Y, Cai QD (2019) Solving differential equation with constrained multilayer feedforward network. arXiv:1904.06619"},{"key":"1871_CR30","unstructured":"Abadi M, Barham P, Chen J, Chen Z, Davis A, Dean J, Devin M, Ghemawat S, Irving G, Isard M et al (2016) $$\\{$$TensorFlow$$\\}$$: a system for $$\\{$$Large-Scale$$\\}$$ machine learning. In: 12th USENIX symposium on operating systems design and implementation (OSDI 16), pp 265\u2013283"},{"key":"1871_CR31","unstructured":"Kingma DP, Ba J (2014) Adam: a method for stochastic optimization. arXiv:1412.6980"},{"issue":"1","key":"1871_CR32","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/BF01589116","volume":"45","author":"DC Liu","year":"1989","unstructured":"Liu DC, Nocedal J (1989) On the limited memory bfgs method for large scale optimization. Math Program 45(1):503\u2013528","journal-title":"Math Program"},{"key":"1871_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2021.110666","volume":"446","author":"W Chen","year":"2021","unstructured":"Chen W, Wang Q, Hesthaven JS, Zhang C (2021) Physics-informed machine learning for reduced-order modeling of nonlinear problems. J Comput Phys 446:110666","journal-title":"J Comput Phys"},{"key":"1871_CR34","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4296-3","volume-title":"Introduction to linear elasticity","author":"PL Gould","year":"1994","unstructured":"Gould PL, Feng Y (1994) Introduction to linear elasticity. Springer, New York"},{"key":"1871_CR35","doi-asserted-by":"publisher","first-page":"52","DOI":"10.1016\/j.cma.2018.10.039","volume":"345","author":"Y Jia","year":"2019","unstructured":"Jia Y, Anitescu C, Zhang YJ, Rabczuk T (2019) An adaptive isogeometric analysis collocation method with a recovery-based error estimator. Comput Methods Appl Mech Eng 345:52\u201374","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"3\u20134","key":"1871_CR36","doi-asserted-by":"publisher","first-page":"576","DOI":"10.1002\/nme.4783","volume":"102","author":"D Schillinger","year":"2015","unstructured":"Schillinger D, Evans JA, Frischmann F, Hiemstra RR, Hsu M-C, Hughes TJ (2015) A collocated c0 finite element method: reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics. Int J Numer Methods Eng 102(3\u20134):576\u2013631","journal-title":"Int J Numer Methods Eng"},{"key":"1871_CR37","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198503675.001.0001","volume-title":"The mathematical theory of plasticity","author":"R Hill","year":"1998","unstructured":"Hill R (1998) The mathematical theory of plasticity, vol 11. Oxford University Press, Oxford"},{"key":"1871_CR38","volume-title":"Plasticity theory","author":"J Lubliner","year":"2008","unstructured":"Lubliner J (2008) Plasticity theory. Courier Corporation, North Chelmsford"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01871-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-023-01871-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01871-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,2]],"date-time":"2024-07-02T09:15:02Z","timestamp":1719911702000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-023-01871-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,7]]},"references-count":38,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2024,6]]}},"alternative-id":["1871"],"URL":"https:\/\/doi.org\/10.1007\/s00366-023-01871-2","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,7]]},"assertion":[{"value":"3 March 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 June 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 September 2023","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 known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}