{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T19:50:00Z","timestamp":1793130600305,"version":"4.5.0"},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T00:00:00Z","timestamp":1684108800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T00:00:00Z","timestamp":1684108800000},"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":[[2024,4]]},"DOI":"10.1007\/s00366-023-01822-x","type":"journal-article","created":{"date-parts":[[2023,5,15]],"date-time":"2023-05-15T14:56:12Z","timestamp":1684162572000},"page":"917-929","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":46,"title":["Deep learning operator network for plastic deformation with variable loads and material properties"],"prefix":"10.1007","volume":"40","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7330-6401","authenticated-orcid":false,"given":"Seid","family":"Koric","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Asha","family":"Viswantah","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Diab W.","family":"Abueidda","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nahil A.","family":"Sobh","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kamran","family":"Khan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,5,15]]},"reference":[{"key":"1822_CR1","doi-asserted-by":"publisher","first-page":"114217","DOI":"10.1016\/j.cma.2021.114217","volume":"388","author":"JN Fuhg","year":"2022","unstructured":"Fuhg JN, Marino M, Bouklas N (2022) Local approximate Gaussian process regression for data-driven constitutive models: development and comparison with neural networks. Comput Methods Appl Mech Eng 388:114217. https:\/\/doi.org\/10.1016\/j.cma.2021.114217","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1822_CR2","doi-asserted-by":"publisher","first-page":"1118","DOI":"10.1016\/j.cma.2018.09.007","volume":"346","author":"DA White","year":"2019","unstructured":"White DA, Arrighi WJ, Kudo J, Watts SE (2019) Multiscale topology optimization using neural network surrogate models. Comput Methods Appl Mech Eng 346:1118\u20131135. https:\/\/doi.org\/10.1016\/j.cma.2018.09.007","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1822_CR3","doi-asserted-by":"publisher","first-page":"104761","DOI":"10.1016\/j.jmbbm.2021.10476","volume":"123","author":"SCY Shen","year":"2021","unstructured":"Shen SCY, Fern\u00e1ndez MP, Tozzi G, Buehler MJ (2021) Deep learning approach to assess damage mechanics of bone tissue. J Mech Behav Biomed Mater 123:104761. https:\/\/doi.org\/10.1016\/j.jmbbm.2021.10476","journal-title":"J Mech Behav Biomed Mater"},{"key":"1822_CR4","doi-asserted-by":"publisher","first-page":"1367","DOI":"10.1016\/j.ijplas.2005.09.002","volume":"22","author":"MS Al-Haik","year":"2006","unstructured":"Al-Haik MS, Hussaini MY, Garmestani H (2006) Prediction of nonlinear viscoelastic behavior of polymeric composites using an artificial neural network. Int J Plast 22:1367\u20131392. https:\/\/doi.org\/10.1016\/j.ijplas.2005.09.002","journal-title":"Int J Plast"},{"key":"1822_CR5","doi-asserted-by":"publisher","first-page":"989","DOI":"10.1002\/nme.905","volume":"59","author":"Y Hashash","year":"2004","unstructured":"Hashash Y, Jung S, Ghaboussi J (2004) Numerical implementation of a neural network based material model in finite element analysis. Int J Numer Methods Eng 59:989\u20131005. https:\/\/doi.org\/10.1002\/nme.905","journal-title":"Int J Numer Methods Eng"},{"key":"1822_CR6","doi-asserted-by":"publisher","first-page":"102919","DOI":"10.1016\/j.ijplas.2020.102919","volume":"138","author":"DP Jang","year":"2021","unstructured":"Jang DP, Piemaan F, Yoon JW, J. W., (2021) Machine learning-based constitutive model for J2- plasticity. Int J Plast 138:102919. https:\/\/doi.org\/10.1016\/j.ijplas.2020.102919","journal-title":"Int J Plast"},{"key":"1822_CR7","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.ijplas.2019.05.001","volume":"120","author":"U Ali","year":"2019","unstructured":"Ali U, Muhammad W, Brahme A, Skiba O, Inal K (2019) Application of artificial neural networks in micromechanics for polycrystalline metals. Int J Plast 120:205\u2013219. https:\/\/doi.org\/10.1016\/j.ijplas.2019.05.001","journal-title":"Int J Plast"},{"key":"1822_CR8","doi-asserted-by":"publisher","first-page":"1138","DOI":"10.1016\/j.cma.2018.09.020","volume":"345","author":"Z Liu","year":"2019","unstructured":"Liu Z, Wu CT, Koishi M (2019) A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials. Comput Methods Appl Mech Eng 345:1138\u20131168. https:\/\/doi.org\/10.1016\/j.cma.2018.09.020","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1822_CR9","doi-asserted-by":"publisher","first-page":"26414","DOI":"10.1073\/pnas.1911815116","volume":"116","author":"M Mozaffar","year":"2019","unstructured":"Mozaffar M, Bostanabad R, Chen W, Ehmann K, Cao J, Bessa M (2019) Deep learning predicts path-dependent plasticity. Proc Natl Acad Sci 116:26414\u201326420. https:\/\/doi.org\/10.1073\/pnas.1911815116","journal-title":"Proc Natl Acad Sci"},{"key":"1822_CR10","doi-asserted-by":"publisher","first-page":"102852","DOI":"10.1016\/j.ijplas.2020.102852","volume":"136","author":"DW Abueidda","year":"2021","unstructured":"Abueidda DW, Koric S, Sobh NA, Sehitoglu H (2021) Deep learning for plasticity and thermo-viscoplasticity. Int J Plast 136:102852. https:\/\/doi.org\/10.1016\/j.ijplas.2020.102852","journal-title":"Int J Plast"},{"issue":"6","key":"1822_CR11","doi-asserted-by":"publisher","first-page":"396","DOI":"10.1080\/10407790.2010.496657","volume":"57","author":"S Koric","year":"2010","unstructured":"Koric S, Thomas BG, Voller VR (2010) Enhanced latent heat method to incorporate superheat effects into fixed-grid multiphysics simulations. Numerical Heat Transfer, Part B: Fundamentals 57(6):396\u2013413. https:\/\/doi.org\/10.1080\/10407790.2010.496657","journal-title":"Numerical Heat Transfer, Part B: Fundamentals"},{"key":"1822_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmatprotec.2019.116469","author":"MLS Zappulla","year":"2019","unstructured":"Zappulla MLS, Cho SM, Koric S, Lee HJ, Kim SH, Thomas BG (2019) Multiphysics modeling of continuous casting of stainless steel. J Mater Process Technol. https:\/\/doi.org\/10.1016\/j.jmatprotec.2019.116469","journal-title":"J Mater Process Technol"},{"key":"1822_CR13","doi-asserted-by":"publisher","first-page":"103072","DOI":"10.1016\/j.ijplas.2021.103072","volume":"146","author":"T Tancogne-Dejean","year":"2021","unstructured":"Tancogne-Dejean T, Gorji MB, Zhu J, Mohr D (2021) Recurrent neural network modeling of the large deformation of lithium-ion battery cells. Int J Plast 146:103072. https:\/\/doi.org\/10.1016\/j.ijplas.2021.103072","journal-title":"Int J Plast"},{"key":"1822_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/s00366-022-01603-y","author":"Z Hajimohammadi","year":"2022","unstructured":"Hajimohammadi Z, Shekarpaz S, Parand K (2022) The novel learning solutions to nonlinear differential models on a semi-infinite domain. Eng Compu. https:\/\/doi.org\/10.1007\/s00366-022-01603-y","journal-title":"Eng Compu"},{"key":"1822_CR15","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. https:\/\/doi.org\/10.1016\/j.jcp.2018.10.045","journal-title":"J Comput Phys"},{"issue":"23","key":"1822_CR16","doi-asserted-by":"publisher","first-page":"7182","DOI":"10.1002\/nme.6828","volume":"122","author":"DW Abueidda","year":"2021","unstructured":"Abueidda DW, Lu Q, Koric S (2021) Meshless physics-informed deep learning method for three-dimensional solid mechanics. Int J Numer Meth Eng 122(23):7182\u20137201. https:\/\/doi.org\/10.1002\/nme.6828","journal-title":"Int J Numer Meth Eng"},{"key":"1822_CR17","doi-asserted-by":"publisher","first-page":"103874","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. Europe J Mech-A\/Solids 80:103874. https:\/\/doi.org\/10.1016\/j.euromechsol.2019.103874","journal-title":"Europe J Mech-A\/Solids"},{"key":"1822_CR18","doi-asserted-by":"publisher","first-page":"112790","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. https:\/\/doi.org\/10.1016\/j.cma.2019.112790","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1822_CR19","doi-asserted-by":"publisher","first-page":"104639","DOI":"10.1016\/j.euromechsol.2022.104639","volume":"95","author":"DW Abueidda","year":"2022","unstructured":"Abueidda DW, Koric S, Abu Al-Rub R, Parrott CM, Kai JA, Sobh NA (2022) A deep learning energy method for hyperelasticity and viscoelasticity. Europe J Mech - A\/Solids 95:104639. https:\/\/doi.org\/10.1016\/j.euromechsol.2022.104639","journal-title":"Europe J Mech - A\/Solids"},{"key":"1822_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2022.105177","author":"S Niu","year":"2023","unstructured":"Niu S, Zhang E, Bazilevs Y, Srivastava V (2023) Modeling finite-strain plasticity using physics-informed neural network and assessment of the network performance. J Mech Phys Solids. https:\/\/doi.org\/10.1016\/j.jmps.2022.105177","journal-title":"J Mech Phys Solids"},{"key":"1822_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113741","author":"E Haghighat","year":"2023","unstructured":"Haghighat E, Raissi M, Moure A, Gomez H, Juanes R (2023) A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput Methods Appl Mech Eng. https:\/\/doi.org\/10.1016\/j.cma.2021.113741","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1822_CR22","doi-asserted-by":"publisher","first-page":"A102","DOI":"10.1017\/jfm.2021.135","volume":"915","author":"S Cai","year":"2021","unstructured":"Cai S, Wang Z, Fuest F, Jeon Y, Gray C, Karniadakis GE (2021) Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks. J Fluid Mech 915:A102. https:\/\/doi.org\/10.1017\/jfm.2021.135","journal-title":"J Fluid Mech"},{"key":"1822_CR23","doi-asserted-by":"publisher","first-page":"218","DOI":"10.1038\/s42256-021-00302-5","volume":"3","author":"L Lu","year":"2021","unstructured":"Lu L, Jin., P., Pang., G.Z. Zhang, Z., Karniadakis, G., E., (2021) Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat Mach Intell 3:218\u2013229. https:\/\/doi.org\/10.1038\/s42256-021-00302-5","journal-title":"Nat Mach Intell"},{"key":"1822_CR24","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1109\/72.392253","volume":"6","author":"T Chen","year":"1995","unstructured":"Chen T, Chen H (1995) Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans Neural Netw 6:911\u2013917. https:\/\/doi.org\/10.1109\/72.392253","journal-title":"IEEE Trans Neural Netw"},{"issue":"40","key":"1822_CR25","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1126\/sciadv.abi8605","volume":"7","author":"S Wang","year":"2021","unstructured":"Wang S, Wang H, Perdikaris P (2021) Learning the solution operator of parametric partial differential equations with physics-informed DeepONets. Sci Adv 7(40):1\u20139. https:\/\/doi.org\/10.1126\/sciadv.abi8605","journal-title":"Sci Adv"},{"key":"1822_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijheatmasstransfer.2022.123809","author":"S Koric","year":"2023","unstructured":"Koric S, Abueidda DW (2023) Data-driven and physics-informed deep learning operators for solution of heat conduction equation with parametric heat source. Int J Heat and Mass Transfer. https:\/\/doi.org\/10.1016\/j.ijheatmasstransfer.2022.123809","journal-title":"Int J Heat and Mass Transfer"},{"key":"1822_CR27","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115027","author":"M Yin","year":"2022","unstructured":"Yin M, Zhang E, Yu Y, Karniadakis G, M., (2022) Interfacing finite elements with deep neural operators for fast multiscale modeling of mechanics problems. Comput Methods Appl Mech Eng. https:\/\/doi.org\/10.1016\/j.cma.2022.115027","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1822_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114587","author":"S Goswami","year":"2022","unstructured":"Goswami S, Yin M, Yu Y, Karniadakis G, M., (2022) A physics-informed variational DeepONet for predicting crack path in quasi-brittle materials. Comput Methods Appl Mech Eng. https:\/\/doi.org\/10.1016\/j.cma.2022.114587","journal-title":"Comput Methods Appl Mech Eng"},{"key":"1822_CR29","unstructured":"GRoT Linear and Nonlinear Finite Element Method solver, https:\/\/github.com\/tutajrobert\/grot, 2022 (accessed 01.30.23)"},{"key":"1822_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/b98904","author":"JC Simo","year":"1998","unstructured":"Simo JC, Hughes TJR (1998) Computational Inelasticity. Springer-Verlag. https:\/\/doi.org\/10.1007\/b98904","journal-title":"Springer-Verlag"},{"key":"1822_CR31","unstructured":"Zienkiewicz, O. C., and R. L. Taylor., 2000. The Finite Element Method. 5th ed. Oxford\u202f;: Butterworth-Heinemann, ISBN 0750650559"},{"key":"1822_CR32","doi-asserted-by":"publisher","first-page":"1955","DOI":"10.1002\/nme.1614","volume":"66","author":"S Koric","year":"2006","unstructured":"Koric S, Thomas BG (2006) Efficient thermo-mechanical model for solidification processes. Int J Numer Meth Engng 66:1955\u20131989. https:\/\/doi.org\/10.1002\/nme.1614","journal-title":"Int J Numer Meth Engng"},{"key":"1822_CR33","unstructured":"Bradbury J., Frostig R., Hawkins P., Johnson M.J., Leary C., Maclaurin D., Necula G.,Paszke, VanderPlas A.J., Wanderman-Milne S., Zhang Q.: JAX: Composable transformations of Python+NumPy programs (2018)."},{"key":"1822_CR34","unstructured":"Commercial finite element software Abaqus V2019 theory guide. http:\/\/www.3ds.com\/products-services\/simulia\/support\/documentation\/"}],"container-title":["Engineering with Computers"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01822-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00366-023-01822-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00366-023-01822-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,21]],"date-time":"2024-04-21T02:42:32Z","timestamp":1713667352000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00366-023-01822-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,15]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,4]]}},"alternative-id":["1822"],"URL":"https:\/\/doi.org\/10.1007\/s00366-023-01822-x","relation":{},"ISSN":["0177-0667","1435-5663"],"issn-type":[{"value":"0177-0667","type":"print"},{"value":"1435-5663","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,15]]},"assertion":[{"value":"21 February 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"15 May 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"}}]}}