{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T16:39:12Z","timestamp":1772728752325,"version":"3.50.1"},"reference-count":72,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T00:00:00Z","timestamp":1748476800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T00:00:00Z","timestamp":1748476800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["DFG 255042459"],"award-info":[{"award-number":["DFG 255042459"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"name":"ETH Zurich Postdoctoral Fellowship"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv. Model. and Simul. in Eng. Sci."],"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>The calibration of constitutive models from full-field data has recently gained increasing interest due to improvements in full-field measurement capabilities. In addition to the experimental characterization of novel materials, continuous structural health monitoring is another application that is of great interest. However, monitoring is usually associated with severe time constraints, difficult to meet with standard numerical approaches. Therefore, parametric physics-informed neural networks (PINNs) for constitutive model calibration from full-field displacement data are investigated. In an offline stage, a parametric PINN can be trained to learn a parameterized solution of the underlying partial differential equation. In the subsequent online stage, the parametric PINN then acts as a surrogate for the parameters-to-state map in calibration. We test the proposed approach for the deterministic least-squares calibration of a linear elastic as well as a hyperelastic constitutive model from noisy synthetic displacement data. We further carry out Markov chain Monte Carlo-based Bayesian inference to quantify the uncertainty. A proper statistical evaluation of the results underlines the high accuracy of the deterministic calibration and that the estimated uncertainty is valid. Finally, we consider experimental data and show that the results are in good agreement with a finite element method-based calibration. Due to the fast evaluation of PINNs, calibration can be performed in near real-time. This advantage is particularly evident in many-query applications such as Markov chain Monte Carlo-based Bayesian inference.<\/jats:p>","DOI":"10.1186\/s40323-025-00285-7","type":"journal-article","created":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T12:10:04Z","timestamp":1748520604000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks"],"prefix":"10.1186","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0888-0220","authenticated-orcid":false,"given":"David","family":"Anton","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4999-4558","authenticated-orcid":false,"given":"Jendrik-Alexander","family":"Tr\u00f6ger","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2542-1130","authenticated-orcid":false,"given":"Henning","family":"Wessels","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1277-7509","authenticated-orcid":false,"given":"Ulrich","family":"R\u00f6mer","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4615-9271","authenticated-orcid":false,"given":"Alexander","family":"Henkes","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1849-0784","authenticated-orcid":false,"given":"Stefan","family":"Hartmann","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,5,29]]},"reference":[{"key":"285_CR1","doi-asserted-by":"publisher","unstructured":"Chang, F.-K.: Structural health monitoring: a summary report on the first stanford workshop on structural health monitoring, September 18-20, 1997. Technical report, Stanford University (1998). https:\/\/doi.org\/10.21236\/ADA350933.","DOI":"10.21236\/ADA350933"},{"key":"285_CR2","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-66259-2","volume-title":"Structural health monitoring by time series analysis and statistical distance measures","author":"A Entezami","year":"2021","unstructured":"Entezami A. Structural health monitoring by time series analysis and statistical distance measures. 1st ed. Cham: SpringerBriefs in Applied Sciences and Technology. Springer; 2021. https:\/\/doi.org\/10.1007\/978-3-030-66259-2.","edition":"1"},{"key":"285_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-78747-3","volume-title":"Image correlation for shape, motion and deformation measurements","author":"MA Sutton","year":"2009","unstructured":"Sutton MA, Orteu J-J, Schreier H. Image correlation for shape, motion and deformation measurements. 1st ed. New York: Springer; 2009. https:\/\/doi.org\/10.1007\/978-0-387-78747-3.","edition":"1"},{"issue":"5","key":"285_CR4","doi-asserted-by":"publisher","first-page":"1257","DOI":"10.1117\/1.1566781","volume":"42","author":"LX Yang","year":"2003","unstructured":"Yang LX, Ettemeyer A. Strain measurement by three-dimensional electronic speckle pattern interferometry: potentials, limitations, and applications. Opt Eng. 2003;42(5):1257\u201366. https:\/\/doi.org\/10.1117\/1.1566781.","journal-title":"Opt Eng"},{"issue":"3","key":"285_CR5","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/0045-7825(96)00991-7","volume":"136","author":"R Mahnken","year":"1996","unstructured":"Mahnken R, Stein E. A unified approach for parameter identification of inelastic material models in the frame of the finite element method. Comput Methods Appl Mech Eng. 1996;136(3):225\u201358. https:\/\/doi.org\/10.1016\/0045-7825(96)00991-7.","journal-title":"Comput Methods Appl Mech Eng"},{"key":"285_CR6","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmat.2019.103292","volume":"145","author":"L Rose","year":"2020","unstructured":"Rose L, Menzel A. Optimisation based material parameter identification using full field displacement and temperature measurements. Mech Mater. 2020;145: 103292. https:\/\/doi.org\/10.1016\/j.mechmat.2019.103292.","journal-title":"Mech Mater"},{"issue":"4","key":"285_CR7","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s11340-008-9148-y","volume":"48","author":"S Avril","year":"2008","unstructured":"Avril S, Bonnet M, Bretelle A-S, Gr\u00e9diac M, Hild F, Ienny P, Latourte F, Lemosse D, Pagano S, Pagnacco E, Pierron F. Overview of identification methods of mechanical parameters based on full-field measurements. Exp Mech. 2008;48(4):381\u2013402. https:\/\/doi.org\/10.1007\/s11340-008-9148-y.","journal-title":"Exp Mech"},{"key":"285_CR8","doi-asserted-by":"publisher","first-page":"330","DOI":"10.1016\/j.ijmecsci.2018.07.013","volume":"145","author":"JMP Martins","year":"2018","unstructured":"Martins JMP, Andrade-Campos A, Thuillier S. Comparison of inverse identification strategies for constitutive mechanical models using full-field measurements. Int J Mech Sci. 2018;145:330\u201345. https:\/\/doi.org\/10.1016\/j.ijmecsci.2018.07.013.","journal-title":"Int J Mech Sci"},{"key":"285_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. Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys. 2019;378:686\u2013707. https:\/\/doi.org\/10.1016\/j.jcp.2018.10.045.","journal-title":"J Comput Phys"},{"issue":"6","key":"285_CR10","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","volume":"3","author":"GE Karniadakis","year":"2021","unstructured":"Karniadakis GE, Kevrekidis IG, Lu L, Perdikaris P, Wang S, Yang L. Physics-informed machine learning. Nat Rev Phys. 2021;3(6):422\u201340. https:\/\/doi.org\/10.1038\/s42254-021-00314-5.","journal-title":"Nat Rev Phys"},{"issue":"10","key":"285_CR11","doi-asserted-by":"publisher","first-page":"1499","DOI":"10.1002\/aic.690381003","volume":"38","author":"DC Psichogios","year":"1992","unstructured":"Psichogios DC, Ungar LH. A hybrid neural network-first principles approach to process modeling. Am Instit Chem Eng J. 1992;38(10):1499\u2013511. https:\/\/doi.org\/10.1002\/aic.690381003.","journal-title":"Am Instit Chem Eng J"},{"issue":"5","key":"285_CR12","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1109\/72.712178","volume":"9","author":"IE Lagaris","year":"1998","unstructured":"Lagaris IE, Likas A, Fotiadis DI. Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Networks. 1998;9(5):987\u20131000. https:\/\/doi.org\/10.1109\/72.712178.","journal-title":"IEEE Trans Neural Networks"},{"issue":"1","key":"285_CR13","first-page":"5595","volume":"18","author":"AG Baydin","year":"2018","unstructured":"Baydin AG, Pearlmutter BA, Radul AA, Siskind JM. Automatic differentiation in machine learning: a survey. J Mach Learn Res. 2018;18(1):5595\u2013637.","journal-title":"J Mach Learn Res"},{"key":"285_CR14","unstructured":"Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G.S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Man\u00e9, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Vi\u00e9gas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., Zheng, X.: TensorFlow: Large-Scale Machine Learning on Heterogeneous Systems. Software available from tensorflow.org. 2015. https:\/\/www.tensorflow.org\/"},{"key":"285_CR15","doi-asserted-by":"publisher","first-page":"8026","DOI":"10.5555\/3454287.3455008","volume-title":"Proceedings of the 33rd International Conference on Neural Information Processing Systems","author":"A Paszke","year":"2019","unstructured":"Paszke A, Gross S, Massa F, Lerer A, Bradbury J, Chanan G, Killeen T, Lin Z, Gimelshein N, Antiga L, Desmaison A, K\u00f6pf A, Yang E, DeVito Z, Raison M, Tejani A, Chilamkurthy S, Steiner B, Fang L, Bai J, Chintala S. PyTorch: an imperative style, high-performance deep learning library. In: Wallach HM, Larochelle H, Beygelzimer A, d\u2019Alch\u00e9-Buc F, Fox EB, editors. Proceedings of the 33rd International Conference on Neural Information Processing Systems. Red Hook: Curran Associates Inc; 2019. p. 8026\u201337. https:\/\/doi.org\/10.5555\/3454287.3455008."},{"key":"285_CR16","doi-asserted-by":"publisher","DOI":"10.1115\/1.4066118","author":"U R\u00f6mer","year":"2024","unstructured":"R\u00f6mer U, Hartmann S, Tr\u00f6ger J-A, Anton D, Wessels H, Flaschel M, De Lorenzis L. Reduced and all-at-once approaches for model calibration and discovery in computational solid mechanics. Appl Mech Rev. 2024. https:\/\/doi.org\/10.1115\/1.4066118.","journal-title":"Appl Mech Rev"},{"issue":"1","key":"285_CR17","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1109\/MSP.2021.3118904","volume":"39","author":"K Shukla","year":"2022","unstructured":"Shukla K, Jagtap AD, Blackshire JL, Sparkman D, Karniadakis GE. A physics-informed neural network for quantifying the microstructural properties of polycrystalline nickel using ultrasound data: a promising approach for solving inverse problems. IEEE Signal Process Mag. 2022;39(1):68\u201377. https:\/\/doi.org\/10.1109\/MSP.2021.3118904.","journal-title":"IEEE Signal Process Mag"},{"issue":"3","key":"285_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.taml.2023.100450","volume":"13","author":"CJG Rojas","year":"2023","unstructured":"Rojas CJG, Boldrini JL, Bittencourt ML. Parameter identification for a damage phase field model using a physics-informed neural network. Theoret Appl Mech Lett. 2023;13(3): 100450. https:\/\/doi.org\/10.1016\/j.taml.2023.100450.","journal-title":"Theoret Appl Mech Lett"},{"issue":"7","key":"285_CR19","doi-asserted-by":"publisher","first-page":"0644","DOI":"10.1126\/sciadv.abk0644","volume":"8","author":"E Zhang","year":"2022","unstructured":"Zhang E, Dao M, Karniadakis GE, Suresh S. Analyses of internal structures and defects in materials using physics-informed neural networks. Sci Adv. 2022;8(7):0644. https:\/\/doi.org\/10.1126\/sciadv.abk0644.","journal-title":"Sci Adv"},{"key":"285_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113741","volume":"379","author":"E Haghighat","year":"2021","unstructured":"Haghighat E, Raissi M, Moure A, Gomez H, Juanes R. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Comput Methods Appl Mech Eng. 2021;379: 113741. https:\/\/doi.org\/10.1016\/j.cma.2021.113741.","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"2","key":"285_CR21","doi-asserted-by":"publisher","first-page":"12431","DOI":"10.1111\/str.12431","volume":"59","author":"CM Hamel","year":"2022","unstructured":"Hamel CM, Long KN, Kramer SLB. Calibrating constitutive models with full-field data via physics informed neural networks. Strain. 2022;59(2):12431. https:\/\/doi.org\/10.1111\/str.12431.","journal-title":"Strain"},{"key":"285_CR22","doi-asserted-by":"publisher","unstructured":"Zhang E, Yin M, Karniadakis GE. Physics-informed neural networks for nonhomogeneous material identification in elasticity imaging. arXiv Preprint (2020)https:\/\/doi.org\/10.48550\/arXiv.2009.04525.","DOI":"10.48550\/arXiv.2009.04525"},{"key":"285_CR23","doi-asserted-by":"publisher","unstructured":"Anton D, Wessels H. Physics-informed neural networks for material model calibration from full-field displacement data. arXiv Preprint (2023) https:\/\/doi.org\/10.48550\/arXiv.2212.07723.","DOI":"10.48550\/arXiv.2212.07723"},{"key":"285_CR24","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2023.116019","volume":"410","author":"E Hosseini","year":"2023","unstructured":"Hosseini E, Scheel P, M\u00fcller O, Molinaro R, Mishra S. Single-track thermal analysis of laser powder bed fusion process: Parametric solution through physics-informed neural networks. Comput Methods Appl Mech Eng. 2023;410: 116019. https:\/\/doi.org\/10.1016\/j.cma.2023.116019.","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"4","key":"285_CR25","doi-asserted-by":"publisher","first-page":"2678","DOI":"10.1109\/TEC.2022.3180295","volume":"37","author":"A Beltr\u00e1n-Pulido","year":"2022","unstructured":"Beltr\u00e1n-Pulido A, Bilionis I, Aliprantis D. Physics-informed neural networks for solving parametric magnetostatic problems. IEEE Trans Energy Convers. 2022;37(4):2678\u201389. https:\/\/doi.org\/10.1109\/TEC.2022.3180295.","journal-title":"IEEE Trans Energy Convers"},{"key":"285_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2023.116042","volume":"411","author":"Y Sun","year":"2023","unstructured":"Sun Y, Sengupta U, Juniper M. Physics-informed deep learning for simultaneous surrogate modeling and PDE-constrained optimization of an airfoil geometry. Comput Methods Appl Mech Eng. 2023;411: 116042. https:\/\/doi.org\/10.1016\/j.cma.2023.116042.","journal-title":"Comput Methods Appl Mech Eng"},{"key":"285_CR27","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-024-02517-w","author":"G Agarwal","year":"2024","unstructured":"Agarwal G, Urrea-Quintero J-H, Wessels H, Wick T. Parameter identification and uncertainty propagation of hydrogel coupled diffusion-deformation using pod-based reduced-order modeling. Comput Mech. 2024. https:\/\/doi.org\/10.1007\/s00466-024-02517-w.","journal-title":"Comput Mech"},{"issue":"6","key":"285_CR28","doi-asserted-by":"publisher","first-page":"1465","DOI":"10.1007\/s10596-022-10164-4","volume":"26","author":"SKF Stoter","year":"2022","unstructured":"Stoter SKF, Jessen E, Niedens V, Schillinger D. A DEIM driven reduced basis method for the diffuse stokes\/Darcy model coupled at parametric phase-field interfaces. Comput Geosci. 2022;26(6):1465\u2013502. https:\/\/doi.org\/10.1007\/s10596-022-10164-4.","journal-title":"Comput Geosci"},{"key":"285_CR29","doi-asserted-by":"publisher","DOI":"10.5281\/zenodo.10447666","author":"IA Baratta","year":"2023","unstructured":"Baratta IA, Dean JP, Dokken JS, Habera M, Hale JS, Richardson CN, Rognes ME, Scroggs MW, Sime N, Wells GN. DOLFINx: the next generation FEniCS problem solving environment. Zenodo. 2023. https:\/\/doi.org\/10.5281\/zenodo.10447666.","journal-title":"Zenodo"},{"key":"285_CR30","doi-asserted-by":"publisher","unstructured":"Anton D. Code for the publication \u201cDeterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks.\u201d Zenodo. 2024. https:\/\/doi.org\/10.5281\/zenodo.11368998.","DOI":"10.5281\/zenodo.11368998"},{"key":"285_CR31","doi-asserted-by":"publisher","DOI":"10.5281\/zenodo.11257192","author":"J-A Tr\u00f6ger","year":"2024","unstructured":"Tr\u00f6ger J-A, Hartmann S, Anton D, Wessels H. Digital image correlation measurement of linear elastic steel specimen. Zenodo Data set. 2024. https:\/\/doi.org\/10.5281\/zenodo.11257192.","journal-title":"Zenodo Data set"},{"key":"285_CR32","volume-title":"Nonlinear solid mechanics: a continuum approach for engineering","author":"GA Holzapfel","year":"2000","unstructured":"Holzapfel GA. Nonlinear solid mechanics: a continuum approach for engineering. 1st ed. Chichester: Wiley; 2000.","edition":"1"},{"key":"285_CR33","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-71001-1","volume-title":"Nonlinear finite element methods","author":"P Wriggers","year":"2008","unstructured":"Wriggers P. Nonlinear finite element methods. 1st ed. Berlin: Springer; 2008. https:\/\/doi.org\/10.1007\/978-3-540-71001-1.","edition":"1"},{"key":"285_CR34","volume-title":"Mathematical foundations of elasticity","author":"J Marsden","year":"1983","unstructured":"Marsden J, Hughes TJR. Mathematical foundations of elasticity. 1st ed. New York: Dover Books on Mathematics. Dover Publications; 1983.","edition":"1"},{"key":"285_CR35","doi-asserted-by":"publisher","DOI":"10.1515\/9783110563214","volume-title":"Geometry of incompatible deformations: differential geometry in continuum mechanics","author":"S Lychev","year":"2019","unstructured":"Lychev S, Koifman K. Geometry of incompatible deformations: differential geometry in continuum mechanics. Berlin: De Gruyter Studies in Mathematical Physics. De Gruyter; 2019. https:\/\/doi.org\/10.1515\/9783110563214."},{"issue":"1","key":"285_CR36","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1007\/BF01187040","volume":"130","author":"W Ehlers","year":"1998","unstructured":"Ehlers W, Eipper G. The simple tension problem at large volumetric strains computed from finite hyperelastic material laws. Acta Mechanica. 1998;130(1):17\u201327. https:\/\/doi.org\/10.1007\/BF01187040.","journal-title":"Acta Mechanica"},{"issue":"11","key":"285_CR37","doi-asserted-by":"publisher","first-page":"2767","DOI":"10.1016\/S0020-7683(03)00086-6","volume":"40","author":"S Hartmann","year":"2003","unstructured":"Hartmann S, Neff P. Polyconvexity of generalized polynomial-type hyperelastic strain energy functions for near-incompressibility. Int J Solids Struct. 2003;40(11):2767\u201391. https:\/\/doi.org\/10.1016\/S0020-7683(03)00086-6.","journal-title":"Int J Solids Struct"},{"key":"285_CR38","volume-title":"Deep learning","author":"I Goodfellow","year":"2016","unstructured":"Goodfellow I, Bengio Y, Courville A. Deep learning. Cambridge: MIT Press; 2016."},{"issue":"4","key":"285_CR39","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/BF02551274","volume":"2","author":"G Cybenko","year":"1989","unstructured":"Cybenko G. Approximation by superpositions of a sigmoidal function. Math Control Signals Syst. 1989;2(4):303\u201314. https:\/\/doi.org\/10.1007\/BF02551274.","journal-title":"Math Control Signals Syst"},{"issue":"5","key":"285_CR40","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","volume":"2","author":"K Hornik","year":"1989","unstructured":"Hornik K, Stinchcombe M, White H. Multilayer feedforward networks are universal approximators. Neural Networks. 1989;2(5):359\u201366. https:\/\/doi.org\/10.1016\/0893-6080(89)90020-8.","journal-title":"Neural Networks"},{"issue":"4","key":"285_CR41","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/0925-2312(95)00070-4","volume":"12","author":"X Li","year":"1996","unstructured":"Li X. Simultaneous approximations of multivariate functions and their derivatives by neural networks with one hidden layer. Neurocomputing. 1996;12(4):327\u201343. https:\/\/doi.org\/10.1016\/0925-2312(95)00070-4.","journal-title":"Neurocomputing"},{"key":"285_CR42","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1016\/j.neucom.2018.06.056","volume":"317","author":"J Berg","year":"2018","unstructured":"Berg J, Nystr\u00f6m K. A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing. 2018;317:28\u201341. https:\/\/doi.org\/10.1016\/j.neucom.2018.06.056.","journal-title":"Neurocomputing"},{"key":"285_CR43","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/978-3-642-35289-8_3","volume-title":"Neural networks: tricks of the Trade","author":"YA LeCun","year":"2012","unstructured":"LeCun YA, Bottou L, Orr GB, M\u00fcller K-R. Efficient BackProp. In: Montavon G, Orr GB, M\u00fcller K-R, editors. Neural networks: tricks of the Trade. 2nd ed. Berlin: Lecture Notes in Computer Science. Springer; 2012. p. 9\u201348.","edition":"2"},{"issue":"5","key":"285_CR44","doi-asserted-by":"publisher","first-page":"3055","DOI":"10.1137\/20M1318043","volume":"43","author":"S Wang","year":"2021","unstructured":"Wang S, Teng Y, Perdikaris P. Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM J Sci Comput. 2021;43(5):3055\u201381. https:\/\/doi.org\/10.1137\/20M1318043.","journal-title":"SIAM J Sci Comput"},{"key":"285_CR45","volume-title":"3rd International conference on learning representations","author":"DP Kingma","year":"2015","unstructured":"Kingma DP, Ba JL. Adam: a method for stochastic optimization. In: Bengio Y, LeCun Y, editors. 3rd International conference on learning representations. San Diego: ICLR; 2015."},{"issue":"1","key":"285_CR46","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/BF01589116","volume":"45","author":"DC Liu","year":"1989","unstructured":"Liu DC, Nocedal J. On the limited memory BFGS method for large scale optimization. Math Programm. 1989;45(1):503\u201328. https:\/\/doi.org\/10.1007\/BF01589116.","journal-title":"Math Programm"},{"issue":"1","key":"285_CR47","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1093\/imamat\/6.1.76","volume":"6","author":"CG Broyden","year":"1970","unstructured":"Broyden CG. The convergence of a class of double-rank minimization algorithms 1. General considerations. IMA J Appl Math. 1970;6(1):76\u201390. https:\/\/doi.org\/10.1093\/imamat\/6.1.76.","journal-title":"IMA J Appl Math"},{"issue":"3","key":"285_CR48","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1093\/comjnl\/13.3.317","volume":"13","author":"R Fletcher","year":"1970","unstructured":"Fletcher R. A new approach to variable metric algorithms. Comput J. 1970;13(3):317\u201322. https:\/\/doi.org\/10.1093\/comjnl\/13.3.317.","journal-title":"Comput J"},{"issue":"109","key":"285_CR49","doi-asserted-by":"publisher","first-page":"23","DOI":"10.2307\/2004873","volume":"24","author":"D Goldfarb","year":"1970","unstructured":"Goldfarb D. A family of variable-metric methods derived by variational means. Math Comput. 1970;24(109):23\u20136. https:\/\/doi.org\/10.2307\/2004873.","journal-title":"Math Comput"},{"issue":"111","key":"285_CR50","doi-asserted-by":"publisher","first-page":"647","DOI":"10.2307\/2004840","volume":"24","author":"DF Shanno","year":"1970","unstructured":"Shanno DF. Conditioning of quasi-Newton methods for function minimization. Math Comput. 1970;24(111):647\u201356. https:\/\/doi.org\/10.2307\/2004840.","journal-title":"Math Comput"},{"key":"285_CR51","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-5981-1","volume-title":"The implicit function theorem: history, theory, and applications","author":"SG Krantz","year":"2013","unstructured":"Krantz SG, Parks HR. The implicit function theorem: history, theory, and applications. 1st ed. New York: Modern Birkh\u00e4user Classics. Birkh\u00e4user; 2013. https:\/\/doi.org\/10.1007\/978-1-4614-5981-1.","edition":"1"},{"key":"285_CR52","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971217","author":"CL Lawson","year":"1995","unstructured":"Lawson CL, Hanson RJ. Solving least squares problems. Classics Appl Math Soc Indust Appl Math. 1995. https:\/\/doi.org\/10.1137\/1.9781611971217.","journal-title":"Classics Appl Math Soc Indust Appl Math"},{"key":"285_CR53","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/9781119176817.ecm2043","volume-title":"Identification of material parameters for constitutive equations","author":"R Mahnken","year":"2017","unstructured":"Mahnken R. Identification of material parameters for constitutive equations. 2nd ed. Hoboken: Wiley; 2017. p. 1\u201321. https:\/\/doi.org\/10.1002\/9781119176817.ecm2043.","edition":"2"},{"key":"285_CR54","doi-asserted-by":"publisher","DOI":"10.1201\/b16018","volume-title":"Bayesian data analysis","author":"A Gelman","year":"2013","unstructured":"Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB. Bayesian data analysis. 3rd ed. New York: Texts in Statistical Science. Chapman and Hall\/CRC; 2013.","edition":"3"},{"issue":"1","key":"285_CR55","doi-asserted-by":"publisher","first-page":"65","DOI":"10.2140\/camcos.2010.5.65","volume":"5","author":"J Goodman","year":"2010","unstructured":"Goodman J, Weare J. Ensemble samplers with affine invariance. Commun Appl Math Comput Sci. 2010;5(1):65\u201380. https:\/\/doi.org\/10.2140\/camcos.2010.5.65.","journal-title":"Commun Appl Math Comput Sci"},{"issue":"925","key":"285_CR56","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1086\/670067","volume":"125","author":"D Foreman-Mackey","year":"2013","unstructured":"Foreman-Mackey D, Hogg D-W, Lang D, Goodman J. emcee: The MCMC Hammer. Publ Astron Soc Pacific. 2013;125(925):306\u201312. https:\/\/doi.org\/10.1086\/670067.","journal-title":"Publ Astron Soc Pacific"},{"key":"285_CR57","doi-asserted-by":"publisher","DOI":"10.1214\/12-EJS675","author":"BJK Kleijn","year":"2012","unstructured":"Kleijn BJK, Vaart AW. The Bernstein-von-mises theorem under misspecification. Electron J Stat. 2012. https:\/\/doi.org\/10.1214\/12-EJS675.","journal-title":"Electron J Stat"},{"key":"285_CR58","first-page":"249","volume-title":"Proceedings of the thirteenth international conference on artificial intelligence and statistics. proceedings of machine learning research","author":"X Glorot","year":"2010","unstructured":"Glorot X, Bengio Y. Understanding the difficulty of training deep feedforward neural networks. In: Teh YW, Titterington M, editors. Proceedings of the thirteenth international conference on artificial intelligence and statistics. proceedings of machine learning research. Sardinia: PMLR; 2010. p. 249\u201356."},{"issue":"4","key":"285_CR59","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1016\/0041-5553(67)90144-9","volume":"7","author":"IM Sobol\u2019","year":"1967","unstructured":"Sobol\u2019 IM. On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput Math Math Phys. 1967;7(4):86\u2013112. https:\/\/doi.org\/10.1016\/0041-5553(67)90144-9.","journal-title":"USSR Comput Math Math Phys"},{"issue":"22","key":"285_CR60","doi-asserted-by":"publisher","first-page":"2993","DOI":"10.1016\/j.ijsolstr.2010.06.022","volume":"47","author":"F Pierron","year":"2010","unstructured":"Pierron F, Avril S, Tran VT. Extension of the virtual fields method to elasto-plastic material identification with cyclic loads and kinematic hardening. Int J Solids Struct. 2010;47(22):2993\u20133010. https:\/\/doi.org\/10.1016\/j.ijsolstr.2010.06.022.","journal-title":"Int J Solids Struct"},{"issue":"1","key":"285_CR61","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/s11340-022-00886-y","volume":"63","author":"S Hartmann","year":"2023","unstructured":"Hartmann S, M\u00fcller-Lohse L, Tr\u00f6ger J-A. Temperature gradient determination with thermography and image correlation in curved surfaces with application to additively manufactured components. Exp Mech. 2023;63(1):43\u201361. https:\/\/doi.org\/10.1007\/s11340-022-00886-y.","journal-title":"Exp Mech"},{"key":"285_CR62","doi-asserted-by":"publisher","DOI":"10.7551\/mitpress\/3206.001.0001","volume-title":"Gaussian processes for machine learning","author":"CE Rasmussen","year":"2005","unstructured":"Rasmussen CE, Williams CKI. Gaussian processes for machine learning. 1st ed. Cambridge: MIT Press; 2005. https:\/\/doi.org\/10.7551\/mitpress\/3206.001.0001.","edition":"1"},{"issue":"3","key":"285_CR63","doi-asserted-by":"publisher","first-page":"425","DOI":"10.1111\/1467-9868.00294","volume":"63","author":"MC Kennedy","year":"2001","unstructured":"Kennedy MC, O\u2019Hagan A. Bayesian calibration of computer models. J Royal Stat Soc Series B Stat Methodol. 2001;63(3):425\u201364. https:\/\/doi.org\/10.1111\/1467-9868.00294.","journal-title":"J Royal Stat Soc Series B Stat Methodol"},{"issue":"4","key":"285_CR64","doi-asserted-by":"publisher","first-page":"1602614","DOI":"10.1126\/sciadv.1602614","volume":"3","author":"SH Rudy","year":"2017","unstructured":"Rudy SH, Brunton SL, Proctor JL, Kutz JN. Data-driven discovery of partial differential equations. Sci Adv. 2017;3(4):1602614. https:\/\/doi.org\/10.1126\/sciadv.1602614.","journal-title":"Sci Adv"},{"key":"285_CR65","doi-asserted-by":"publisher","first-page":"239","DOI":"10.1016\/j.jcp.2019.01.036","volume":"384","author":"J Berg","year":"2019","unstructured":"Berg J, Nystr\u00f6m K. Data-driven discovery of PDEs in complex datasets. J Comput Phys. 2019;384:239\u201352. https:\/\/doi.org\/10.1016\/j.jcp.2019.01.036.","journal-title":"J Comput Phys"},{"key":"285_CR66","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113852","volume":"381","author":"M Flaschel","year":"2021","unstructured":"Flaschel M, Kumar S, De Lorenzis L. Unsupervised discovery of interpretable hyperelastic constitutive laws. Comput Methods Appl Mech Eng. 2021;381: 113852. https:\/\/doi.org\/10.1016\/j.cma.2021.113852.","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"2","key":"285_CR67","doi-asserted-by":"publisher","first-page":"692","DOI":"10.1177\/1475921720935585","volume":"20","author":"C-Z Dong","year":"2021","unstructured":"Dong C-Z, Catbas FN. A review of computer vision-based structural health monitoring at local and global levels. Struct Health Monitor. 2021;20(2):692\u2013743. https:\/\/doi.org\/10.1177\/1475921720935585.","journal-title":"Struct Health Monitor"},{"key":"285_CR68","first-page":"1545","volume-title":"Digital image correlation techniques for NDE and SHM","author":"C Niezrecki","year":"2019","unstructured":"Niezrecki C, Baqersad J, Sabato A. Digital image correlation techniques for NDE and SHM. Cham: Springer; 2019. p. 1545\u201390."},{"issue":"5","key":"285_CR69","doi-asserted-by":"publisher","first-page":"1056","DOI":"10.1177\/1475921717735326","volume":"17","author":"D Reagan","year":"2018","unstructured":"Reagan D, Sabato A, Niezrecki C. Feasibility of using digital image correlation for unmanned aerial vehicle structural health monitoring of bridges. Struct Health Monitor. 2018;17(5):1056\u201372. https:\/\/doi.org\/10.1177\/1475921717735326.","journal-title":"Struct Health Monitor"},{"key":"285_CR70","doi-asserted-by":"publisher","unstructured":"Moreno-Gomez A, Perez-Ramirez CA, Dominguez-Gonzalez A, Valtierra-Rodriguez M, Chavez-Alegria O, Amezquita-Sanchez JP. Sensors used in structural health monitoring. Archiv Comput Methods Eng. 2018;25(4):901\u201318. https:\/\/doi.org\/10.1007\/s11831-017-9217-4.","DOI":"10.1007\/s11831-017-9217-4"},{"key":"285_CR71","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. A physics-guided neural network framework for elastic plates: comparison of governing equations-based and energy-based approaches. Comput Methods Appl Mech Eng. 2021;383: 113933. https:\/\/doi.org\/10.1016\/j.cma.2021.113933.","journal-title":"Comput Methods Appl Mech Eng"},{"key":"285_CR72","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.114790","volume":"393","author":"A Henkes","year":"2022","unstructured":"Henkes A, Wessels H, Mahnken R. Physics informed neural networks for continuum micromechanics. Comput Methods Appl Mech Eng. 2022;393: 114790. https:\/\/doi.org\/10.1016\/j.cma.2022.114790.","journal-title":"Comput Methods Appl Mech Eng"}],"container-title":["Advanced Modeling and Simulation in Engineering Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s40323-025-00285-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1186\/s40323-025-00285-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1186\/s40323-025-00285-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T12:10:12Z","timestamp":1748520612000},"score":1,"resource":{"primary":{"URL":"https:\/\/amses-journal.springeropen.com\/articles\/10.1186\/s40323-025-00285-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,29]]},"references-count":72,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["285"],"URL":"https:\/\/doi.org\/10.1186\/s40323-025-00285-7","relation":{},"ISSN":["2213-7467"],"issn-type":[{"value":"2213-7467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,29]]},"assertion":[{"value":"28 May 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 February 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 May 2025","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 competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"12"}}