Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

doi:10.1007/s10483-023-2995-8

Abstract

Abstract: Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials.

Key words: solid mechanics, material identification, physics-informed neural network (PINN), data sampling, boundary condition (BC) constraint

2010 MSC Number:

W. WU, M. DANEKER, M. A. JOLLEY, K. T. TURNER, L. LU. Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics. Applied Mathematics and Mechanics (English Edition), 2023, 44(7): 1039-1068.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-023-2995-8

https://www.amm.shu.edu.cn/EN/Y2023/V44/I7/1039

[1] BRÜLS, O. and GOLINVAL, J. C. On the numerical damping of time integrators for coupled mechatronic systems. Computer Methods in Applied Mechanics and Engineering, 197, 577-588(2008)
[2] CHUNG, J. and HULBERT, G. M. A time integration algorithm for structural dynamics with improved numerical dissipation:the generalized-α method. Journal of Applied Mechanics, 60, 371-375(1993)
[3] HILBER, H. M., HUGHES, T. J. R., and TAYLOR, R. L. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering & Structural Dynamics, 5, 283-292(1977)
[4] NEWMARK, N. M. A method of computation for structural dynamics. Journal of the Engineering Mechanics Division, 85, 67-94(1959)
[5] HORNIK, K., STINCHCOMBE, M., and WHITE, H. Multilayer feedforward networks are universal approximators. Neural Networks, 2, 359-366(1989)
[6] KARNIADAKIS, G. E., KEVREKIDIS, I. G., LU, L., PERDIKARIS, P., WANG, S., and YANG, L. Physics-informed machine learning. Nature Reviews Physics, 3, 422-440(2021)
[7] CAI, S., WANG, Z., WANG, S., PERDIKARIS, P., and KARNIADAKIS, G. E. Physics-informed neural networks for heat transfer problems. Journal of Heat Transfer, 143, 060801(2021)
[8] LU, L., JIN, P., PANG, G., ZHANG, Z., and KARNIADAKIS, G. E. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature Machine Intelligence, 3, 218-229(2021)
[9] MAO, Z., JAGTAP, A. D., and KARNIADAKIS, G. E. Physics-informed neural networks for high-speed flows. Computer Methods in Applied Mechanics and Engineering, 360, 112789(2020)
[10] RAISSI, M. Deep hidden physics models:deep learning of nonlinear partial differential equations. The Journal of Machine Learning Research, 19, 932-955(2018)
[11] GUO, K., YANG, Z., YU, C. H., and BUEHLER, M. J. Artificial intelligence and machine learning in design of mechanical materials. Materials Horizons, 8, 1153-1172(2021)
[12] CHEN, G., LI, T., CHEN, Q., REN, S., WANG, C., and LI, S. Application of deep learning neural network to identify collision load conditions based on permanent plastic deformation of shell structures. Computational Mechanics, 64, 435-449(2019)
[13] LINKA, K., HILLGÄRTNER, M., ABDOLAZIZI, K. P., AYDIN, R. C., ITSKOV, M., and CYRON, C. J. Constitutive artificial neural networks:a fast and general approach to predictive data-driven constitutive modeling by deep learning. Journal of Computational Physics, 429, 110010(2021)
[14] LU, L., DAO, M., KUMAR, P., RAMAMURTY, U., KARNIADAKIS, G. E., and SURESH, S. Extraction of mechanical properties of materials through deep learning from instrumented indentation. Proceedings of the National Academy of Sciences, 117, 7052-7062(2020)
[15] STERN, M., ARINZE, C., PEREZ, L., PALMER, S. E., and MURUGAN, A. Supervised learning through physical changes in a mechanical system. Proceedings of the National Academy of Sciences, 117, 14843-14850(2020)
[16] WU, L., NGUYEN, V. D., KILINGAR, N. G., and NOELS, L. A recurrent neural network-accelerated multi-scale model for elasto-plastic heterogeneous materials subjected to random cyclic and non-proportional loading paths. Computer Methods in Applied Mechanics and Engineering, 369, 113234(2020)
[17] YOU, H., ZHANG, Q., ROSS, C. J., LEE, C. H., and YU, Y. Learning deep implicit Fourier neural operators (IFNOs) with applications to heterogeneous material modeling. Computer Methods in Applied Mechanics and Engineering, 398, 115296(2022)
[18] CONDUIT, B., JONES, N., STONE, H., and CONDUIT, G. Design of a nickel-base superalloy using a neural network. Materials & Design, 131, 358-365(2017)
[19] GU, G. X., CHEN, C. T., RICHMOND, D. J., and BUEHLER, M. J. Bioinspired hierarchical composite design using machine learning:simulation, additive manufacturing, and experiment. Materials Horizons, 5, 939-945(2018)
[20] KIM, Y., KIM, Y., YANG, C., PARK, K., GU, G. X., and RYU, S. Deep learning framework for material design space exploration using active transfer learning and data augmentation. NPJ Computational Materials, 7, 140(2021)
[21] LING, J., HUTCHINSON, M., ANTONO, E., PARADISO, S., and MEREDIG, B. High-dimensional materials and process optimization using data-driven experimental design with well-calibrated uncertainty estimates. Integrating Materials and Manufacturing Innovation, 6, 207-217(2017)
[22] SPEAR, A. D., KALIDINDI, S. R., MEREDIG, B., KONTSOS, A., and LE GRAVEREND, J. B. Data-driven materials investigations:the next frontier in understanding and predicting fatigue behavior. JOM, 70, 1143-1146(2018)
[23] XUE, D., YUAN, R., ZHOU, Y., XUE, D., LOOKMAN, T., ZHANG, G., DING, X., and SUN, J. Design of high temperature Ti-Pd-Cr shape memory alloys with small thermal hysteresis. Scientific Reports, 6, 28244(2016)
[24] JIN, P., LU, L., TANG, Y., and KARNIADAKIS, G. E. Quantifying the generalization error in deep learning in terms of data distribution and neural network smoothness. Neural Networks, 130, 85-99(2020)
[25] LU, L., MENG, X., MAO, Z., and KARNIADAKIS, G. E. DeepXDE:a deep learning library for solving differential equations. SIAM Review, 63, 208-228(2021)
[26] RAISSI, M., PERDIKARIS, P., and KARNIADAKIS, G. Physics-informed neural networks:a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686-707(2019)
[27] HAGHIGHAT, E., RAISSI, M., MOURE, A., GOMEZ, H., and JUANES, R. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 379, 113741(2021)
[28] ALMAJID, M. M. and ABU-AL-SAUD, M. O. Prediction of porous media fluid flow using physics informed neural networks. Journal of Petroleum Science and Engineering, 208, 109205(2022)
[29] CHEN, Y., LU, L., KARNIADAKIS, G. E., and NEGRO, L. D. Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Optics Express, 28, 11618-11633(2020)
[30] CUOMO, S., DI COLA, V. S., GIAMPAOLO, F., ROZZA, G., RAISSI, M., and PICCIALLI, F. Scientific machine learning through physics-informed neural networks:where we are and what's next. Journal of Scientific Computing, 92, 88(2022)
[31] DANEKER, M., ZHANG, Z., KARNIADAKIS, G. E., and LU, L. Systems biology:identifiability analysis and parameter identification via systems-biology informed neural networks. arXiv Preprint, arXiv:2202.01723(2022) https://doi.org/10.48550/arXiv.2202.01723
[32] JIN, X., CAI, S., LI, H., and KARNIADAKIS, G. E. NSFnets (Navier-Stokes flow nets):physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 426, 109951(2021)
[33] YAZDANI, A., LU, L., RAISSI, M., and KARNIADAKIS, G. E. Systems biology informed deep learning for inferring parameters and hidden dynamics. PLoS Computational Biology, 16, e1007575(2020)
[34] PANG, G., LU, L., and KARNIADAKIS, G. E. fPINNs:fractional physics-informed neural networks. SIAM Journal on Scientific Computing, 41, A2603-A2626(2019)
[35] YUAN, L., NI, Y. Q., DENG, X. Y., and HAO, S. A-PINN:auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations. Journal of Computational Physics, 462, 111260(2022)
[36] ZHANG, D., LU, L., GUO, L., and KARNIADAKIS, G. E. Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems. Journal of Computational Physics, 397, 108850(2019)
[37] YU, J., LU, L., MENG, X., and KARNIADAKIS, G. E. Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems. Computer Methods in Applied Mechanics and Engineering, 393, 114823(2022)
[38] SAMANIEGO, E., ANITESCU, C., GOSWAMI, S., NGUYEN-THANH, V., GUO, H., HAMDIA, K., ZHUANG, X., and RABCZUK, T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning:concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 362, 112790(2020)
[39] NGUYEN-THANH, V. M., ANITESCU, C., ALAJLAN, N., RABCZUK, T., and ZHUANG, X. Parametric deep energy approach for elasticity accounting for strain gradient effects. Computer Methods in Applied Mechanics and Engineering, 386, 114096(2021)
[40] NGUYEN-THANH, V. M., ZHUANG, X., and RABCZUK, T. A deep energy method for finite deformation hyperelasticity. European Journal of Mechanics-A/Solids, 80, 103874(2020)
[41] ABUEIDDA, D. W., LU, Q., and KORIC, S. Meshless physics-informed deep learning method for three-dimensional solid mechanics. International Journal for Numerical Methods in Engineering, 122, 7182-7201(2021)
[42] FUHG, J. N. and BOUKLAS, N. The mixed deep energy method for resolving concentration features in finite strain hyperelasticity. Journal of Computational Physics, 451, 110839(2022)
[43] HENKES, A., WESSELS, H., and MAHNKEN, R. Physics informed neural networks for continuum micromechanics. Computer Methods in Applied Mechanics and Engineering, 393, 114790(2022)
[44] WU, C., ZHU, M., TAN, Q., KARTHA, Y., and LU, L. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 403, 115671(2023)
[45] RAO, C. P., SUN, H., and LIU, Y. Physics-informed deep learning for computational elastodynamics without labeled data. Journal of Engineering Mechanics, 147, 04021043(2021)
[46] ZHANG, E., DAO, M., KARNIADAKIS, G. E., and SURESH, S. Analyses of internal structures and defects in materials using physics-informed neural networks. Science Advances, 8, eabk0644(2022)
[47] LU, L., PESTOURIE, R., YAO, W., WANG, Z., VERDUGO, F., and JOHNSON, S. G. Physicsinformed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing, 43, B1105-B1132(2021)
[48] ROHRHOFER, F. M., POSCH, S., GÖSSNITZER, C., and GEIGER, B. C. On the role of fixed points of dynamical systems in training physics-informed neural networks. arXiv Preprint, arXiv:2203.13648(2022) https://doi.org/10.48550/arXiv.2203.13648
[49] MCCLENNY, L. D. and BRAGA-NETO, U. M. Self-adaptive physics-informed neural networks using a soft attention mechanism. arXiv Preprint, arXiv:2009.04544(2020) https://doi.org/10.48550/arXiv.2009.04544
[50] RUMELHART, D. E., HINTON, G. E., and WILLIAMS, R. J. Learning representations by back- propagating errors. nature, 323, 533-536(1986)
[51] BAYDIN, A. G., PEARLMUTTER, B. A., RADUL, A. A., and SISKIND, J. M. Automatic differentiation in machine learning:a survey. The Journal of Machine Learning Research, 18, 5595-5637(2017)
[52] MARGOSSIAN, C. C. A review of automatic differentiation and its efficient implementation. Wires Data Mining and Knowledge Discovery, 9, e1305(2019)
[53] MATTEY, R. and GHOSH, S. A novel sequential method to train physics informed neural net- works for Allen Cahn and Cahn Hilliard equations. Computer Methods in Applied Mechanics and Engineering, 390, 114474(2022)
[54] STEPHANY, R. and EARLS, C. PDE-read:human-readable partial differential equation discovery using deep learning. Neural Networks, 154, 360-382(2022)
[55] LOGG, A. and WELLS, G. N. DOLFIN:automated finite element computing. ACM Transactions on Mathematical Software, 37, 20(2011)
[56] BERG, J. and NYSTRÖM, K. Neural networks as smooth priors for inverse problems for PDEs. Journal of Computational Mathematics and Data Science, 1, 100008(2021)
[57] THAKOLKARAN, P., JOSHI, A., ZHENG, Y., FLASCHEL, M., DE LORENZIS, L., and KUMAR, S. NN-Euclid:deep-learning hyperelasticity without stress data. Journal of the Mechanics and Physics of Solids, 169, 105076(2022)
[58] FREUTEL, M., SCHMIDT, H., DÜRSELEN, L., IGNATIUS, A., and GALBUSERA, F. Finite element modeling of soft tissues:material models, tissue interaction and challenges. Clinical Biomechanics, 29, 363-372(2014)
[59] LIN, S., MORGANT, M. C., MARÍN-CASTRILLÓN, D. M., WALKER, P. M., AHO GLÉLÉ, L. S., BOUCHER, A., PRESLES, B., BOUCHOT, O., and LALANDE, A. Aortic local biomechanical properties in ascending aortic aneurysms. Acta Biomaterialia, 149, 40-50(2022)
[60] LIU, M., LIANG, L., SULEJMANI, F., LOU, X., IANNUCCI, G., CHEN, E., LESHNOWER, B., and SUN, W. Identification of in vivo nonlinear anisotropic mechanical properties of ascending thoracic aortic aneurysm from patient-specific CT scans. Scientific Reports, 9, 12983(2019)
[61] LEE, C. H., LAURENCE, D. W., ROSS, C. J., KRAMER, K. E., BABU, A. R., JOHNSON, E. L., HSU, M. C., AGGARWAL, A., MIR, A., BURKHART, H. M., TOWNER, R. A., BAUMWART, R., and WU, Y. Mechanics of the tricuspid valve——from clinical diagnosis/treatment, in-vivo and in-vitro investigations, to patient-specific biomechanical modeling. Bioengineering, 6, 2(2019)
[62] NARANG, H., REGO, B. V., KHALIGHI, A. H., ALY, A., POUCH, A. M., GORMAN, R. C., GORMAN, J. H., III, and SACKS, M. S. Pre-surgical prediction of ischemic mitral regurgitation recurrence using in vivo mitral valve leaflet strains. Annals of Biomedical Engineering, 49, 3711-3723(2021)
[63] WU, W., CHING, S., MAAS, S. A., LASSO, A., SABIN, P., WEISS, J. A., and JOLLEY, M. A. A computational framework for atrioventricular valve modeling using open-source software. Journal of Biomechanical Engineering, 144, 101012(2022)

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

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