A robust solution of a statistical inverse problem in multiscale computational mechanics using an artificial neural network. (English) Zbl 1506.74488
Summary: This work addresses the inverse identification of apparent elastic properties of random heterogeneous materials using machine learning based on artificial neural networks. The proposed neural network-based identification method requires the construction of a database from which an artificial neural network can be trained to learn the nonlinear relationship between the hyperparameters of a prior stochastic model of the random compliance field and some relevant quantities of interest of an ad hoc multiscale computational model. An initial database made up with input and target data is first generated from the computational model, from which a processed database is deduced by conditioning the input data with respect to the target data using the nonparametric statistics. Two- and three-layer feedforward artificial neural networks are then trained from each of the initial and processed databases to construct an algebraic representation of the nonlinear mapping between the hyperparameters (network outputs) and the quantities of interest (network inputs). The performances of the trained artificial neural networks are analyzed in terms of mean squared error, linear regression fit and probability distribution between network outputs and targets for both databases. An ad hoc probabilistic model of the input random vector is finally proposed in order to take into account uncertainties on the network input and to perform a robustness analysis of the network output with respect to the input uncertainties level. The capability of the proposed neural network-based identification method to efficiently solve the underlying statistical inverse problem is illustrated through two numerical examples developed within the framework of 2D plane stress linear elasticity, namely a first validation example on synthetic data obtained through computational simulations and a second application example on real experimental data obtained through a physical experiment monitored by digital image correlation on a real heterogeneous biological material (beef cortical bone).
MSC:
| 74S60 | Stochastic and other probabilistic methods applied to problems in solid mechanics |
| 62M45 | Neural nets and related approaches to inference from stochastic processes |
| 62M40 | Random fields; image analysis |
| 62P30 | Applications of statistics in engineering and industry; control charts |
| 65C05 | Monte Carlo methods |
| 65C20 | Probabilistic models, generic numerical methods in probability and statistics |
| 74G75 | Inverse problems in equilibrium solid mechanics |
| 74Q05 | Homogenization in equilibrium problems of solid mechanics |
| 68T05 | Learning and adaptive systems in artificial intelligence |
Keywords:
machine learning; uncertainty quantification; probabilistic modeling; statistical inverse problem; random heterogeneous materials; random multiscale modelingReferences:
| [1] | Soize, C., Non-Gaussian positive-definite matrix-valued random fields for elliptic stochastic partial differential operators, Comput. Methods Appl. Mech. Engrg., 195, 1-3, 26-64 (2006) · Zbl 1093.74065 |
| [2] | Soize, C., Tensor-valued random fields for meso-scale stochastic model of anisotropic elastic microstructure and probabilistic analysis of representative volume element size, Probab. Eng. Mech., 23, 2-3, 307-323 (2008), 5th International Conference on Computational Stochastic Mechanics |
| [3] | Nguyen, M.-T.; Desceliers, C.; Soize, C.; Allain, J.-M.; Gharbi, H., Multiscale identification of the random elasticity field at mesoscale of a heterogeneous microstructure using multiscale experimental observations, Int. J. Multiscale Comput. Eng., 13, 4, 281-295 (2015) |
| [4] | Zhang, T.; Pled, F.; Desceliers, C., Robust multiscale identification of apparent elastic properties at mesoscale for random heterogeneous materials with multiscale field measurements, Materials, 13, 12 (2020), URL https://www.mdpi.com/1996-1944/13/12/2826 |
| [5] | Desceliers, C.; Ghanem, R.; Soize, C., Maximum likelihood estimation of stochastic chaos representations from experimental data, Internat. J. Numer. Methods Engrg., 66, 6, 978-1001 (2006) · Zbl 1110.74826 |
| [6] | Ghanem, R. G.; Doostan, A., On the construction and analysis of stochastic models: Characterization and propagation of the errors associated with limited data, J. Comput. Phys., 217, 1, 63-81 (2006) · Zbl 1102.65004 |
| [7] | Desceliers, C.; Soize, C.; Ghanem, R., Identification of chaos representations of elastic properties of random media using experimental vibration tests, Comput. Mech., 39, 6, 831-838 (2007) · Zbl 1161.74016 |
| [8] | Marzouk, Y. M.; Najm, H. N.; Rahn, L. A., Stochastic spectral methods for efficient Bayesian solution of inverse problems, J. Comput. Phys., 224, 2, 560-586 (2007) · Zbl 1120.65306 |
| [9] | Arnst, M.; Clouteau, D.; Bonnet, M., Inversion of probabilistic structural models using measured transfer functions, Comput. Methods Appl. Mech. Engrg., 197, 6, 589-608 (2008) · Zbl 1169.74422 |
| [10] | Das, S.; Ghanem, R.; Spall, J. C., Asymptotic sampling distribution for polynomial chaos representation of data: A maximum entropy and Fisher information approach, (Proceedings of the 45th IEEE Conference on Decision and Control (2006)), 4139-4144 |
| [11] | Das, S.; Ghanem, R.; Finette, S., Polynomial chaos representation of spatio-temporal random fields from experimental measurements, J. Comput. Phys., 228, 23, 8726-8751 (2009) · Zbl 1177.65019 |
| [12] | Desceliers, C.; Soize, C.; Grimal, Q.; Talmant, M.; Naili, S., Determination of the random anisotropic elasticity layer using transient wave propagation in a fluid-solid multilayer: Model and experiments, J. Acoust. Soc. Am., 125, 4, 2027-2034 (2009), arXiv:https://doi.org/10.1121/1.3087428 |
| [13] | Guilleminot, J.; Soize, C.; Kondo, D., Mesoscale probabilistic models for the elasticity tensor of fiber reinforced composites: Experimental identification and numerical aspects, Mech. Mater., 41, 12, 1309-1322 (2009) |
| [14] | Ma, X.; Zabaras, N., An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method, Inverse Problems, 25, 3, Article 035013 pp. (2009), URL http://stacks.iop.org/0266-5611/25/i=3/a=035013 · Zbl 1161.62011 |
| [15] | Marzouk, Y. M.; Najm, H. N., Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems, J. Comput. Phys., 228, 6, 1862-1902 (2009) · Zbl 1161.65308 |
| [16] | Arnst, M.; Ghanem, R.; Soize, C., Identification of Bayesian posteriors for coefficients of chaos expansions, J. Comput. Phys., 229, 9, 3134-3154 (2010) · Zbl 1184.62034 |
| [17] | Das, S.; Spall, J. C.; Ghanem, R., Efficient Monte Carlo computation of Fisher information matrix using prior information, Comput. Statist. Data Anal., 54, 2, 272-289 (2010) · Zbl 1464.62052 |
| [18] | Ta, Q.-A.; Clouteau, D.; Cottereau, R., Modeling of random anisotropic elastic media and impact on wave propagation, Eur. J. Comput. Mech., 19, 1-3, 241-253 (2010), arXiv:http://www.tandfonline.com/doi/pdf/10.3166/ejcm.19.241-253 · Zbl 1426.74170 |
| [19] | Soize, C., Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data, Comput. Methods Appl. Mech. Engrg., 199, 33-36, 2150-2164 (2010) · Zbl 1231.74501 |
| [20] | Soize, C., A computational inverse method for identification of non-Gaussian random fields using the Bayesian approach in very high dimension, Comput. Methods Appl. Mech. Engrg., 200, 45-46, 3083-3099 (2011) · Zbl 1230.74241 |
| [21] | Cottereau, R.; Clouteau, D.; Dhia, H. B.; Zaccardi, C., A stochastic-deterministic coupling method for continuum mechanics, Comput. Methods Appl. Mech. Engrg., 200, 47-48, 3280-3288 (2011) · Zbl 1230.74239 |
| [22] | Desceliers, C.; Soize, C.; Naili, S.; Haiat, G., Probabilistic model of the human cortical bone with mechanical alterations in ultrasonic range, Mech. Syst. Signal Process., 32, 170-177 (2012), Uncertainties in Structural Dynamics |
| [23] | Perrin, G.; Soize, C.; Duhamel, D.; Funfschilling, C., Identification of polynomial chaos representations in high dimension from a set of realizations, SIAM J. Sci. Comput., 34, 6, A2917-A2945 (2012), arXiv:https://doi.org/10.1137/11084950X · Zbl 1262.60067 |
| [24] | Clouteau, D.; Cottereau, R.; Lombaert, G., Dynamics of structures coupled with elastic media—A review of numerical models and methods, J. Sound Vib., 332, 10, 2415-2436 (2013) |
| [25] | Haykin, S., Neural Networks: A Comprehensive Foundation (1994), Prentice Hall PTR: Prentice Hall PTR Upper Saddle River, NJ, USA · Zbl 0828.68103 |
| [26] | Hagan, M. T.; Demuth, H. B.; Beale, M. H., Neural Network Design (1996), PWS Publishing Co.: PWS Publishing Co. Boston, MA, USA |
| [27] | Demuth, H. B.; Beale, M. H.; De Jess, O.; Hagan, M. T., Neural Network Design (2014), Martin Hagan: Martin Hagan USA |
| [28] | Bowman, A. W.; Azzalini, A., Applied Smoothing Techniques for Data Analysis (1997), Oxford University Press: Oxford University Press Oxford · Zbl 0889.62027 |
| [29] | Horová, I.; Koláček, J.; Zelinka, J., Kernel Smoothing in MATLAB: Theory and Practice of Kernel Smoothing (2012), World Scientific Publishing Co. Pte. Ltd.: World Scientific Publishing Co. Pte. Ltd. Singapore, URL http://www.worldscientific.com/worldscibooks/10.1142/8468#t=aboutBook · Zbl 1273.62019 |
| [30] | Givens, G. H.; Hoeting, J. A., Computational Statistics (2013), John Wiley & Sons: John Wiley & Sons Hoboken, New Jersey · Zbl 1267.62003 |
| [31] | Scott, D. W., Multivariate Density Estimation: Theory, Practice, and Visualization (2015), John Wiley & Sons, Inc. · Zbl 1311.62004 |
| [32] | Soize, C., Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering, Interdisciplinary Applied Mathematics (2017), Springer International Publishing · Zbl 1377.60002 |
| [33] | Jaynes, E. T., Information theory and statistical mechanics, Phys. Rev., 106, 620-630 (1957), URL http://link.aps.org/doi/10.1103/PhysRev.106.620 · Zbl 0084.43701 |
| [34] | Jaynes, E. T., Information theory and statistical mechanics. II, Phys. Rev., 108, 171-190 (1957), URL http://link.aps.org/doi/10.1103/PhysRev.108.171 · Zbl 0084.43701 |
| [35] | Sobezyk, K.; Trȩbicki, J., Maximum entropy principle in stochastic dynamics, Probab. Eng. Mech., 5, 3, 102-110 (1990) |
| [36] | Kapur, J. N.; Kesavan, H. K., Entropy Optimization Principles and Their Applications, 3-20 (1992), Springer Netherlands: Springer Netherlands Dordrecht |
| [37] | Jumarie, G., Maximum Entropy, Information Without Probability and Complex Fractals: Classical and Quantum Approach, Fundamental Theories of Physics (2000), Springer Science & Business Media: Springer Science & Business Media Dordrecht · Zbl 0982.94001 |
| [38] | Jaynes, E. T., Probability Theory: The Logic of Science (2003), Cambridge university press · Zbl 1045.62001 |
| [39] | Cover, T. M.; Thomas, J. A., Elements of Information Theory, A Wiley-Interscience publication (2006), Wiley: Wiley New York, NY, USA · Zbl 1140.94001 |
| [40] | Hughes, T. J.R., The Finite Element Method : Linear Static and Dynamic Finite Element Analysis (1987), Prentice Hall: Prentice Hall Englewood Cliffs, New Jersey · Zbl 0634.73056 |
| [41] | Zienkiewicz, O. C.; Taylor, R. L.; Zhu, J. Z., The Finite Element Method: Its Basis and Fundamentals (2005), Butterworth-Heinemann · Zbl 1307.74005 |
| [42] | Zhang, T., Multiscale statistical inverse problem for the identification of random fields of elastic properties (in french) (2019), Université Paris-Est, URL https://tel.archives-ouvertes.fr/tel-02506242 |
| [43] | Nemat-Nasser, S.; Hori, M., Micromechanics: Overall Properties of Heterogeneous Materials, North-Holland Series in Applied Mathematics and Mechanics, 3-687 (1993), North-Holland: North-Holland Amsterdam, The Netherlands · Zbl 0924.73006 |
| [44] | Bornert, M.; Bretheau, T.; Gilormini, P., Homogénéisation en mécanique des matériaux 1. Matériaux aléatoires élastiques et milieux périodiques (2001), Hermès Science publications: Hermès Science publications Paris · Zbl 1067.74501 |
| [45] | Torquato, S., Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Vol. 16 (2002), Springer-Verlag: Springer-Verlag New York, NY, USA · Zbl 0988.74001 |
| [46] | Zaoui, A., Continuum micromechanics: Survey, J. Eng. Mech., 128, 8, 808-816 (2002), arXiv:https://ascelibrary.org/doi/pdf/10.1061/(ASCE)0733-9399(2002)128:8(808) |
| [47] | Bourgeat, A.; Piatnitski, A., Approximations of effective coefficients in stochastic homogenization, Ann. Inst. Henri Poincare (B) Probab. Stat., 40, 2, 153-165 (2004) · Zbl 1058.35023 |
| [48] | Nguyen, M.-T.; Allain, J.-M.; Gharbi, H.; Desceliers, C.; Soize, C., Experimental multiscale measurements for the mechanical identification of a cortical bone by digital image correlation, J. Mech. Behav. Biomed. Mater., 63, 125-133 (2016) |
| [49] | Shinozuka, M., Simulation of multivariate and multidimensional random processes, J. Acoust. Soc. Am., 49, 1B, 357-368 (1971), arXiv:https://doi.org/10.1121/1.1912338 |
| [50] | Shinozuka, M.; Wen, Y. K., Monte Carlo solution of nonlinear vibrations, AIAA J.. AIAA J., AIAA J., 10, 1, 37-40 (1972) · Zbl 0236.70032 |
| [51] | Shinozuka, M.; Jan, C.-M., Digital simulation of random processes and its applications, J. Sound Vib., 25, 1, 111-128 (1972) |
| [52] | Poirion, F.; Soize, C., Numerical simulation of homogeneous and inhomogeneous Gaussian stochastic vector fields, La Rech. Aerosp. (English edition), 1, -, 41-61 (1989), URL https://hal-upec-upem.archives-ouvertes.fr/hal-00770316 · Zbl 0679.76062 |
| [53] | Poirion, F.; Soize, C., Numerical methods and mathematical aspects for simulation of homogeneous and non homogeneous gaussian vector fields, (Krée, P.; Wedig, W., Probabilistic Methods in Applied Physics (1995), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 17-53 · Zbl 0839.65152 |
| [54] | Silverman, B. W., Density Estimation for Statistics and Data Analysis (1986), Chapman and Hall: Chapman and Hall London · Zbl 0617.62042 |
| [55] | Robert, C.; Casella, G., Monte Carlo Statistical Methods, Springer Texts in Statistics (2004), Springer-Verlag: Springer-Verlag New York, NY, USA · Zbl 1096.62003 |
| [56] | Kaipio, J.; Somersalo, E., Statistical and Computational Inverse Problems, Applied Mathematical Sciences (2005), Springer-Verlag: Springer-Verlag New York, NY, USA · Zbl 1068.65022 |
| [57] | Spall, J. C., Introduction to Stochastic Search and Optimization: Estimation, Simulation, and Control (2005), John Wiley & Sons |
| [58] | Cybenko, G., Approximation by superpositions of a sigmoidal function, Math. Control Signals Systems, 2, 4, 303-314 (1989) · Zbl 0679.94019 |
| [59] | Hornik, K.; Stinchcombe, M.; White, H., Multilayer feedforward networks are universal approximators, Neural Netw., 2, 5, 359-366 (1989) · Zbl 1383.92015 |
| [60] | Hornik, K., Approximation capabilities of multilayer feedforward networks, Neural Netw., 4, 2, 251-257 (1991) |
| [61] | Leshno, M.; Lin, V. Y.; Pinkus, A.; Schocken, S., Multilayer feedforward networks with a nonpolynomial activation function can approximate any function, Neural Netw., 6, 6, 861-867 (1993) |
| [62] | Barron, A. R., Universal approximation bounds for superpositions of a sigmoidal function, IEEE Trans. Inform. Theory, 39, 3, 930-945 (1993) · Zbl 0818.68126 |
| [63] | LeCun, Y.; Bengio, Y.; Hinton, G., Deep learning, Nature, 521, 7553, 436-444 (2015) |
| [64] | Goodfellow, I.; Bengio, Y.; Courville, A., Deep Learning (2016), The MIT Press: The MIT Press Cambridge, MA, USA, http://www.deeplearningbook.org · Zbl 1373.68009 |
| [65] | Vogl, T. P.; Mangis, J. K.; Rigler, A. K.; Zink, W. T.; Alkon, D. L., Accelerating the convergence of the back-propagation method, Biol. Cybernet., 59, 4, 257-263 (1988) |
| [66] | Nguyen, D.; Widrow, B., Improving the learning speed of 2-layer neural networks by choosing initial values of the adaptive weights, (1990 IJCNN International Joint Conference on Neural Networks, Vol. 3 (1990)), 21-26 |
| [67] | Beale, M. H.; Hagan, M. T.; Demuth, H. B., Neural Network Toolbox User’s Guide (1992), The MathWorks Inc |
| [68] | Soize, C., Random matrix theory for modeling uncertainties in computational mechanics, Comput. Methods Appl. Mech. Engrg., 194, 12-16, 1333-1366 (2005), Special Issue on Computational Methods in Stochastic Mechanics and Reliability Analysis · Zbl 1083.74052 |
| [69] | Soize, C., Random matrix models and nonparametric method for uncertainty quantification, (Handbook of Uncertainty Quantification (2016), Springer International Publishing: Springer International Publishing Cham), 1-69 |
| [70] | Lawson, C.; Hanson, R., Solving Least Squares Problems (1995), Society for Industrial and Applied Mathematics, arXiv:https://epubs.siam.org/doi/pdf/10.1137/1.9781611971217 · Zbl 0860.65029 |
| [71] | Serfling, R., Approximation Theorems of Mathematical Statistics (1980), Wiley: Wiley New York, NY, USA · Zbl 0538.62002 |
| [72] | Papoulis, A.; Pillai, S. U., Probability, Random Variables, and Stochastic Processes (2002), McGraw-Hill Higher Education: McGraw-Hill Higher Education New York, NY, USA |
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