**Abstract** : For many materials, as for instance the biological materials, the microstructure is complex and highly heterogenous. An efficient approach for constructing the model of such materials consists in modeling their apparent elasticity properties at mesoscale by a tensor-valued random field [1]. Nevertheless, an important challenge is related to the identification of the hyperparameters of such a probabilistic mesoscopic model with limited experimental measurements. An efficient methodology has been recently proposed in [2,3] to address this statistical inverse problem, which consists in solving a multiscale and multi-objective optimization problem with limited experimental information at both macroscale and mesoscale. The multiobjective cost functions that are used rely on four experimentally measured indicators that are sensitive to the values of the hyperparameters even with a very low number of experimental specimens: good results are obtained with only one specimen. In this work, we propose to train an artificial neural network in using in silico data generated by a multiscale computational model and a probabilistic mesoscopic model of the random material, for which the output layer corresponds to the values of the hyperparameters and the input layer corresponds to the values of the three experimentally measured indicators at mesoscale and the six algebraically independent components of the effective elasticity tensor at macroscale. Nevertheless, training an artificial neural network with such in silico data usually fails to solve the statistical inverse problem [4] and large discrepancies are observed between the values of the network targets and the outputs due to the stochastic nature of the mapping between the hyperparameters and the indicators. To circumvent this issue, the in silico data corresponding to the network input features are statistically processed by conditional probability using kernel smoothing techniques. Such a statistical processing of the network input data introduces uncertainties on the values that have to be presented at the input of the network since such processed input features are not directly observable by experimental measurements. A probabilistic model of the network inputs is then introduced to take into account such uncertainties. Consequently, for given experimental values obtained at both macroscale and mesoscale of the indicators, we finally propose to design an additional artificial neural network for which the outputs model the uncertainties on the optimal hyperparameters of the probabilistic mesoscopic model.
REFERENCES
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computational mechanics using an artificial neural network, Comput. Methods Appl. Mech.
Engrg. 373 (2021):113540