  Artificial Neural Networks (NNWs) are appealing functions to substitute high
dimensional and non-linear history-dependent problems in computational
mechanics since they offer the possibility to drastically reduce the
computational time. This feature has recently been exploited in the context of
multi-scale simulations, in which the NNWs serve as surrogate model of
micro-scale finite element resolutions. Nevertheless, in the literature, mainly
the macro-stress-macro-strain response of the meso-scale boundary value problem
was considered and the micro-structure information could not be recovered in a
so-called localization step. In this work, we develop Recurrent Neural Networks
(RNNs) as surrogates of the RVE response while being able to recover the
evolution of the local micro-structure state variables for complex loading
scenarios. The main difficulty is the high dimensionality of the RNNs output
which consists in the internal state variable distribution in the
micro-structure. We thus propose and compare several surrogate models based on
a dimensionality reduction: i) direct RNN modeling with implicit NNW
dimensionality reduction, ii) RNN with PCA dimensionality reduction, and iii)
RNN with PCA dimensionality reduction and dimensionality break down, i.e. the
use of several RNNs instead of a single one. Besides, we optimize the
sequential training strategy of the latter surrogate for GPU usage in order to
speed up the process. Finally, through RNN modeling of the principal components
coefficients, the connection between the physical state variables and the
hidden variables of the RNN is revealed, and exploited in order to select the
hyper-parameters of the RNN-based surrogate models in their design stage.
