Recurrent Neural Networks (RNNs) with dimension reduction and break down in the context of high dimensional localization step in multi-scale analysis

Artificial Neural Networks (NNWs) are appealing tools to serve as surrogate model
of high dimensional and non-linear history-dependent problems in computational mechanics, in particular in multi-scale methods. Indeed, as shown in [1] once properly
trained, they replace expensive micro-scale finite element resolutions allowing reducing
the computational time by more than 4 orders of magnitude.
Nevertheless, such a paradigm is essentially used as a mapping between the macro-stress
and macro-strain tensors of the micro-scale boundary value response and the micro-structure information could not be recovered in a so-called localization step.
In this talk, we present a Recurrent Neural Networks (RNNs)-based surrogate of the
micro-scale boundary value problem, while being able to recover the evolution of the
local micro-structure state variables for complex loading scenarios [2]. In order to
address the curse of dimensionality arising because of the large amount of internal
state variables in the micro-structure, we enrich the RNN with PCA dimension reduction
and dimension break down, i.e. the use of several RNNs instead of a single one.
Besides, the sequential training strategy is optimized in order to allow for GPU usage.
[1] L. Wu, V.-D. Nguyen, N.G. Kilingar and L. Noels, Computer Methods in Applied
Mechanics and Engineering, 360:113234, 2020.
[2] L. Wu, L. Noels, Recurrent Neural Networks (RNNs) in computational mechanics
for high dimensional problems: RNNs structure design through dimension reduction
and break down; application to multi-scale localization step. Submitted.