  Many important multi-component crystalline solids undergo mechanochemical
spinodal decomposition: a phase transformation in which the compositional
redistribution is coupled with structural changes of the crystal, resulting in
dynamically evolving microstructures. The ability to rapidly compute the
macroscopic behavior based on these detailed microstructures is of paramount
importance for accelerating material discovery and design. Here, our focus is
on the macroscopic, nonlinear elastic response of materials harboring
microstructure. Because of the diversity of microstructural patterns that can
form, there is interest in taking a purely computational approach to predicting
their macroscopic response. However, the evaluation of macroscopic, nonlinear
elastic properties purely based on direct numerical simulations (DNS) is
computationally very expensive, and hence impractical for material design when
a large number of microstructures need to be tested. A further complexity of a
hierarchical nature arises if the elastic free energy and its variation with
strain is a small-scale fluctuation on the dominant trajectory of the total
free energy driven by microstructural dynamics. To address these challenges, we
present a data-driven approach, which combines advanced neural network (NN)
models with DNS to predict the homogenized, macroscopic, mechanical free energy
and stress fields arising in a family of multi-component crystalline solids
that develop microstructure. The hierarchical structure of the free energy's
evolution induces a multi-resolution character to the machine learning
paradigm: We construct knowledge-based neural networks (KBNNs) with either
pre-trained fully connected deep neural networks (DNNs), or pre-trained
convolutional neural networks (CNNs) that describe the dominant characteristic
of the data to fully represent the hierarchically evolving free energy.
