  This paper is the first attempt to use geometric deep learning and Sobolev
training to incorporate non-Euclidean microstructural data such that
anisotropic hyperelastic material machine learning models can be trained in the
finite deformation range. While traditional hyperelasticity models often
incorporate homogenized measures of microstructural attributes, such as
porosity averaged orientation of constitutes, these measures cannot reflect the
topological structures of the attributes. We fill this knowledge gap by
introducing the concept of weighted graph as a new mean to store topological
information, such as the connectivity of anisotropic grains in assembles. Then,
by leveraging a graph convolutional deep neural network architecture in the
spectral domain, we introduce a mechanism to incorporate these non-Euclidean
weighted graph data directly as input for training and for predicting the
elastic responses of materials with complex microstructures. To ensure
smoothness and prevent non-convexity of the trained stored energy functional,
we introduce a Sobolev training technique for neural networks such that stress
measure is obtained implicitly from taking directional derivatives of the
trained energy functional. By optimizing the neural network to approximate both
the energy functional output and the stress measure, we introduce a training
procedure the improves efficiency and generalize the learned energy functional
for different microstructures. The trained hybrid neural network model is then
used to generate new stored energy functional for unseen microstructures in a
parametric study to predict the influence of elastic anisotropy on the
nucleation and propagation of fracture in the brittle regime.
