  We develop a new neural network architecture that strictly enforces
constitutive constraints such as polyconvexity, frame-indifference, and the
symmetry of the stress and material stiffness. Additionally, we show that the
accuracy of the stress and material stiffness predictions is significantly
improved for this neural network by using a Sobolev minimization strategy that
includes derivative terms. Using our neural network, we model the constitutive
behavior of fibrous-type discrete network material. With Sobolev minimization,
we obtain a normalized mean square error of 0.15% for the strain energy
density, 0.815% averaged across the components of the stress, and 5.4% averaged
across the components of the stiffness tensor. This machine-learned
constitutive model was deployed in a finite element simulation of a facet
capsular ligament. The displacement fields and stress-strain curves were
compared to a multiscale simulation that required running on a GPU-based
supercomputer. The new approach maintained upward of 85% accuracy in stress up
to 70% strain while reducing the computation cost by orders of magnitude.
