A Machine Learning Approach for Multiscale Modeling of the Facet Capsular Ligament

We develop a new neural network architecture that strictly enforces constitutive constraints such as polyconvexity, frame-indifference, zero strain energy with zero deformations, and the symmetry of the stress and material stiffness. Additionally, we show that for this neural network, the accuracy is significantly improved by using a Sobolev minimization strategy that includes derivative terms. Using our network and Sobolev minimization, we obtain a NMSE of 0.15% for the energy, 0.815% averaged across the components of the stress, and 5.4% averaged across the components of the stiffness. This machine learned constitutive model was deployed in a finite element simulation of a facet capsular ligament. The displacement fields and stress-strain curves where compared to a multiscale simulation that required running on a GPU based supercomputer. At 70% strain, the model using the neural network had less than 10% relative error in the mean stress value.