  In this work, we develop Gaussian process regression (GPR) models of
hyperelastic material behavior. First, we consider the direct approach of
modeling the components of the Cauchy stress tensor as a function of the
components of the Finger stretch tensor in a Gaussian process. We then consider
an improvement on this approach that embeds rotational invariance of the
stress-stretch constitutive relation in the GPR representation. This approach
requires fewer training examples and achieves higher accuracy while maintaining
invariance to rotations exactly. Finally, we consider an approach that recovers
the strain-energy density function and derives the stress tensor from this
potential. Although the error of this model for predicting the stress tensor is
higher, the strain-energy density is recovered with high accuracy from limited
training data. The approaches presented here are examples of physics-informed
machine learning. They go beyond purely data-driven approaches by embedding the
physical system constraints directly into the Gaussian process representation
of materials models.
