  We present a new approach for predictive modeling and its uncertainty
quantification for mechanical systems, where coarse-grained models such as
constitutive relations are derived directly from observation data. We explore
the use of a neural network to represent the unknown constitutive relations,
compare the neural networks with piecewise linear functions, radial basis
functions, and radial basis function networks, and show that the neural network
outperforms the others in certain cases. We analyze the approximation error of
the neural networks using a scaling argument. The training and predicting
processes in our framework combine the finite element method, automatic
differentiation, and neural networks (or other function approximators). Our
framework also allows uncertainty quantification in the form of confidence
intervals. Numerical examples on a multiscale fiber-reinforced plate problem
and a nonlinear rubbery membrane problem from solid mechanics demonstrate the
effectiveness of our framework.
