  We propose a deep neural network (DNN) as a fast surrogate model for local
stress (and in principle strain) calculation in inhomogeneous non-linear
material systems. We show that the DNN predicts the local stresses with about
3.8% mean absolute percentage error (MAPE) for the case of heterogeneous
elastic media and a mechanical phase contrast of up to factor of 1.5 among
neighboring domains, while performing 103 times faster than spectral solvers.
The speed-up arises from the fact that after training, the DNN predicts the
stress without any iterations, as opposed to the iterative nature of standard
non-linear solvers. The new DNN surrogate model also proves suited for general
purposes: it is capable to reproduce the stress distribution in geometries
topologically far different from those used for training, implying effective
learning of scenarios described by the underlying partial differential
equations. Even in the case of elasto-plastic materials with up to 2 times
mechanical contrast in elastic stiffness and 4 times in yield stress among
adjacent regions, where conventional solvers typically require a substantial
number of iterations to arrive at stress predictions, the trained model
simulates the micromechanics with a MAPE of 6.4% in one single forward
evaluation step of the network, i.e. without any iterative calculations even
for the case of such a non-linear problem. The results reveal a completely new
and highly efficient approach to solve non-linear mechanical boundary value
problems and/or augment existing solution methods in corresponding hybrid
variants, with an acceleration up to factor of 8300 for heterogeneous
elastic-plastic materials in comparison to the currently fastest available
solvers.
