  The macroscopic properties of materials that we observe and exploit in
engineering application result from complex interactions between physics at
multiple length and time scales: electronic, atomistic, defects, domains etc.
Multiscale modeling seeks to understand these interactions by exploiting the
inherent hierarchy where the behavior at a coarser scale regulates and averages
the behavior at a finer scale. This requires the repeated solution of
computationally expensive finer-scale models, and often a priori knowledge of
those aspects of the finer-scale behavior that affect the coarser scale (order
parameters, state variables, descriptors, etc.). We address this challenge in a
two-scale setting where we learn the fine-scale behavior from off-line
calculations and then use the learnt behavior directly in coarse scale
calculations. The approach draws from recent successes of deep neural networks,
in combination with ideas from model reduction. The approach builds on the
recent success of deep neural networks by combining their approximation power
in high dimensions with ideas from model reduction. It results in a neural
network approximation that has high fidelity, is computationally inexpensive,
is independent of the need for a priori knowledge, and can be used directly in
the coarse scale calculations. We demonstrate the approach on problems
involving the impact of magnesium, a promising light-weight structural and
protective material.
