  Fully resolving dynamics of materials with rapidly-varying features involves
expensive fine-scale computations which need to be conducted on macroscopic
scales. The theory of homogenization provides an approach to derive effective
macroscopic equations which eliminates the small scales by exploiting scale
separation. An accurate homogenized model avoids the computationally-expensive
task of numerically solving the underlying balance laws at a fine scale,
thereby rendering a numerical solution of the balance laws more computationally
tractable.
  In complex settings, homogenization only defines the constitutive model
implicitly, and machine learning can be used to learn the constitutive model
explicitly from localized fine-scale simulations. In the case of
one-dimensional viscoelasticity, the linearity of the model allows for a
complete analysis. We establish that the homogenized constitutive model may be
approximated by a recurrent neural network (RNN) that captures the memory. The
memory is encapsulated in the evolution of an appropriate finite set of
internal variables, discovered through the learning process and dependent on
the history of the strain. Simulations are presented which validate the theory.
Guidance for the learning of more complex models, such as arise in plasticity,
by similar techniques, is given.
