  The FE$^2$ homogenization algorithm for multiscale modeling iterates between
the macroscale and the microscale (represented by a representative volume
element) till convergence is achieved at every increment of macroscale loading.
The information exchange between the two scales occurs at the gauss points of
the macroscale finite element discretization. The microscale problem is also
solved using finite elements on-the-fly thus rendering the algorithm
computationally expensive for complex microstructures. We invoke machine
learning to establish the input-output causality of the RVE boundary value
problem using a neural network framework. This renders the RVE as a blackbox
which gets the information from the macroscale as an input and gives
information back to the macroscale as output, thereby eliminating the need for
on-the-fly finite element solves at the RVE level. This framework has the
potential to significantly accelerate the FE$^2$ algorithm.
