  The $L^2$ gradient flow of the Ginzburg-Landau free energy functional leads
to the Allen Cahn equation that is widely used for modeling phase separation.
Machine learning methods for solving the Allen-Cahn equation in its strong form
suffer from inaccuracies in collocation techniques, errors in computing
higher-order spatial derivatives through automatic differentiation, and the
large system size required by the space-time approach. To overcome these
limitations, we propose a separable neural network-based approximation of the
phase field in a minimizing movement scheme to solve the aforementioned
gradient flow problem. At each time step, the separable neural network is used
to approximate the phase field in space through a low-rank tensor decomposition
thereby accelerating the derivative calculations. The minimizing movement
scheme naturally allows for the use of Gauss quadrature technique to compute
the functional. A `$tanh$' transformation is applied on the neural
network-predicted phase field to strictly bounds the solutions within the
values of the two phases. For this transformation, a theoretical guarantee for
energy stability of the minimizing movement scheme is established. Our results
suggest that bounding the solution through this transformation is the key to
effectively model sharp interfaces through separable neural network. The
proposed method outperforms the state-of-the-art machine learning methods for
phase separation problems and is an order of magnitude faster than the finite
element method.
