  Optimal Mass Transport (OMT) is a well studied problem with a variety of
applications in a diverse set of fields ranging from Physics to Computer Vision
and in particular Statistics and Data Science. Since the original formulation
of Monge in 1781 significant theoretical progress been made on the existence,
uniqueness and properties of the optimal transport maps. The actual numerical
computation of the transport maps, particularly in high dimensions, remains a
challenging problem. By Brenier's theorem, the continuous OMT problem can be
reduced to that of solving a non-linear PDE of Monge-Ampere type whose solution
is a convex function. In this paper, building on recent developments of input
convex neural networks and physics informed neural networks for solving PDE's,
we propose a Deep Learning approach to solve the continuous OMT problem.
  To demonstrate the versatility of our framework we focus on the ubiquitous
density estimation and generative modeling tasks in statistics and machine
learning. Finally as an example we show how our framework can be incorporated
with an autoencoder to estimate an effective probabilistic generative model.
