  The term "Normalizing Flows" is related to the task of constructing
invertible transport maps between probability measures by means of deep neural
networks. In this paper, we consider the problem of recovering the
$W_2$-optimal transport map $T$ between absolutely continuous measures
$\mu,\nu\in\mathcal{P}(\mathbb{R}^n)$ as the flow of a linear-control neural
ODE. We first show that, under suitable assumptions on $\mu,\nu$ and on the
controlled vector fields, the optimal transport map is contained in the
$C^0_c$-closure of the flows generated by the system. Assuming that discrete
approximations $\mu_N,\nu_N$ of the original measures $\mu,\nu$ are available,
we use a discrete optimal coupling $\gamma_N$ to define an optimal control
problem. With a $\Gamma$-convergence argument, we prove that its solutions
correspond to flows that approximate the optimal transport map $T$. Finally,
taking advantage of the Pontryagin Maximum Principle, we propose an iterative
numerical scheme for the resolution of the optimal control problem, resulting
in an algorithm for the practical computation of the approximated optimal
transport map.
