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.

In this paper we propose a Deep Learning architecture to approximate diffeomorphisms diffeotopic to the identity. We consider a control system of the form $\dot x = \sum_{i=1}^lF_i(x)u_i$, with linear dependence in the controls, and we use the corresponding flow to approximate the action of a diffeomorphism on a compact ensemble of points. Despite the simplicity of the control system, it has been recently shown that a Universal Approximation Property holds. The problem of minimizing the sum of the training error and of a regularizing term induces a gradient flow in the space of admissible controls. A possible training procedure for the discrete-time neural network consists in projecting the gradient flow onto a finite-dimensional subspace of the admissible controls. An alternative approach relies on an iterative method based on Pontryagin Maximum Principle for the numerical resolution of Optimal Control problems. Here the maximization of the Hamiltonian can be carried out with an extremely low computational effort, owing to the linear dependence of the system in the control variables.

Optimal control problems can be solved by first applying the Pontryagin maximum principle, followed by computing a solution of the corresponding unconstrained Hamiltonian dynamical system. In this paper, and to achieve a balance between robustness and efficiency, we learn a reduced Hamiltonian of the unconstrained Hamiltonian. This reduced Hamiltonian is learned by going backward in time and by minimizing the loss function resulting from application of the Pontryagin maximum principle's conditions. The robustness of our learning process is then further improved by progressively learning a posterior distribution of reduced Hamiltonians. This leads to a more efficient sampling of the generalized coordinates (position, velocity) of our phase space. Our solution framework applies to not only optimal control problems with finite-dimensional phase (state) spaces but also the infinite-dimensional case.