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.

通过深度神经网络构建概率测度之间的可逆传输映射，本文考虑了恢复绝对连续测度之间$W_2$-最优传输映射作为线性控制神经ODE流的问题，并提出了一种基于离散最优耦合的离散控制问题，以近似求解最优传输映射。最后，通过Pontryagin最大值原理，提出了一种迭代数值方法用于实际计算近似最优传输映射的算法。

线性控制神经ODE 中将正规化流视为最优传输映射的近似