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.

翻译：“归一化流”这一术语涉及通过深度神经网络构建概率测度间可逆传输映射的任务。本文考虑将绝对连续测度μ,ν∈P(ℝⁿ)间的W₂-最优传输映射T恢复为线性控制神经ODE的流。我们首先证明，在μ、ν及受控向量场的适当假设下，最优传输映射包含于该系统生成流的C⁰_c闭包中。假设原始测度μ、ν的离散近似μ_N、ν_N已知，我们利用离散最优耦合γ_N定义最优控制问题。通过Γ-收敛论证，我们证明其解对应于逼近最优传输映射T的流。最后，借助庞特里亚金最大值原理，我们提出一种求解该最优控制问题的迭代数值方案，从而得到近似最优传输映射的实际计算算法。