  This paper approaches the unsupervised learning problem by minimizing the
second-order Wasserstein loss (the $W_2$ loss) through a distribution-dependent
ordinary differential equation (ODE), whose dynamics involves the Kantorovich
potential associated with the true data distribution and a current estimate of
it. A main result shows that the time-marginal laws of the ODE form a gradient
flow for the $W_2$ loss, which converges exponentially to the true data
distribution. An Euler scheme for the ODE is proposed and it is shown to
recover the gradient flow for the $W_2$ loss in the limit. An algorithm is
designed by following the scheme and applying persistent training, which
naturally fits our gradient-flow approach. In both low- and high-dimensional
experiments, our algorithm outperforms Wasserstein generative adversarial
networks by increasing the level of persistent training appropriately.
