  We study the long time behavior of an underdamped mean-field Langevin (MFL)
equation, and provide a general convergence as well as an exponential
convergence rate result under different conditions. The results on the MFL
equation can be applied to study the convergence of the Hamiltonian gradient
descent algorithm for the overparametrized optimization. We then provide a
numerical example of the algorithm to train a generative adversarial networks
(GAN).
