  We study a simplification of GAN training: the problem of transporting
particles from a source to a target distribution. Starting from the Sobolev GAN
critic, part of the gradient regularized GAN family, we show a strong relation
with Optimal Transport (OT). Specifically with the less popular dynamic
formulation of OT that finds a path of distributions from source to target
minimizing a ``kinetic energy''. We introduce Sobolev descent that constructs
similar paths by following gradient flows of a critic function in a kernel
space or parametrized by a neural network. In the kernel version, we show
convergence to the target distribution in the MMD sense. We show in theory and
experiments that regularization has an important role in favoring smooth
transitions between distributions, avoiding large gradients from the critic.
This analysis in a simplified particle setting provides insight in paths to
equilibrium in GANs.
