  Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals with
non-smooth Riesz kernels show a rich structure as singular measures can become
absolutely continuous ones and conversely. In this paper we contribute to the
understanding of such flows. We propose to approximate the backward scheme of
Jordan, Kinderlehrer and Otto for computing such Wasserstein gradient flows as
well as a forward scheme for so-called Wasserstein steepest descent flows by
neural networks (NNs). Since we cannot restrict ourselves to absolutely
continuous measures, we have to deal with transport plans and velocity plans
instead of usual transport maps and velocity fields. Indeed, we approximate the
disintegration of both plans by generative NNs which are learned with respect
to appropriate loss functions. In order to evaluate the quality of both neural
schemes, we benchmark them on the interaction energy. Here we provide analytic
formulas for Wasserstein schemes starting at a Dirac measure and show their
convergence as the time step size tends to zero. Finally, we illustrate our
neural MMD flows by numerical examples.
