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Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals withnon-smooth Riesz kernels show a rich structure as singular measures can becomeabsolutely continuous ones and conversely. In this paper we contribute to theunderstanding of such flows. We propose to approximate the backward scheme ofJordan, Kinderlehrer and Otto for computing such Wasserstein gradient flows aswell as a forward scheme for so-called Wasserstein steepest descent flows byneural networks (NNs). Since we cannot restrict ourselves to absolutelycontinuous measures, we have to deal with transport plans and velocity plansinstead of usual transport maps and velocity fields. Indeed, we approximate thedisintegration of both plans by generative NNs which are learned with respectto appropriate loss functions. In order to evaluate the quality of both neuralschemes, we benchmark them on the interaction energy. Here we provide analyticformulas for Wasserstein schemes starting at a Dirac measure and show theirconvergence as the time step size tends to zero. Finally, we illustrate ourneural MMD flows by numerical examples.

Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernels