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.

翻译：具有非光滑Riesz核的最大均值差异（MMD）泛函的Wasserstein梯度流展现了丰富的结构，其中奇异测度可以变为绝对连续测度，反之亦然。本文旨在深化对此类流形的理解。我们提出用神经网络逼近Jordan、Kinderlehrer和Otto的向后格式以计算此类Wasserstein梯度流，同时提出一种用于计算所谓Wasserstein最速下降流的前向格式。由于我们无法将自身限制于绝对连续测度，必须处理传输计划和速度计划而非常规的传输映射和速度场。具体而言，我们通过生成式神经网络来近似这两种计划的分解，并针对适当的损失函数进行学习。为评估两种神经格式的质量，我们以相互作用能为基准进行测试。对此，我们提供了始于狄拉克测度的Wasserstein格式的解析公式，并证明了当时步步长趋于零时其收敛性。最后，通过数值算例展示了我们的神经MMD流。