Neural Wasserstein Gradient Flows for Maximum Mean Discrepancies with Riesz Kernels

TL;DR

This paper develops neural network-based schemes to approximate Wasserstein gradient flows of MMD functionals with Riesz kernels, enabling analysis of singular measures and providing convergence and numerical illustrations.

Contribution

It introduces neural network methods for approximating Wasserstein gradient flows with Riesz kernels, including backward and forward schemes, and provides convergence analysis and benchmarks.

Findings

Neural schemes effectively approximate Wasserstein gradient flows.

Convergence of schemes shown as time step approaches zero.

Numerical examples demonstrate practical applicability.

Abstract

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.