Parameterized Wasserstein Gradient Flow

TL;DR

This paper introduces a scalable numerical method for Wasserstein gradient flows using neural networks to parameterize push-forward maps, enabling efficient high-dimensional solutions without spatial discretization.

Contribution

It proposes the parameterized Wasserstein gradient flow (PWGF) framework that leverages neural networks for efficient high-dimensional WGF solutions, avoiding traditional discretization and nonconvex optimization.

Findings

Effective approximation of WGFs for various energy functionals

High computational efficiency demonstrated in experiments

Accurate solutions verified across multiple examples

Abstract

We develop a fast and scalable numerical approach to solve Wasserstein gradient flows (WGFs), particularly suitable for high-dimensional cases. Our approach is to use general reduced-order models, like deep neural networks, to parameterize the push-forward maps such that they can push a simple reference density to the one solving the given WGF. The new dynamical system is called parameterized WGF (PWGF), and it is defined on the finite-dimensional parameter space equipped with a pullback Wasserstein metric. Our numerical scheme can approximate the solutions of WGFs for general energy functionals effectively, without requiring spatial discretization or nonconvex optimization procedures, thus avoiding some limitations of classical numerical methods and more recent deep-learning-based approaches. A comprehensive analysis of the approximation errors measured by Wasserstein distance is also provided in this work. Numerical experiments show promising computational efficiency and verified accuracy on various WGF examples using our approach.