Parameterized Wasserstein gradient flow

Yijie Jin

doi:10.1016/j.jcp.2024.113660

Parameterized Wasserstein gradient flow

2025-03-01

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AI

DOI：10.1016/j.jcp.2024.113660

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We develop a fast and scalable numerical approach to solve Wasserstein gradient flows (WGFs), which is 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 a variety of WGF examples using our approach.

Keyword:

Wasserstein gradient flow

Fokker-Planck equation

Porous medium equation

Deep neural networks

Numerical analysis

Parameterized Wasserstein gradient flow

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