Parameterized Wasserstein Gradient Flow
CoRR(2024)
摘要
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.
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