A Dynamical Approximation Scheme on the Stiefel manifold for Wasserstein Gradient Flows

Abstract

We propose a meshless Lagrangian dynamical method for approximating Wasserstein gradient flows (WGFs). The evolving measure is represented as the pushforward of the initial measure $μ_0$ through a transport map in the weighted Hilbert space $L^2_{μ_0}$. We approximate this map in time-dependent linear subspaces of $L^2_{μ_0}$, whose orthonormal frames are evolved by a Dirac--Frenkel dynamical principle on a Stiefel manifold constrained to a finite-dimensional background space, adaptively constructed via local approximations of the WGF velocity field. We prove that the resulting transport map induces an absolutely continuous curve of probability measures in Wasserstein space, whose velocity is obtained by projecting the exact WGF velocity onto the background space, and we show that the approximation preserves the energy dissipation structure up to the projection error of the velocity. Moreover, for geodesically convex energies, we derive an a posteriori estimate controlling such projection error through the adaptive construction of the background space, yielding as well a bound on the approximation error of the pushforward measure in the Wasserstein metric. Numerical experiments on linear and nonlinear Fokker--Planck equations, porous-medium diffusion, and interaction energies demonstrate the accuracy of the method, its energy-dissipation properties, and the advantages of the adaptive construction.