Entropy dissipation of Fokker-Planck equations on graphs

Abstract

We study the nonlinear Fokker-Planck equation on graphs, which is the gradient flow in the space of probability measures supported on the nodes with respect to the discrete Wasserstein metric. The energy functional driving the gradient flow consists of a Boltzmann entropy, a linear potential and a quadratic interaction energy. We show that the solution converges to the Gibbs measures exponentially fast with a rate that can be given analytically. The continuous analog of this asymptotic rate is related to the Yano's formula.

Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula

“…We further discretize the above variational problem following the same discretization as in optimal transport on graphs [9,10,11,16,23]. We then apply a multi-level block stochastic gradient descent method to optimize the discretized problem.…”

Natural gradient via optimal transport

“…Equation (14) is the generalization of Wasserstein gradient flow in probability simplex to the one on parameter space. If p is an identity map with the parameter space Θ equal to the entire probability simplex, then (14) iṡ…”

“…which is the Wasserstein gradient flow on Ω. From now on, we call (14) the Wasserstein gradient flow on parameter space.…”

“…The L 2 -Wasserstein metric on discrete states was introduced in [12,26,29]. Following the settings from [13,14,17,22], the probability simplex forms the Riemannian manifold called Wasserstein probability manifold. The Wasserstein metric on the probability simplex can be pulled back to the parameter space of a probability model.…”

“…We note that for the static formulation, there is no Riemannian metric tensor for the discrete probability simplex. See [14,26] for a detailed discussion.…”