{Euclidean, metric, and Wasserstein} gradient flows: an overview

Abstract

This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan-Kinderlehrer-Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

Cardinality estimation of support of the global minimizer for the interaction energy with mildly repulsive potentials

“…In this paper, we consider the following interaction energy It is known that the equation (1.2) has the structure of a gradient flow of the interaction energy (1.1) [1,9,21].…”

Proximal Recursion for Solving the Fokker-Planck Equation

“…The resulting variational recursion (3) has since been known as the Jordan-Kinderlehrer-Otto (JKO) scheme [18], and we will refer the FPK operator with such assumptions on f and g to be in "JKO canonical form". Similar gradient descent schemes have been derived for many other PDEs; see e.g., [19] for a recent survey.…”

“…Specifically, the solution of (14) can be recovered from the following proximal recursion of the form (3): (14)) as h ↓ 0. Next, we develop a framework to numerically solve (19).…”

Hopfield Neural Network Flow: A Geometric Viewpoint

“…In particular, k → ρ(x, t = kh) in strong L 1 (M) sense as h ↓ 0. That a discrete time-stepping procedure like (54) can approximate the solution of FPK PDE, was first proved in [41] for G ≡ I, and has since become a topic of burgeoning research; see e.g., [39], [42], [43]. The RHS in (54) is an infinite dimensional version of the Moreau-Yosida proximal operator [44]- [46], denoted as prox W G hF (·), i.e., (54) can be written succinctly as…”

“…Furthermore, the FPK PDE (43) reduces to the Liouville PDE ∂ρ ∂t = ∇ · (G(x)) −1 ρ∇f , whose stationary PDF becomes an weighted sum of Diracs located at the stationary points of f . The stationary solution of (56) converges to this stationary PDF.…”