Entropic Regularization of NonGradient Systems

Abstract

The theory of Wasserstein gradient flows in the space of probability measures has made enormous progress over the last 20 years. It constitutes a unified and powerful framework in the study of dissipative partial differential equations (PDEs) providing the means to prove well-posedness, regularity, stability, and quantitative convergence to the equilibrium. The recently developed entropic regularization technique paves the way for fast and efficient numerical methods for solving these gradient flows. However, many PDEs of interest do not have a gradient flow structure and, a priori, the theory is not applicable. In this paper, we develop a time-discrete entropy regularized variational scheme for a general class of such nongradient PDEs. We prove the convergence of the scheme and illustrate the breadth of the proposed framework with concrete examples including the nonlinear kinetic Fokker--Planck (Kramers) equation and a nonlinear degenerate diffusion of Kolmogorov type. Numerical simulations are also provided.