Computational Mean-field information dynamics associated with Reaction diffusion equations

TL;DR

This paper develops a mean-field information dynamics framework for reaction-diffusion equations, utilizing optimal transport and control methods, and introduces efficient numerical schemes validated by numerical examples.

Contribution

It formulates a novel mean-field control approach for reaction-diffusion equations using gradient flows and optimal transport, with an innovative numerical algorithm.

Findings

Effective computation of mean-field control problems

New variational scheme for implicit time schemes

Numerical demonstrations of the proposed method

Abstract

We formulate and compute a class of mean-field information dynamics for reaction-diffusion equations. Given a class of nonlinear reaction-diffusion equations and entropy type Lyapunov functionals, we study their gradient flows formulations with generalized optimal transport metrics and mean-field control problems. We apply the primal-dual hybrid gradient algorithm to compute the mean-field control problems with potential energies. A byproduct of the proposed method contains a new and efficient variational scheme for solving implicit in time schemes of mean-field control problems. Several numerical examples demonstrate the solutions of mean-field control problems.