Mean-field optimal control for biological pattern formation

TL;DR

This paper introduces a mean-field optimal control framework utilizing Wasserstein distance for identifying parameters in biological pattern formation, with proven convergence and demonstrated numerical effectiveness.

Contribution

It develops a novel mean-field optimal control method with gradient descent for biological pattern parameter identification, including convergence analysis and particle-level discretization.

Findings

Effective parameter identification for biological patterns.

Convergence rate established for particle discretization.

Numerical results confirm approach feasibility.

Abstract

We propose a mean-field optimal control problem for the parameter identification of a given pattern. The cost functional is based on the Wasserstein distance between the probability measures of the modeled and the desired patterns. The first-order optimality conditions corresponding to the optimal control problem are derived using a Lagrangian approach on the mean-field level. Based on these conditions we propose a gradient descent method to identify relevant parameters such as angle of rotation and force scaling which may be spatially inhomogeneous. We discretize the first-order optimality conditions in order to employ the algorithm on the particle level. Moreover, we prove a rate for the convergence of the controls as the number of particles used for the discretization tends to infinity. Numerical results for the spatially homogeneous case demonstrate the feasibility of the approach.