Aggregation-Diffusion Equations for Collective Behaviour in the Sciences

TL;DR

This survey reviews the current state of aggregation-diffusion equations, highlighting their analysis, numerical methods, and diverse applications in sciences like biology, optimization, and machine learning.

Contribution

It provides a comprehensive overview of recent advances in the analysis, numerical simulation, and applications of aggregation-diffusion equations, including biological and computational contexts.

Findings

Aggregation-diffusion models effectively describe cell sorting in tissue growth.

Numerical methods enable accurate simulation of complex aggregation behaviors.

Applications extend to optimization and machine learning tasks.

Abstract

This is a survey article based on the content of the plenary lecture given by Jos\'e A. Carrillo at the ICIAM23 conference in Tokyo. It is devoted to produce a snapshot of the state of the art in the analysis, numerical analysis, simulation, and applications of the vast area of aggregation-diffusion equations. We also discuss the implications in mathematical biology explaining cell sorting in tissue growth as an example of this modelling framework. This modelling strategy is quite successful in other timely applications such as global optimisation, parameter estimation and machine learning.