Interacting particle- or agent-based systems are ubiquitous in science. They arise in an extremely wide variety of applications including materials science, biology, economics and social sciences. Several mathematical models exist to account for the evolution of such systems at different scales: stochastic differential equations and optimal transport problems (microscopic models), kinetic equations (mesoscopic models) or mean-field models such as mean-field games systems (macroscopic models). However, all of them suffer from severe limitations when it comes to the simulation of high-dimensional problems, the high-dimensionality character coming either from the large number of particles or agents in the system, the huge quantity of parameters entering the model or the high amount of features of each agent or particle.
The objective of this project is to provide a new mathematical framework for the development and analysis of efficient and accurate numerical methods for the simulation of high-dimensional particle or agent systems, stemming from applications in materials science and stochastic game theory.
The main challenges which will be addressed in this project are:
(1) sparse optimization problems for multi-marginal optimal transport problems, using moment constraints;
(2) efficient approximation of parametric stochastic differential equations, by means of reduced-order modeling approaches;
(3) numerical resolution of high-dimensional partial differential equations, with provably convergent deep learning methods.
The potential impacts of the project are huge: making possible such extreme-scale simulations will enable to gain precious insights on the predictive power of agent- or particle-based models, with applications in various fields, such as quantum chemistry, molecular dynamics, neutron transport, crowd motion or urban traffic.
Madame Virginie Ehrlacher (Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
CERMICS Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
Help of the ANR 113,000 euros
Beginning and duration of the scientific project: May 2022 - 24 Months