This was part of

**Mathematical Advances in Mean Field Games**## On machine learning methods for mean field master equations

**
Mathieu Laurière, Google Brain ParisTuesday, December 14, 2021
**

**Abstract**: Mean field games have been introduced to study games with a very large number of players. They combine mean field approximation techniques borrowed from statistical physics to describe the population with optimal control techniques to describe the behavior of a representative player. Master equations are partial differential equations introduced by Pierre-Louis Lions to characterize Nash equilibria in such games. The unknown is a function taking the population distribution as an input. These methods typically rely on three ingredients: (1) representation of the distribution, (2) approximation of the function of interest, and (3) training algorithm. Since we cannot solve such equations for every possible distribution, the question of generalisation is unavoidable. In this respect, deep neural networks seem to be a promising tool. We will present approaches that are based on the full knowledge of the model, as well as approaches that are model-free.