Simulating numerically the Krusell-Smith model with neural networks

TL;DR

This paper introduces a neural network-based numerical method to approximate solutions of the high-dimensional master equation in the Krusell-Smith mean field game model, leveraging low-dimensional structures for efficiency.

Contribution

It proposes a novel semi-Lagrangian neural network approach to solve the master equation and identify low-dimensional variables in the Krusell-Smith model.

Findings

Successfully approximates the master equation solutions.

Identifies low-dimensional variables that capture essential information.

Demonstrates efficiency of the neural network-based method.

Abstract

The celebrated Krusel-Smith growth model is an important example of a Mean Field Game with a common noise. The Mean Field Game is encoded in the master equation, a partial differential equation satisfied by the value of the game which depends on the whole distribution of states. The latter equation is therefore posed in an infinite dimensional space. This makes the numerical simulations quite challenging. However, Krusell and Smith conjectured that the value function of the game mostly depends on the state distribution through low dimensional quantities. In this paper, we wish to propose a numerical method for approximating the solutions of the master equation arising in Krusell-Smith model, and for adaptively identifying low-dimensional variables which retain an important part of the information. This new numerical framework is based on a semi-Lagrangian method and uses neural networks as an important ingredient.