Deep Galerkin Method for Mean Field Control Problem

TL;DR

This paper introduces the Deep Galerkin Method (DGM) to solve mean-field control problems, demonstrating neural network-based approximation of value functions and distributions with proven convergence.

Contribution

It applies DGM to mean-field control problems, providing a neural network approach with convergence guarantees for solving Hamilton-Jacobi-Bellman equations.

Findings

Neural network approximations converge to analytical solutions.

DGM effectively estimates value functions and distributions.

The method handles finite-state, continuous-time, finite-horizon settings.

Abstract

We consider an optimal control problem where the average welfare of weakly interacting agents is of interest. We examine the mean-field control problem as the fluid approximation of the N-agent control problem with the setup of finite-state space, continuous-time, and finite-horizon. The value function of the mean-field control problem is characterized as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation in the simplex. We apply the DGM to estimate the value function and the evolution of the distribution. We also prove the numerical solution approximated by a neural network converges to the analytical solution.