Convergence of the Deep Galerkin Method for Mean Field Control Problems

TL;DR

This paper proves the convergence of a deep learning method for solving high-dimensional Hamilton-Jacobi-Bellman equations in mean field control problems, supported by theoretical analysis and numerical experiments.

Contribution

It establishes the first convergence results for the deep Galerkin method applied to high-dimensional HJB equations in mean field control contexts.

Findings

DGM loss functional can be made arbitrarily small with sufficient regularity.

Neural network approximators converge uniformly to the true value function when the loss tends to zero.

Numerical experiments show DGM effectively generalizes to high-dimensional problems.

Abstract

We establish the convergence of the deep Galerkin method (DGM), a deep learning-based scheme for solving high-dimensional nonlinear PDEs, for Hamilton-Jacobi-Bellman (HJB) equations that arise from the study of mean field control problems (MFCPs). Based on a recent characterization of the value function of the MFCP as the unique viscosity solution of an HJB equation on the simplex, we establish both an existence and convergence result for the DGM. First, we show that the loss functional of the DGM can be made arbitrarily small given that the value function of the MFCP possesses sufficient regularity. Then, we show that if the loss functional of the DGM converges to zero, the corresponding neural network approximators must converge uniformly to the true value function on the simplex. We also provide numerical experiments demonstrating the DGM's ability to generalize to high-dimensional HJB equations.