Extensions of the Deep Galerkin Method

TL;DR

This paper extends the Deep Galerkin Method to solve complex PDEs in stochastic control and mean field games, including constrained Fokker-Planck and Hamilton-Jacobi-Bellman equations, using neural networks and novel sampling techniques.

Contribution

It introduces reparameterization and importance sampling for constrained PDEs and extends DGM to solve primal HJB equations with simultaneous neural network approximation.

Findings

Successfully handles positivity and normalization constraints in PDE solutions.

Solves primal HJB equations with integrated optimization using neural networks.

Demonstrates effectiveness on stochastic control and mean field game problems.

Abstract

We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018)} to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to unity, as is the case with Fokker-Planck equations. Our approach involves reparameterizing the solution as the exponential of a neural network appropriately normalized to ensure both requirements are satisfied. This then gives rise to nonlinear a partial integro-differential equation (PIDE) where the integral appearing in the equation is handled by a novel application of importance sampling. Secondly, we tackle a number of Hamilton-Jacobi-Bellman (HJB) equations that appear in stochastic optimal control problems. The key contribution is that these equations are approached in their unsimplified primal form which includes an optimization problem as part of the equation. We extend the DGM algorithm to solve for the value function and the optimal control \simultaneously by characterizing both as deep neural networks. Training the networks is performed by taking alternating stochastic gradient descent steps for the two functions, a technique inspired by the policy improvement algorithms (PIA).