A novel control method for solving high-dimensional Hamiltonian systems through deep neural networks

TL;DR

This paper introduces a new deep neural network-based control method for efficiently solving high-dimensional stochastic Hamiltonian systems, outperforming previous approaches in convergence speed and stability.

Contribution

The paper presents a novel control-based approach with two algorithms for high-dimensional Hamiltonian systems, improving convergence speed and stability over existing methods.

Findings

Faster convergence compared to Deep FBSDE method

More stable convergence across different systems

Requires fewer training steps

Abstract

In this paper, we mainly focus on solving high-dimensional stochastic Hamiltonian systems with boundary condition, which is essentially a Forward Backward Stochastic Differential Equation (FBSDE in short), and propose a novel method from the view of the stochastic control. In order to obtain the approximated solution of the Hamiltonian system, we first introduce a corresponding stochastic optimal control problem such that the extended Hamiltonian system of the control problem is exactly what we need to solve, then we develop two different algorithms suitable for different cases of the control problem and approximate the stochastic control via deep neural networks. From the numerical results, comparing with the Deep FBSDE method developed previously from the view of solving FBSDEs, the novel algorithms converge faster, which means that they require fewer training steps, and demonstrate more stable convergences for different Hamiltonian systems.