A novel control method for solving high-dimensional Hamiltonian systems through deep neural networks
This work addresses computational challenges in high-dimensional systems for researchers in applied mathematics and machine learning, though it appears incremental as it builds on existing FBSDE methods.
The paper tackles solving high-dimensional stochastic Hamiltonian systems by proposing a novel control method using deep neural networks, resulting in algorithms that converge faster and more stably than prior methods, requiring fewer training steps.
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.