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In this paper, we mainly focus on solving high-dimensional stochasticHamiltonian systems with boundary condition, which is essentially a ForwardBackward Stochastic Differential Equation (FBSDE in short), and propose a novelmethod from the view of the stochastic control. In order to obtain theapproximated solution of the Hamiltonian system, we first introduce acorresponding stochastic optimal control problem such that the extendedHamiltonian system of the control problem is exactly what we need to solve,then we develop two different algorithms suitable for different cases of thecontrol problem and approximate the stochastic control via deep neuralnetworks. From the numerical results, comparing with the Deep FBSDE methoddeveloped previously from the view of solving FBSDEs, the novel algorithmsconverge faster, which means that they require fewer training steps, anddemonstrate more stable convergences for different Hamiltonian systems.

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