Learning Deep Stochastic Optimal Control Policies using Forward-Backward SDEs

In this paper we propose a new methodology for decision-making under uncertainty using recent advancements in the areas of nonlinear stochastic optimal control theory, applied mathematics, and machine learning. Grounded on the fundamental relation between certain nonlinear partial differential equations and forward-backward stochastic differential equations, we develop a control framework that is scalable and applicable to general classes of stochastic systems and decision-making problem formulations in robotics and autonomy. The proposed deep neural network architectures for stochastic control consist of recurrent and fully connected layers. The performance and scalability of the aforementioned algorithm are investigated in three non-linear systems in simulation with and without control constraints. We conclude with a discussion on future directions and their implications to robotics

Deep Forward-Backward SDEs for Min-max Control

This paper presents a novel approach to numerically solve stochastic differential games for nonlinear systems. The proposed approach relies on the nonlinear Feynman-Kac theorem that establishes a connection between parabolic deterministic partial differential equations and forward-backward stochastic differential equations. Using this theorem the Hamilton-Jacobi-Isaacs partial differential equation associated with differential games is represented by a system of forward-backward stochastic differential equations. Numerical solution of the aforementioned system of stochastic differential equations is performed using importance sampling and a Long-Short Term Memory recurrent neural network, which is trained in an offline fashion. The resulting algorithm is tested on two example systems in simulation and compared against the standard risk neutral stochastic optimal control formulations