Deep 2FBSDEs for Systems with Control Multiplicative Noise

We present a deep recurrent neural network architecture to solve a class of stochastic optimal control problems described by fully nonlinear Hamilton Jacobi Bellman partial differential equations. Such PDEs arise when one considers stochastic dynamics characterized by uncertainties that are additive and control multiplicative. Stochastic models with the aforementioned characteristics have been used in computational neuroscience, biology and aerospace systems and provide a more accurate representation of actuation than models with additive uncertainty. Previous literature has established the inadequacy of the linear HJB theory and and instead rely on a non-linear Feynman-Kac lemma resulting in a second order forward-backward stochastic differential equations representation. However, the proposed solutions that use this representation suffer from compounding errors and computational complexity leading to lack of scalability. In this paper, we propose a deep learning based algorithm that leverages the second order Forward-Bacward SDE representation along with importance sampling and LSTM based recurrent neural networks to not only solve such Stochastic Optimal Control problems but also overcome the problems faced by previous approaches and scales well to high dimensional systems. The resulting control algorithm is tested on three non-linear systems to demonstrate feasibility and out-performance against previous methods.