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We present a deep recurrent neural network architecture to solve a class ofstochastic optimal control problems described by fully nonlinear HamiltonJacobi Bellmanpartial differential equations. Such PDEs arise when oneconsiders stochastic dynamics characterized by uncertainties that are additiveand control multiplicative. Stochastic models with the aforementionedcharacteristics have been used in computational neuroscience, biology, financeand aerospace systems and provide a more accurate representation of actuationthan models with additive uncertainty. Previous literature has established theinadequacy of the linear HJB theory and instead rely on a non-linearFeynman-Kac lemma resulting in a second order forward-backward stochasticdifferential equations representation. However, the proposed solutions that usethis representation suffer from compounding errors and computational complexityleading to lack of scalability. In this paper, we propose a deep learning basedalgorithm that leverages the second order Forward-Backward SDE representationand LSTM based recurrent neural networks to not only solve such StochasticOptimal Control problems but also overcome the problems faced by previousapproaches and scales well to high dimensional systems. The resulting controlalgorithm is tested on non-linear systems in robotics and biomechanics todemonstrate feasibility and out-performance against previous methods.

Deep 2FBSDEs For Systems With Control Multiplicative Noise