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A modified Deep BSDE (backward differential equation) learning method withmeasurability loss, called Deep BSDE-ML method, is introduced in this paper tosolve a kind of linear decoupled forward-backward stochastic differentialequations (FBSDEs), which is encountered in the policy evaluation of learningthe optimal feedback policies of a class of stochastic control problems. Themeasurability loss is characterized via the measurability of BSDE's state atthe forward initial time, which differs from that related to terminal state ofthe known Deep BSDE method. Though the minima of the two loss functions areshown to be equal, this measurability loss is proved to be equal to theexpected mean squared error between the true diffusion term of BSDE and itsapproximation. This crucial observation extends the application of the DeepBSDE method -- approximating the gradients of the solution of a partialdifferential equation (PDE) instead of the solution itself. Simultaneously, alearning-based framework is introduced to search an optimal feedback control ofa deterministic nonlinear system. Specifically, by introducing Gaussianexploration noise, we are aiming to learn a robust optimal controller underthis stochastic case. This reformulation sacrifices the optimality to someextent, but as suggested in reinforcement learning (RL) exploration noise isessential to enable the model-free learning.

Deep BSDE-ML Learning and Its Application to Model-Free Optimal Control