  Many optimal control problems are formulated as two point boundary value
problems (TPBVPs) with conditions of optimality derived from the
Hamilton-Jacobi-Bellman (HJB) equations. In most cases, it is challenging to
solve HJBs due to the difficulty of guessing the adjoint variables. This paper
proposes two learning-based approaches to find the initial guess of adjoint
variables in real-time, which can be applied to solve general TPBVPs. For cases
with database of solutions and corresponding adjoint variables of a TPBVP under
varying boundary conditions, a supervised learning method is applied to learn
the HJB solutions off-line. After obtaining a trained neural network from
supervised learning, we are able to find proper initial adjoint variables for
given boundary conditions in real-time. However, when validated solutions of
TPBVPs are not available, the reinforcement learning method is applied to solve
HJB by constructing a neural network, defining a reward function, and setting
appropriate super parameters. The reinforcement learning based HJB method can
learn how to find accurate adjoint variables via an updating neural network.
Finally, both learning approaches are implemented in classical optimal control
problems to verify the effectiveness of the learning based HJB methods.
