  Optimal control problem is typically formulated by Hamilton-Jacobi-Bellman
(HJB) equation for the value function and it is well-known that the value
function is the viscosity solution of the HJB equation. Once the HJB solution
is known, it can be used to construct the optimal control by taking the
minimizer of the Hamiltonian. In this work, instead of focusing on the value
function, we propose a new formulation for the components of the gradient of
the value function (value-gradient) as a decoupled system of partial
differential equations in the context of continuous-time deterministic
discounted optimal control problem. We develop an efficient iterative scheme
for this system of equations in parallel by utilizing the properties that they
share the same characteristic curves as the HJB equation for the value
function. This property allows us to generalize prior successive approximation
algorithms of policy iteration from the value function to the value-gradient
functions. To be compatible with the high dimensional control problem, we
generate multiple characteristic curves at each policy iteration from an
ensemble of initial states, and compute both the value function and its
gradient simultaneously on each curve as the labelled data. Then supervised
machine learning strategy is applied to minimize the weighted squared loss for
both the value function and its gradients. Experimental results of various
examples demonstrate that this new strategy of jointly learning both the value
function and its gradient not only significantly increases the accuracy but
also improves the efficiency and robustness, particularly with less amount of
characteristics data or fewer training steps.
