  We propose a neural network approach for solving high-dimensional optimal
control problems arising in real-time applications. Our approach yields
controls in a feedback form and can therefore handle uncertainties such as
perturbations to the system's state. We accomplish this by fusing the
Pontryagin Maximum Principle (PMP) and Hamilton-Jacobi-Bellman (HJB) approaches
and parameterizing the value function with a neural network. We train our
neural network model using the objective function of the control problem and
penalty terms that enforce the HJB equations. Therefore, our training algorithm
does not involve data generated by another algorithm. By training on a
distribution of initial states, we ensure the controls' optimality on a large
portion of the state-space. Our grid-free approach scales efficiently to
dimensions where grids become impractical or infeasible. We demonstrate the
effectiveness of our approach on several multi-agent collision-avoidance
problems in up to 150 dimensions. Furthermore, we empirically observe that the
number of parameters in our approach scales linearly with the dimension of the
control problem, thereby mitigating the curse of dimensionality.
