  Computing optimal feedback controls for nonlinear systems generally requires
solving Hamilton-Jacobi-Bellman (HJB) equations, which are notoriously
difficult when the state dimension is large. Existing strategies for
high-dimensional problems often rely on specific, restrictive problem
structures, or are valid only locally around some nominal trajectory. In this
paper, we propose a data-driven method to approximate semi-global solutions to
HJB equations for general high-dimensional nonlinear systems and compute
candidate optimal feedback controls in real-time. To accomplish this, we model
solutions to HJB equations with neural networks (NNs) trained on data generated
without discretizing the state space. Training is made more effective and
data-efficient by leveraging the known physics of the problem and using the
partially-trained NN to aid in adaptive data generation. We demonstrate the
effectiveness of our method by learning solutions to HJB equations
corresponding to the attitude control of a six-dimensional nonlinear rigid
body, and nonlinear systems of dimension up to 30 arising from the
stabilization of a Burgers'-type partial differential equation. The trained NNs
are then used for real-time feedback control of these systems.
