  The framework of deep operator network (DeepONet) has been widely exploited
thanks to its capability of solving high dimensional partial differential
equations. In this paper, we incorporate DeepONet with a recently developed
policy iteration scheme to numerically solve optimal control problems and the
corresponding Hamilton--Jacobi--Bellman (HJB) equations. A notable feature of
our approach is that once the neural network is trained, the solution to the
optimal control problem and HJB equations with different terminal functions can
be inferred quickly thanks to the unique feature of operator learning.
Furthermore, a quantitative analysis of the accuracy of the algorithm is
carried out via comparison principles of viscosity solutions. The effectiveness
of the method is verified with various examples, including 10-dimensional
linear quadratic regulator problems (LQRs).
