  The aim of this work is to develop deep learning-based algorithms for
high-dimensional stochastic control problems based on physics-informed learning
and dynamic programming. Unlike classical deep learning-based methods relying
on a probabilistic representation of the solution to the
Hamilton--Jacobi--Bellman (HJB) equation, we introduce a pathwise operator
associated with the HJB equation so that we can define a problem of
physics-informed learning. According to whether the optimal control has an
explicit representation, two numerical methods are proposed to solve the
physics-informed learning problem. We provide an error analysis on how the
truncation, approximation and optimization errors affect the accuracy of these
methods. Numerical results on various applications are presented to illustrate
the performance of the proposed algorithms.
