  This paper addresses distributional offline continuous-time reinforcement
learning (DOCTR-L) with stochastic policies for high-dimensional optimal
control. A soft distributional version of the classical Hamilton-Jacobi-Bellman
(HJB) equation is given by a semilinear partial differential equation (PDE).
This `soft HJB equation' can be learned from offline data without assuming that
the latter correspond to a previous optimal or near-optimal policy. A
data-driven solution of the soft HJB equation uses methods of Neural PDEs and
Physics-Informed Neural Networks developed in the field of Scientific Machine
Learning (SciML). The suggested approach, dubbed `SciPhy RL', thus reduces
DOCTR-L to solving neural PDEs from data. Our algorithm called Deep DOCTR-L
converts offline high-dimensional data into an optimal policy in one step by
reducing it to supervised learning, instead of relying on value iteration or
policy iteration methods. The method enables a computable approach to the
quality control of obtained policies in terms of both their expected returns
and uncertainties about their values.
