# Distributional Offline Continuous-Time Reinforcement Learning with Neural Physics-Informed PDEs (SciPhy RL for DOCTR-L)

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.

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