Abstract: In this talk, I will explore the connection between differential equations and Reinforcement learning. In the first half, I will build the connection between Hamilton-Jacobi-Bellman equations and the multi-armed bandit (MAB) problems. MAB is a widely used paradigm for studying the exploration-exploitation trade-off in sequential decision making under uncertainty. This is the first work that establishes this connection in a general setting. I will present an efficient algorithm for solving MAB problems based on this connection and demonstrate its practical applications. In the second half, I will focus on the reinforcement learning problem in smooth environment. We propose a novel method to alleviate the double sampling problem in model-free reinforcement learning, and PDE will be used to do error analysis for the algorithm.