##### Department of Mathematics,

University of California San Diego

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### Center for Computational Mathematics Seminar

## Yuhua Zhu

#### UCSD

## A PDE based Bellman equation for Continuous-time Reinforcement Learning

##### Abstract:

In this paper, we address the problem of continuous-time reinforcement learning in scenarios where the dynamics follow a stochastic differential equation. When the underlying dynamics remain unknown and we have access only to discrete-time information, how can we effectively perform policy evaluation? We first demonstrate that the commonly used Bellman equation is a first-order approximation to the true value function. We then introduce a higher order PDE-based Bellman equation called PhiBE. We show that the solution to the i-th order PhiBE is an i-th order approximation to the true value function. Additionally, even the first-order PhiBE outperforms the Bellman equation in approximating the true value function when the system dynamics change slowly. We develop a numerical algorithm based on Galerkin method to solve PhiBE when we possess only discrete-time trajectory data. Numerical experiments are provided to validate the theoretical guarantees we propose.

### February 6, 2024

### 11:00 AM

Zoom Only, ID 990 3560 4352

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