# Optimal-PhiBE: A PDE-based Model-free framework for Continuous-time Reinforcement Learning

arXiv: https://arxiv.org/abs/2506.05208 · HTML: https://mlsanity.com/paper/2506.05208/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- First posted on arXiv 5 Jun 2025.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Built on by (most cited first, 4 of 4)

- [Discretization error from regularized Reinforcement Learning to continuous-time stochastic control](https://mlsanity.com/paper/2604.21179/) (2026)
- [A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients](https://mlsanity.com/paper/2606.24999/) (2026)
- [Mean-Field PhiBE: Continuous-Time Mean-Field Reinforcement Learning from Discrete-Time Data](https://mlsanity.com/paper/2606.26498/) (2026)
- [PhiBE-Q-Learning: Bridging Off-Policy Reinforcement Learning and Continuous-Time Control](https://mlsanity.com/paper/2606.21925/) (2026)

## Abstract (arXiv)

This paper addresses continuous-time reinforcement learning (CTRL) where the system dynamics are governed by an unknown stochastic differential equation, and only discrete-time observations are available. Existing approaches face limitations: model-based PDE methods suffer from non-identifiability, while model-free methods based on the discrete-time optimal Bellman equation (Optimal-BE) suffer from large discretization errors that are highly sensitive to both the system dynamics and the reward structure. To overcome these challenges, we introduce Optimal-PhiBE, a formulation that integrates discrete-time information into a continuous-time PDE, combining the strength of both existing frameworks while mitigating their limitations. Optimal-PhiBE exhibits smaller discretization errors when the uncontrolled system evolves slowly, and demonstrates reduced sensitivity to oscillatory reward structures, and enables model-free algorithms that bypass explicit dynamics estimation. In the linear-quadratic regulator (LQR) setting, sharp error bounds are established for both Optimal-PhiBE and Optimal-BE. The results show that Optimal-PhiBE exactly recovers the optimal policy in the undiscounted case and substantially outperforms Optimal-BE when the problem is weakly discounted or control-dominant. Furthermore, we extend Optimal-PhiBE to higher orders, providing increasingly accurate approximations. A model-free policy iteration algorithm is proposed to solve the Optimal-PhiBE directly from trajectory data. Numerical experiments are conducted to verify the theoretical findings.