{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/2506-05208","title":"Optimal-PhiBE: A PDE-based Model-free framework for Continuous-time Reinforcement Learning","arxiv_id":"2506.05208","date":"2025-06-05","proceeding":null,"authors":["Yuhua Zhu","Yuming Zhang","Haoyu Zhang"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2506.05208v1","url_pdf":"https://arxiv.org/pdf/2506.05208v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"2506-05208","repo_url":"https://github.com/haz053ucsd/Optimal_PhiBE","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}