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DPO: A Differential and Pointwise Control Approach to Reinforcement Learning

Minh Nguyen, Chandrajit Bajaj

TL;DR

The paper addresses sample-inefficiency and lack of physical consistency in reinforcement learning for scientific computing by reframing RL as a continuous-time control problem via a differential dual and Hamiltonian structure. It introduces Differential Policy Optimization (DPO), a stagewise, pointwise update rule that learns a local trajectory operator, promoting trajectory-consistent learning aligned with system dynamics. The authors establish pointwise convergence guarantees and a regret bound of $O(K^{5/6})$, and demonstrate empirically that DPO outperforms standard baselines on surface modeling, grid-based modeling, and molecular dynamics under low data. This approach integrates physics priors into RL through a differential dual, enabling more data-efficient learning in physics-constrained environments with broad potential impact in scientific computing. The work also provides a foundation for future extensions to adaptive discretization and broader domains beyond physics-informed control.

Abstract

Reinforcement learning (RL) in continuous state-action spaces remains challenging in scientific computing due to poor sample efficiency and lack of pathwise physical consistency. We introduce Differential Reinforcement Learning (Differential RL), a novel framework that reformulates RL from a continuous-time control perspective via a differential dual formulation. This induces a Hamiltonian structure that embeds physics priors and ensures consistent trajectories without requiring explicit constraints. To implement Differential RL, we develop Differential Policy Optimization (DPO), a pointwise, stage-wise algorithm that refines local movement operators along the trajectory for improved sample efficiency and dynamic alignment. We establish pointwise convergence guarantees, a property not available in standard RL, and derive a competitive theoretical regret bound of $O(K^{5/6})$. Empirically, DPO outperforms standard RL baselines on representative scientific computing tasks, including surface modeling, grid control, and molecular dynamics, under low-data and physics-constrained conditions.

DPO: A Differential and Pointwise Control Approach to Reinforcement Learning

TL;DR

The paper addresses sample-inefficiency and lack of physical consistency in reinforcement learning for scientific computing by reframing RL as a continuous-time control problem via a differential dual and Hamiltonian structure. It introduces Differential Policy Optimization (DPO), a stagewise, pointwise update rule that learns a local trajectory operator, promoting trajectory-consistent learning aligned with system dynamics. The authors establish pointwise convergence guarantees and a regret bound of , and demonstrate empirically that DPO outperforms standard baselines on surface modeling, grid-based modeling, and molecular dynamics under low data. This approach integrates physics priors into RL through a differential dual, enabling more data-efficient learning in physics-constrained environments with broad potential impact in scientific computing. The work also provides a foundation for future extensions to adaptive discretization and broader domains beyond physics-informed control.

Abstract

Reinforcement learning (RL) in continuous state-action spaces remains challenging in scientific computing due to poor sample efficiency and lack of pathwise physical consistency. We introduce Differential Reinforcement Learning (Differential RL), a novel framework that reformulates RL from a continuous-time control perspective via a differential dual formulation. This induces a Hamiltonian structure that embeds physics priors and ensures consistent trajectories without requiring explicit constraints. To implement Differential RL, we develop Differential Policy Optimization (DPO), a pointwise, stage-wise algorithm that refines local movement operators along the trajectory for improved sample efficiency and dynamic alignment. We establish pointwise convergence guarantees, a property not available in standard RL, and derive a competitive theoretical regret bound of . Empirically, DPO outperforms standard RL baselines on representative scientific computing tasks, including surface modeling, grid control, and molecular dynamics, under low-data and physics-constrained conditions.
Paper Structure (19 sections, 12 theorems, 58 equations, 4 figures, 4 tables, 1 algorithm)

This paper contains 19 sections, 12 theorems, 58 equations, 4 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Differential reinforcement learning
  3. Problem formulation
  4. Differential policy optimization (DPO) algorithm
  5. Application to scientific computing
  6. Theoretical analysis
  7. Pointwise convergence and sample complexity
  8. Regret bound analysis
  9. Experiments
  10. Evaluation tasks
  11. Experimental results
  12. Conclusion
  13. Basic algorithmic learning theory
  14. Proofs of theorems and corollaries in Section 3
  15. Computational details
  16. ...and 4 more sections

Key Result

Theorem 3.2

Suppose that we are given a threshold error $\epsilon$, a probability threshold $\delta$, and a number of steps per episode $H$. Assume that $\left\{ N_k \right\}_{k = 1}^{H-1}$ is the sequence of numbers of samples used at each stage in alg:dpo (DPO) so that: Here $\delta_k = \delta/3^{H-k} = 3 \delta_{k-1}$. We further assume that there exists a Lipschitz constant $L > 0$ such that both the tru

Figures (4)

  • Figure : (a) Surface modeling
  • Figure : (a) Surface modeling
  • Figure : (b) Grid-based modeling
  • Figure : (c) Molecular dynamics

Theorems & Definitions (25)

  • Definition 2.1
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Corollary 3.4
  • Corollary 3.5
  • proof
  • Corollary 3.6
  • proof
  • ...and 15 more