# A Differential and Pointwise Control Approach to Reinforcement Learning

arXiv: https://arxiv.org/abs/2404.15617 · HTML: https://mlsanity.com/paper/2404.15617/

## Key facts

- Cited by 1 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.
- Builds on 20 of its 38 references that are in the map.
- First posted on arXiv 24 Apr 2024.

## Placed in

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

## Builds on (most cited first, 20 of 20)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Continuous control with deep reinforcement learning](https://mlsanity.com/paper/1509.02971/) (2015)
- [Fast R-CNN](https://mlsanity.com/paper/1504.08083/) (2015)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [QT-Opt: Scalable Deep Reinforcement Learning for Vision-Based Robotic Manipulation](https://mlsanity.com/paper/1806.10293/) (2018)
- [Introduction to Multi-Armed Bandits](https://mlsanity.com/paper/1904.07272/) (2019)
- [Neural Network Dynamics for Model-Based Deep Reinforcement Learning with Model-Free Fine-Tuning](https://mlsanity.com/paper/1708.02596/) (2017)
- [Provably Efficient Exploration in Policy Optimization](https://mlsanity.com/paper/1912.05830/) (2019)
- [Controlling Overestimation Bias with Truncated Mixture of Continuous Distributional Quantile Critics](https://mlsanity.com/paper/2005.04269/) (2020)
- [Learning Continuous Control Policies by Stochastic Value Gradients](https://mlsanity.com/paper/1510.09142/) (2015)
- [Benchmarking Model-Based Reinforcement Learning](https://mlsanity.com/paper/1907.02057/) (2019)
- [CrossQ: Batch Normalization in Deep Reinforcement Learning for Greater Sample Efficiency and Simplicity](https://mlsanity.com/paper/1902.05605/) (2019)
- [q-Learning in Continuous Time](https://mlsanity.com/paper/2207.00713/) (2022)
- [Continuous-Time Model-Based Reinforcement Learning](https://mlsanity.com/paper/2102.04764/) (2021)
- [Policy Optimization for Continuous Reinforcement Learning](https://mlsanity.com/paper/2305.18901/) (2023)
- [Kernelized Reinforcement Learning with Order Optimal Regret Bounds](https://mlsanity.com/paper/2306.07745/) (2023)
- [PowerGridworld: A Framework for Multi-Agent Reinforcement Learning in Power Systems](https://mlsanity.com/paper/2111.05969/) (2021)
- [Faster Policy Learning with Continuous-Time Gradients](https://mlsanity.com/paper/2012.06684/) (2020)
- [Reinforcement Learning for Molecular Dynamics Optimization: A Stochastic Pontryagin Maximum Principle Approach](https://mlsanity.com/paper/2212.03320/) (2022)

## Built on by (most cited first, 1 of 1)

- [PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting](https://mlsanity.com/paper/2602.17998/) (2026)

## Abstract (arXiv)

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 (dfPO), 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(K5/6). Empirically, dfPO outperforms standard RL baselines on representative scientific computing tasks, including surface modeling, grid control, and molecular dynamics, under low-data and physics-constrained conditions.