Title: A Differential and Pointwise Control Approach to Reinforcement Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Hamiltonian Dynamics

Policy Optimization

Pointwise Convergence

Continuous-Time Control

Physics-Informed Learning

Scientific Computing
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 (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 $\mathcal{O}(K^{5/6})$. 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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present DPO (Differential Policy Optimization), which is a continuous generalization of RL aimed at improving performance for continuous control problems where many other RL methods deliver subpar performance. DPO is build on the Pontryagin Maximum Principle and the foundations of classical Hamiltonian mechanics and enforces a symplectic structure on the evolution of the state. This makes the algorithm particularly suited for solving scientific computing problems where the system in question can be typically modeled with Hamiltonian mechanics. It also gives the algorithm point wise convergence guarantees and enables the derivation of upper bounds on the regret. The empirical performance of DPO is evaluated on a set of scientific computing problems and delivers state-of-the-art results.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n1.) Weaknesses of current state of the art are well-explained and work presented is well justified.\n\n2.) Mathematically well founded algorithm whose derivation is sound with the goal it intends to achieve. Proofs in the main body are kept at a reasonable length and provide enough intuition to grasp the core idea.\n\n3.) For two scenarios, the algorithm and its theoretical bounds enable the computation of the number of necessary episodes until convergence.\n\n4.) Good experimental evaluation and documentation of procedure.\n\n5.) Code is included and results are reproducible.\n\n**Weaknesses:**\n\n1.) Lack of a Related Works section.  This work could benefit from a related works section in order to showcase its advantages and compare to other algorithms and make its results stand out more, e.g. the theoretically derived regret bound. \n\n2.) Notation is often non-standard, inconsistent and confusing. I added my proposed changes under styles.\n\n3.) Lack of investigation for computational requirements/cost of running the algorithm compared to others. This could potentially make the algorithm stand out further. However I assume that DPO has high computational requirements than most other algorithms it compares to due to the extensive use of Automatic Differentiation.\n\nQUESTIONS:\n1.) Why is $\\rho_0$ termed an adversarial distribution?\n\n2.) Why $L_1$ loss in Step 6 of the algorithm?\n\n3.) Symplectic for does not really encode a physical prior but rather a particular form of evolution of the system that is consistent with the classical Hamilton equations. So the derived method in the given shape only works for classical physics systems. Also for some systems, the symplectic form alone might not be enough, e.g. fluid dynamics or quantum mechanics.\nWould it be possible to do something similar for fluid dynamics or quantum mechanics by incorporating other physical priors and not only the symplectic behavior?\n\n4.) In the surface modeling task, the performance improvement of DPO seems to much less impressive compared to the other tasks. Why is that?\n\n5.) The examples you chose have a high mathematical similarity to the proposed algorithm, especially the reward functions and the forward dynamics already look like what the algorithm does (which is fine, right tool for right problem). But what happens when you use the algorithm on a problem with less similarity? Say with more complex forward dynamics that involve a decay on state s for example or dynamics that are not Lipschitz? How much is this going to deteriorate the performance of your algorithm?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper develops a novel reinforcement learning algorithm: differential policy optimisation. The algorithm shifts from a “global” formulation to a local one following physical trajectories. The authors use Pontryagin’s maximum principle to update the classical RL formulation. The authors also offer a strong theoretical background regarding the algorithm’s convergence properties. Finally, several experiments are run on three different benchmarks, where DPO outperforms other DRL algorithms.\n\nSTRENGTHS AND WEAKNESSES:\n*Strengths*\n\n- The authors find and expand a new RL formulation with good physical foundations, in my opinion.\n- A strong theoretical foundation is provided for pointwise convergence and sample efficiency. The proofs in the appendix are sound. \n- The authors demonstrate strong performance against already well-known DRL algorithms on three different benchmarks.\n- A strong learning efficiency is also demonstrated.\n\n*Weaknesses*\n\n- The paper only focuses on 3 (low number) benchmarks that fit their method very well in terms of pure physical formulation.\n- The paper would greatly benefit from more standardized DRL benchmarks.\n- The paper lacks a proper ablation study of the hyperparameters for all algorithms. \n- While I know page limits make it hard to have not too dense mathematical sections, the appendix sections could benefit from more readability and being easier to follow.\n\nQUESTIONS:\n1. Did you perform any sort of ablation studies on your algorithm? On the other? How sensitive are the results to such parameters? From experience, one can “make or break” the results from a DRL algorithm for a specific environment just by tweaking some hyperparameters.\n2. Would it be possible for you to extend the validation section with other standard DRL environments (non-physics related)?\n3. How is DPO performing on more standard physics DRL benchmarks? \n\n\nAlso, on a less important note:\n\n- In the field of DRL, DPO is already widely used for Direct Preference Optimisation (Direct Preference Optimization:\nYour Language Model is Secretly a Reward Model). It would be a good idea to change it.\n- I would strongly advice the authors to update the timestep notation to a standard one such as $\\Delta_t$ or $\\Delta t$ or $\\delta t$. It will also help readibility in terms of pure differential calculus ($\\Delta$ being a laplacian). \n- line 126 seems like a repetition\n- typo for $\\mathbb{R}$ line 413\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces Differential Reinforcement Learning, a reformulation of RL through a continuous-time control perspective using a differential dual formulation. While standard RL maximizes cumulative rewards from discrete steps, Differential RL instead maximizes the time integral of rewards. To instantiate a practical algorithm within the Differential RL framework, the authors propose Differential Policy Optimization (DPO), an algorithm that optimizes a local trajectory operator by updating pointwise behavior along a trajectory. The authors theoretically analyze DPO by establishing pointwise convergence guarantees, properties not available in standard RL, and show that DPO has a regret bound of $O(K^{5/6})$, where $K$ is the number of training episodes. DPO is evaluated on a set of three tasks spanning surface modeling, grid-based modeling, and molecular dynamics.\n\nSTRENGTHS AND WEAKNESSES:\n[S1] Continuous-time formulations of reinforcement learning is an important topic, given that standard RL algorithms are broadly developed for the discrete-time setting (such as video games), but straightforwardly applied to continuous-time settings. Studying a continuous-time formulation may provide insight into new algorithm designs that improve on sample complexity requirements. \n\n[S2] The paper is generally well written, the differential dual formulation is clearly introduced, and the theoretical analysis in Section 3 is easy to follow. \n\n[W1] There is no section in the main paper or appendix that discusses prior work on continuous-time formulations in RL. I am not too familiar with the prior work in continuous-time RL, but from a cursory scan of the literature: [[Ainsworth et al., 2020]](https://arxiv.org/abs/2012.06684), [[Wang et al., 2020]](https://jmlr.org/papers/v21/19-144.html), [[Yildiz et al., 2021]](https://arxiv.org/abs/2102.04764), [[Jia and Zhou, 2022]](https://arxiv.org/abs/2207.00713), [[Treven et al., 2023]](https://arxiv.org/abs/2310.19848), [[Zhao et al., 2023]](https://arxiv.org/abs/2305.18901), [[Wallace and Si., 2024]](https://www.jmlr.org/papers/v25/24-0017.html) …\nThe authors should conduct a thorough literature review and contextualize their differential dual formulation to existing frameworks for continuous-time RL. \n\n[W2] DPO relies on pointwise (local) updates, and two of the evaluation tasks (surface modeling, grid-based modeling) are designed to directly update states based on actions: $s_{k+1} = s_{k} + \\delta a_k$. It is difficult to evaluate potential practical applications of DPO given these simplified dynamics. The authors may consider validating DPO on more standard control benchmarks, such as double pendulum.\n\n[W3] There is no discussion on computational cost or hyperparameter sensitivity / tuning for the proposed DPO algorithm. \n\n- - -\n\nI am willing to increase my score if [W1] is properly addressed.\n\nQUESTIONS:\nL30: I am not sure if this “re-planning assumption” claim is valid? These planners usually evaluate trajectories using a learned model, rather than assuming oracle resets for the environment. In practice, the planned trajectories may be used for a few time steps, and then the planner may be queried again to generate new trajectories at the latest time step.\n\nL283: 5000 sample steps is very likely too small to properly evaluate the (asymptotic) performance of the standard RL baselines. \n\nL152-161: The forward dynamics are exactly $s_{k+1}=s_{k}+\\delta a_k$ for the surface modeling and grid-based modeling tasks? Is this also true for molecular dynamics? Given these simplified dynamics, it is not clear to me why the surface modeling or grid-based modeling tasks are proper evaluations. \n\nHow does the Differential RL formulation apply to partially-observable MDPs?\n\nTable 4: Why is the standard deviation close to 0 for some methods on the molecular dynamics task?\n\nLIMITATIONS:\nYes, Appendix D.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes Differential Reinforcement Learning (Differential RL), a new framework for continuous state-action RL that emphasizes the _dynamics of the optimal trajectory_ rather than the cumulative reward. The authors introduce an algorithm called Differential Policy Optimization (DPO), which optimizes the policy in a pointwise, stage-wise manner along the trajectory. This approach induces a Hamiltonian structure that can incorporate physics priors, ensuring that learned trajectories remain physically consistent without adding explicit constraints.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper demonstrated a noval framework that could be potentially very interesting. The authors proved pointwise convergence of DPO, meaning the policy at each stage of the trajectory converges to the optimal action for that state. By enforcing trajectory-wide optimality and embedding physics knowledge, I believe that this approach is interesting. However, it is pending testify on how the algorithms are going to perform with real-world/simulation tasks, and in the experiements section it is unclear on how many cases are tested. \nAll experimental domains are physics-based tasks with specific Lagrangian energy reward structures. While appropriate for showing off DPO’s strengths, these tasks are niche and structured. It remains unclear how DPO would perform on more conventional RL benchmarks.\nWhile the $O(K^{5/6})$ regret bound is comparable to some prior continuous-RL results, it is still a fairly slow rate (sublinear but not as favorable as $O(\\sqrt{K})$) and only holds under the aforementioned special conditions.\n\nQUESTIONS:\nFor the current set of evaluations, more reasonings on the selection of the specific tasks should be provided. Also more concrete examples should be given for all 3 evaluation tasks, and reasonings behind the selections of the hyperparameters.\n\nTo strengthen the paper, the authors should consider evaluating DPO on more standard RL environments or a wider variety of tasks.\n\nLIMITATIONS:\nThe notion of “differential RL” and the dual formulation is rooted in control theory, and readers not familiar with Hamiltonian systems or Pontryagin’s principle might struggle without additional intuitive explanation.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-03"}