Action Learning for Scientific Simulations
Scientific inference often requires evaluating and optimizing complete trajectories, yet existing methods typically learn local one-step dynamics or train black-box generators that lack a reusable path objective. This paper proposes action learning, the task of recovering a stochastic action from trajectory observations, and introduces LEAP, which learns a path-space correction relative to a tractable reference process via noise-contrastive trajectory comparisons. By constructing a differentiable objective on complete paths, LEAP captures nonadditive temporal dependencies and enables downstream reuse for ranking, constrained inference, and sampling without retraining.
The paper makes a compelling case for learning path-space objectives directly, backing it with strong theory and diverse experiments. LEAP's decomposition into a reference action plus a learned correction is principled, and the population recovery guarantees together with the double-well and orbital benchmarks show that the method can identify both local drifts and nonadditive trajectory dependencies. However, the practical scope is bounded by the need for a well-chosen reference with sufficient data overlap, and by the small scale of the molecular experiments.
The theoretical framework is unusually thorough for a learning-based simulation paper. The population recovery proof, the structured drift identification result under shared diffusion (Proposition A.2), and the stability bound for downstream inference (Theorem A.6) together ensure that LEAP targets a well-defined object and that errors propagate controllably. Empirically, the double-well recovery is striking: LEAP nearly matches the analytic action on new endpoints and horizons, and the orbital experiment rigorously demonstrates that global path information is necessary to capture latent-induced nonadditive effects that local models miss entirely.
A principal limitation is the reliance on hand-engineered accumulated features in the orbital benchmark: without them, complete-path binary NRE collapses to NRMSE $1.00000\pm 3\times 10^{-5}$, which suggests the global model is not automatically discovering the necessary path-level structure from raw states alone. The molecular experiments are also limited in scope: training sets contain only 130--463 trajectories, and the baselines are not directly comparable because conditional DDPM and flow matching are trained with completion masks whereas LEAP is not. More fundamentally, drift identification fails when endpoints are fully fixed, as noted in Remark A.4, and the effective-coordinate projection in the alanine system introduces memory and state-dependent friction that the learned action does not explicitly model.
The double-well benchmark is carefully constructed with an exact analytic action, and LEAP's inferred path costs ($0.2533$ vs analytic $0.2532$ at $T=4$) strongly validate the approach. The orbital nonadditive correction is derived analytically and matched empirically, providing rare direct evidence that a global path model is necessary. Comparisons to generative baselines in the molecular domain are less convincing: LEAP only marginally outperforms conditional DDPM on dense completion and underperforms on sparse completion, making the claim of broad reuse across inference objectives partially supported but not dominant.
The authors release code and provide detailed training hyperparameters in Appendix B, including batch sizes, learning rates, and compute (Table 4). The double-well and orbital protocols are described with enough precision to replicate drift metrics and path costs. Nevertheless, full architectural specifications for the orbital and molecular networks reside in Appendices E and F, which are referenced but not fully contained in the provided text, leaving some gaps for independent reproduction. The reference process choice is shown to matter significantly, yet practical guidance for selecting references in new domains remains high-level.
Understanding and controlling how systems evolve is central to science and engineering, where many problems require reasoning about complete trajectories. Existing approaches typically learn local dynamics or generate trajectories, but objectives constructed from individual transitions can miss dependencies across time, while generative models do not necessarily provide a tractable objective for evaluating and optimizing arbitrary paths. We study $\textit{action learning}$, the problem of recovering a stochastic action from trajectory observations, yielding a common objective for reasoning about dynamical systems. We introduce $\textbf{LEAP}$ ($\textbf{L}$earned $\textbf{E}$nergy on $\textbf{A}$ction $\textbf{P}$aths), which anchors the unknown action to a tractable reference process and learns the remaining correction by comparing observed trajectories with reference trajectories. This construction yields a differentiable objective on path space that can capture both local dynamical structure and nonadditive dependencies across time. Once learned, the same action can be flexibly reused for path ranking and inference, mechanism steering, constrained optimization and sampling. Across stochastic physical systems, many-body orbital dynamics and molecular simulations, LEAP recovers underlying dynamics, captures dependencies across time and enables transfer across inference objectives and physical conditions. We show that $\textit{action learning}$ is a promising framework for turning trajectory observations into reusable objectives for scientific inference and control.
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