发表机构
University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对亚稳态间罕见过渡采样问题，提出基于Koopman算子与退出时间最优控制的新方法，通过闭式控制器在RKHS中近似，显著提升过渡轨迹比例。
AI 中文摘要
在动力系统理论，特别是分子动力学中，亚稳态之间的过渡采样是一个核心问题。关键挑战在于分隔这些状态的高自由能势垒，使得过渡事件极为罕见。近期基于机器学习的方法将过渡路径采样（TPS）视为固定时间范围内的最优随机控制（OSC）问题，并通过模拟在环训练的神经网络参数化漂移偏置，这需要重复的有偏 rollout。为解决这些模型的计算与性能保证问题，我们提出了一种基于Koopman算子的新方法。由于Koopman算子是线性的，其主导特征函数揭示了亚稳态集合，并在无需过渡路径信息的情况下提供了committor函数的估计。此外，我们将TPS表述为直至退出时间的OSC问题。我们的时间范围是首次命中目标集合的时间，运行代价通过编码估计的committor函数来惩罚在非反应区域花费的时间。我们推导了闭式最优控制器，并在再生核希尔伯特空间（RKHS）中近似它。这将构建最优控制器的问题简化为求解一个等式约束的二次规划，其解可由线性Karush-Kuhn-Tucker（KKT）系统表征。在双通道双阱和丙氨酸二肽上，我们的控制器分别在1000步内将到达目标的轨迹比例从0%提升至99.8%，在1ps内从0%提升至93%。
英文摘要
Sampling transitions between metastable states is a central problem in dynamical systems theory and molecular dynamics in particular. A key challenge is the existence of high free-energy barriers that separate the states, making transitions extremely rare. Recent machine learning-based methods cast transition path sampling (TPS) as an optimal stochastic control (OSC) problem over a fixed time horizon, and parameterize the drift bias via a neural network trained by simulation-in-the-loop, requiring repeated biased rollouts. To address computational and performance guarantee issues of these models, we propose a new approach for the problem based on Koopman operators. Because Koopman operators are linear, their leading eigenfunctions reveal the metastable sets and provide an estimate of the committor function with no transition path information required. Furthermore, we formulate TPS as an OSC problem up to an exit time. Our time horizon is the first hitting time of the target set, and our running cost penalizes time spent in nonreactive regions by encoding the estimated committor function. We derive the optimal controller in closed form and approximate it in a reproducing kernel Hilbert space (RKHS). This reduces the problem of constructing the optimal controller to solving a single equality-constrained quadratic program, whose solution can be characterized by a linear Karush-Kuhn-Tucker (KKT) system. On the two-channel double well and alanine dipeptide, our controller increases the fraction of trajectories reaching the target from 0% to 99.8% within 1000 steps, and from 0% to 93% within 1ps, respectively.
Comments30 pages, 6 figures
