Optimal control for sampling the transition path process and estimating rates
Optimal control for sampling the transition path process and estimating rates
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DOI：
10.1016/j.cnsns.2023.107701
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发表时间：
2023-05
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影响因子：
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通讯作者：
Jiaxin Yuan;Amar Shah;Channing Bentz;M. Cameron
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文献类型：
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作者：
Jiaxin Yuan;Amar Shah;Channing Bentz;M. Cameron
自然界中的许多过程，如生物分子和相互作用粒子簇的保形变化、遗传开关、带有附加噪声的机械或机电振荡器以及许多其他过程，都是使用带有小白噪声的随机微分方程来建模的。研究这类系统中亚稳态之间的罕见跃迁具有极大的意义和重要性。由于等待时间长，对罕见跃迁的直接模拟是困难的。过渡路径理论是定量描述罕见事件的数学框架。它的关键部分是提交函数，它是后向Kolmogorov方程边值问题的解。在这项工作中利用的关键事实是，由提交者构造的最优控制器只导致过渡轨迹的生成。我们对一类广泛的随机微分方程证明了这一事实。此外，我们还证明了对降维系统进行提交计算，然后提升到原始相位空间，仍然允许我们构造有效的控制器并以合理的精度估计转换率。此外，我们提出了一种通过神经网络计算提交者，采样转移轨迹并估计转移率的全程方案，而无需网格化空间。我们将提出的方法应用于四个测试问题：10D中具有穆勒势和崎岖穆勒势的过阻尼朗格万动力学，嘈杂的双稳态Duffing振荡器和2D中的Lennard-Jones-7。
Many processes in nature such as conformal changes in biomolecules and clusters of interacting particles, genetic switches, mechanical or electromechanical oscillators with added noise, and many others are modeled using stochastic differential equations with small white noise. The study of rare transitions between metastable states in such systems is of great interest and importance. The direct simulation of rare transitions is difficult due to long waiting times. Transition path theory is a mathematical framework for the quantitative description of rare events. Its crucial component is the committor function, the solution to a boundary value problem for the backward Kolmogorov equation. The key fact exploited in this work is that the optimal controller constructed from the committor leads to the generation of transition trajectories exclusively. We prove this fact for a broad class of stochastic differential equations. Moreover, we demonstrate that the committor computed for a dimensionally reduced system and then lifted to the original phase space still allows us to construct an effective controller and estimate the transition rate with reasonable accuracy. Furthermore, we propose an all-the-way-through scheme for computing the committor via neural networks, sampling the transition trajectories, and estimating the transition rate without meshing the space. We apply the proposed methodology to four test problems: the overdamped Langevin dynamics with Mueller’s potential and the rugged Mueller potential in 10D, the noisy bistable Duffing oscillator, and Lennard-Jones-7 in 2D.