Memory matters: Beyond Markovian models of rare event kinetics

摘要：罕见的事件涉及系统的两个状态之间快速但不频繁的转变，例如亚稳的母液A和稳定的晶体B。这个项目将通过两个关键的发展来扩展系统从状态A到状态B转变频率的最新模型；(1)从信号处理到结合了记忆的集体动力学模型的拟合技术(2)提交者函数的机器学习，即微观状态在返回到A之前演化到B的概率。这两个发展结合在一起，将使我们能够从原子模拟中做出更好的预测。背景：在许多异质系统中，原子尺度上的行为受到罕见事件的支配，我们指的是两个状态A和B之间的转变，这些转变发生得很快，但足够不频繁，以至于在可实现持续时间的模拟中永远不会被观察到。对这些进行建模是具有挑战性的，通常依赖于简单模型的参数化，这些模型通过单个集体自由度的动力学来描述罕见事件。这种描述可以做出准确的预测，但依赖于识别最佳自由度并选择其动力学的适当模型。在大多数情况下，使用粗略的(但不昂贵的)近似。在这个项目中，我们将结合两种最先进的数值技术来系统地改进这些技术。最优集体变量是提交者已知的，即微状态在状态A之前到达状态B的概率。我们将扩展小组中现有的使用机器学习来预测提交者(以及相关的不确定性)的工作。我们还将使用信号处理文献中的自回归技术对非最优集合变量的时间演化进行建模，以建立具有记忆的动力学模型，即超越通常的马尔可夫动力学假设。这两种技术将在应用于由罕见事件控制的简单格子模型时进行比较，然后扩展到未来几年的原子模拟。该项目将适合对计算方法、统计力学和数据科学感兴趣的学生。

Summary: Rare events involve rapid but infrequent transitions between two states of a system, e.g. a metastable parent liquid A and a stable crystal B. This project will extend the state-of-the-art for models of how often a system transitions from state A to state B with two key developments; (1) techniques from signal processing to fit models of collective dynamics which incorporate memory (2) machine learning of committor functions, the probability that a microstate will evolve to B before returning to A. Combined, these two developments will allow us to make enhanced predictions from atomistic simulations.Background: In many heterogenous systems, behaviour at the atomistic scale is governed by rare events, by which we mean transitions between two states A and B that occur rapidly but sufficiently infrequently that they will never be observed during a simulation of achievable duration. Modelling these is challenging, and often relies on parameterisation of simple models which describe the rare event via the kinetics of a single collective degree of freedom.Such descriptions can make accurate predictions, but rely on identifying the optimal degree of freedom and choosing an appropriate model of its kinetics. In most cases crude (but inexpensive) approximations are used. In this project we will combine two state-of-the-art numerical techniques to improve on these systematically. The optimal collective variable is known to the committor, the probability that a microstate will reach state B before state A. We will extend existing work in the group on using machine learning to predict the committor (and associated uncertainty). We will also model the time evolution of less-optimal collective variables using autoregressive techniques from the signal processing literature to build kinetic models with memory, i.e. beyond the usual assumption of Markovian dynamics.The two techniques will be compared and contrasted when applied to simple on-lattice models governed by rare events, before extending to atomistic simulations in later years. The project would suit a student interested in computational methods, statistical mechanics and data science.