Data-driven Modeling of Equilibrium and Non-equilibrium Statistics

从高维复杂动力系统中寻找预测感兴趣变量的本质简化模型是应用科学和计算科学中的一个重要问题。鉴于我们收集大数据的高级能力，一个重要的挑战是利用数据携带的信息来改进建模工作。从计算上讲，这需要对适当的参数进行充分的推断，使其不确定性是可以量化的。一个更具挑战性但也更重要的问题是能够在存在外部干扰的情况下进行预测。这一问题有着广泛的应用，如在气候变化科学中，人们感兴趣的是预测与火山喷发等外源强迫相对应的气候变化统计数据，甚至是人类活动等人为因素。本提案中的项目就是为了解决这些问题。虽然所开发的方法旨在对多尺度现象进行一般建模，但我们的重点将是提高对石墨烯变形行为的理解和预测。提出了两个项目：1.数据驱动的简化建模范例，以获取底层动态的粗粒度统计解决方案。该方法包括Mori-Zwanzig形式，对记忆效应的精确描述以考虑发生在不同物理尺度上的过程之间的相互作用，以及用于估计随机简化模型参数的数据驱动的数值方案。2.估计简化模型中的参数，以预测在存在小的外部干扰时统计解的变化。这个项目涉及对适当的积分算子使用Padè近似，并设计有效的算法来求解非线性方程系统，该系统尊重未扰动数据的适当的平衡统计，利用波动耗散理论的公式。

An important issue in applied and computational sciences is to find the essential reduced models to predict variables of interests from high-dimensional complex dynamical systems. Given our advanced capability to collect big data, an important challenge is to leverage the information carried by the data to improve the modeling effort. Computationally, this requires adequate inference of appropriate parameters such that their uncertainties are quantifiable. A much more challenging yet important issue is to be able to make prediction in the presence of external disturbances. This problem has a wide range of applications such as in climate change science where one is interested to predict the climate change statistics corresponding to exogenous forcing such as the volcanic eruptions or even the anthropogenic factor such as the human activities. The projects in this proposal are to address these issues. While the developed methodology is aimed for general modeling of multi-scale phenomena, our focus will be to improve the understanding and prediction of the deformation behavior of graphene. Two projects are proposed: 1. Data-driven reduced modeling paradigms to capture coarse grained statistical solutions of the underlying dynamics. The methodology involves the Mori-Zwanzig formalism, a precise description of the memory effect to take into account the interactions between processes occurring on different physical scales, and a data-driven numerical scheme for estimating the parameters of the stochastic reduced model. 2. Estimation of parameters in the reduced models to predict changes on the statistical solutions in the presence of small external disturbances. This project involves employing the Padè approximation on appropriate integral operators and designing efficient algorithm to solve a system of nonlinear equations that respect appropriate equilibrium statistics of the unperturbed data, leveraging the formulation from the fluctuation-dissipation theory.