Data-driven Modeling of Equilibrium and Non-equilibrium Statistics

均衡和非均衡统计的数据驱动建模

• 批准号: 1619661
• 负责人: John Harlim
• 金额: $ 30.09万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1619661&HistoricalAwards=false
• 关键词: Data; driven; Modeling; Equilibrium; Non

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.

应用和计算科学中的一个重要问题是找到必要的简化模型来预测高维复杂动力系统中感兴趣的变量。鉴于我们收集大数据的先进能力，一个重要的挑战是利用数据携带的信息来改进建模工作。在计算上，这需要对适当的参数进行充分的推断，以便它们的不确定性是可量化的。一个更具挑战性但更重要的问题是能够在存在外部干扰的情况下进行预测。这一问题具有广泛的应用，例如在气候变化科学中，人们有兴趣预测与火山爆发等外源强迫甚至人类活动等人为因素相对应的气候变化统计数据。本提案中的项目旨在解决这些问题。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，以捕获底层动态的粗粒度统计解决方案。该方法涉及 Mori-Zwanzig 形式主义、对记忆效应的精确描述，以考虑不同物理尺度上发生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，以预测存在小的外部干扰时统计解的变化。该项目涉及在适当的积分算子上采用 Padè 近似，并设计有效的算法来求解非线性方程组，该方程组尊重未扰动数据的适当平衡统计，利用波动耗散理论的公式。