概率性能模式分解（PPPD）：分析框架及其在随机力学系统中的应用
Probabilistic Performance-Pattern Decomposition (PPPD): analysis framework and applications to stochastic mechanical systems
摘要：
自20世纪初以来，许多研究工作一直致力于发展定量解决方案，随机力学系统。一般来说，当确定了关于感兴趣的量（QoI）的完整或部分概率描述时，问题被认为已经解决。然而，在复杂的系统行为的存在下，有一个关键的需要超越纯粹的概率描述。事实上，为了全面了解系统，从QoI的概率结构中提取物理特征是至关重要的，特别是当以数据驱动的方式获得QoI解决方案时。基于这一观点，本文提出了一个结构化描述随机系统行为的框架。该框架被命名为概率性能模式分解（PPPD）。PPPD分析的目的是将复杂的响应行为分解为系统响应空间中有意义的模式，并研究这些模式在基本随机变量空间中是如何被触发的。为了说明PPPD的应用，本文研究了三个数值例子：1）一个假设的随机过程输入和输出的说明性例子; 2）一个随机Lorenz系统的周期以及混沌行为;和3）一个简化的剪切建筑模型受到随机地面运动激励。
英文摘要：
Since the early 1900s, numerous research efforts have been devoted to developing quantitative solutions to stochastic mechanical systems. In general, the problem is perceived as solved when a complete or partial probabilistic description on the quantity of interest (QoI) is determined. However, in the presence of complex system behavior, there is a critical need to go beyond mere probabilistic descriptions. In fact, to gain a full understanding of the system, it is crucial to extract physical characterizations from the probabilistic structure of the QoI, especially when the QoI solution is obtained in a data-driven fashion. Motivated by this perspective, the paper proposes a framework to obtain structuralized characterizations on behaviors of stochastic systems. The framework is named Probabilistic Performance-Pattern Decomposition (PPPD). PPPD analysis aims to decompose complex response behaviors, conditional to a prescribed performance state, into meaningful patterns in the space of system responses, and to investigate how the patterns are triggered in the space of basic random variables. To illustrate the application of PPPD, the paper studies three numerical examples: 1) an illustrative example with hypothetical stochastic processes input and output; 2) a stochastic Lorenz system with periodic as well as chaotic behaviors; and 3) a simplified shear-building model subjected to a stochastic ground motion excitation.
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