A fundamental limitation of various Equivalent Linearization Methods (ELMs) in nonlinear random vibration analysis is that they are approximate by their nature. A quantity of interest estimated from an ELM has no guarantee to be the same as the solution of the original nonlinear system. In this study, we tackle this fundamental limitation. We sequentially address the following two questions: (i) given an equivalent linear system obtained from any ELM, how to construct an estimator such that, as the linear system simulations are guided by a limited number of nonlinear system simulations, the estimator converges on the nonlinear system solution? (ii) how to construct an optimized equivalent linear system such that the estimator approaches the nonlinear system solution as quickly as possible? The first question is theoretically straightforward since classic Monte Carlo techniques, such as the control variates and importance sampling, can improve upon the solution of any surrogate model. We adapt the well-known Monte Carlo theories into the specific context of equivalent linearization. The second question is challenging, especially when rare event probabilities are of interest. We develop specialized methods to construct and optimize linear systems. In the context of uncertainty quantification (UQ), the proposed optimized ELM can be viewed as a physical surrogate model-based UQ method. The embedded physical equations endow the surrogate model with the capability to handle high-dimensional uncertainties in stochastic dynamics analysis.

非线性随机振动分析中各种等效线性化方法 (ELM) 的一个基本限制是它们本质上是近似的。ELM 估计的感兴趣量不能保证与原始非线性系统的解相同。在这项研究中，我们解决了这个基本限制。我们依次解决以下两个问题：（i）给定从任何 ELM 获得的等效线性系统，如何构造一个估计器，使得当线性系统模拟由有限数量的非线性系统模拟引导时，估计器收敛于非线性系统的解？(ii) 如何构造一个优化的等效线性系统，使得估计器尽快逼近非线性系统解？第一个问题在理论上很简单，因为经典的蒙特卡罗技术（例如控制变量和重要性采样）可以改进任何替代模型的解决方案。我们将著名的蒙特卡洛理论应用到等效线性化的特定背景中。第二个问题具有挑战性，特别是当人们对罕见事件的概率感兴趣时。我们开发专门的方法来构建和优化线性系统。在不确定性量化 (UQ) 的背景下，所提出的优化 ELM 可以被视为基于物理代理模型的 UQ 方法。嵌入的物理方程赋予代理模型处理随机动力学分析中高维不确定性的能力。