Time-variant reliability analysis is a critical task for ensuring the safety of engineering dynamical systems subjected to stochastic excitations. However, assessing failure probability for realistic systems with Monte-Carlo simulation-based methods is often computationally intractable due to the high cost of the underlying models and the large number of simulations required. While surrogate models such as polynomial chaos expansions or Kriging are well-established for time-invariant reliability problems, their direct application to time-dependent systems remains challenging. This chapter introduces two advanced surrogate modeling frameworks designed specifically for dynamical systems: manifold-NARX (mNARX) and functional NARX (F-NARX). The mNARX approach constructs the surrogate on a reduced-order manifold of auxiliary state variables, enabling the efficient handling of high-dimensional inputs by embedding physical insight into a regression formulation. Conversely, the F-NARX framework exploits the functional nature of system trajectories, extracting principal component features from continuous time windows to mitigate issues associated with discrete lag selection and long-memory effects. We demonstrate the efficacy of these methods on two benchmark reliability problems: a stochastic quarter-car model and a hysteretic Bouc-Wen oscillator. The results highlight that, when combined with suitably biased experimental designs, both frameworks accurately capture the tail behavior of the system response, enabling precise and efficient estimation of first-passage probabilities.

翻译：时变可靠性分析是确保受随机激励的工程动力系统安全的关键任务。然而，基于蒙特卡洛仿真的方法在评估实际系统失效概率时，由于底层模型的高计算成本和所需大量仿真次数，往往难以实现计算可行性。尽管多项式混沌展开或克里金等代理模型已在时不变可靠性问题中成熟应用，但其在时变系统中的直接应用仍面临挑战。本章介绍两种专门针对动力系统设计的高级代理建模框架：流形NARX（mNARX）和函数型NARX（F-NARX）。mNARX方法在辅助状态变量的降阶流形上构建代理模型，通过将物理机理嵌入回归公式实现高维输入的高效处理；而F-NARX框架则利用系统轨迹的函数特性，从连续时间窗口中提取主成分特征，以缓解离散延迟选取和长记忆效应带来的问题。我们以随机四分之一车辆模型和滞回Bouc-Wen振子两种基准可靠性问题验证了这些方法的有效性。结果表明，在结合适当偏置的实验设计时，两种框架均能准确捕捉系统响应的尾部行为，从而实现首次穿越概率的精确高效估计。