Structural and multidisciplinary design under uncertainty for high reliability or equivalently small probability of failure is a challenging task owing to the high computational cost associated with generating the samples at the extreme (tail) of the underlying distribution. Among other approaches, statistics of extremes based techniques are usually suitable for small probability estimation. However, typically only 10% of the samples generated that correspond to the tail of the distribution are used for probability estimation. If apriori information about regions in the design space that corresponds to the tail is available, additional samples in the identified region permit better tail fit and hence better probability estimation. In the current work, we propose iSOM (interpretable Self-Organising Map) to identify region/s in the design space, that corresponds to the extremes. An initial sample is used to map (visualize) the limit state function and random/design variables using iSOM which permits the designer to identify the region(s) that corresponds to the tail of the response. Adaptive sampling is performed in the identified region of interest to obtain additional samples. Next, the cumulative distribution function (CDF) of the response using initial as well as adaptive samples is evaluated for probability estimation. The effectiveness of the proposed approach is evident from its successful implementation on benchmark examples, real-world engineering examples, and a multi-objective reliability-based design optimization (MORBDO) case. The proposed method showcases the capability of iSOM to perform adaptive sampling for limit-state functions characterized by non-linearity and multiple modes. iSOM-enabled sampling in conjunction with log-TPNT provides better estimates of small failure probabilities than log-TPNT alone. The results from the proposed approach is compared with results from state-of-the-art (SOTA) sampling and surrogate-based techniques. For a given number of limit state evaluations, the proposed approach estimates probabilities of the order 1e−4, with lesser variance, compared to other SOTA approaches. Hence, the proposed approach is likely to encourage further research into employing iSOM-assisted sampling for other reliability estimation methods as well.

中文翻译：

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使用可解释的自组织映射进行基于自适应采样的小故障概率估计

在不确定性下实现高可靠性或同等小故障概率的结构和多学科设计是一项具有挑战性的任务，因为与在基础分布的极端（尾部）生成样本相关的计算成本很高。在其他方法中，基于极值统计的技术通常适用于小概率估计。然而，通常只有 10% 的生成样本（对应于分布的尾部）用于概率估计。如果有关设计空间中对应于尾部的区域的先验信息可用，则所识别区域中的附加样本允许更好的尾部拟合，从而实现更好的概率估计。在当前的工作中，我们提出 iSOM（可解释的自组织图）来识别设计空间中对应于极值的区域。初始样本用于使用 iSOM 映射（可视化）极限状态函数和随机/设计变量，这允许设计者识别与响应尾部相对应的区域。在识别的感兴趣区域中执行自适应采样以获得额外的样本。接下来，使用初始样本和自适应样本评估响应的累积分布函数（CDF）以进行概率估计。该方法的有效性从其在基准示例、实际工程示例和基于多目标可靠性的设计优化（MORBDO）案例的成功实施中可见一斑。所提出的方法展示了 iSOM 对非线性和多模式特征的极限状态函数执行自适应采样的能力。支持 iSOM 的采样与 log-TPNT 结合使用，可以比单独使用 log-TPNT 更好地估计小故障概率。将所提出方法的结果与最先进（SOTA）采样和基于替代技术的结果进行比较。对于给定数量的极限状态评估，所提出的方法估计 1e−4 阶的概率，与其他 SOTA 方法相比，方差更小。因此，所提出的方法可能会鼓励进一步研究将 iSOM 辅助抽样应用于其他可靠性估计方法。