Adaptive sampling based estimation of small probability of failure using interpretable Self-Organising Map

Abstract

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.