Polynomial chaos expansions (PCEs) have been widely employed to estimate failure probabilities in geotechnical engineering. However, PCEs suffer from two deficiencies: (a) PCE coefficients are solved by the least‐square minimization method which easily causes overfitting issues; (b) building a high order PCE is often computationally expensive. In order to overcome the aforementioned drawbacks, the Bayesian regression technique is employed to evaluate PCE coefficients, which not only provides a sparse solution but also avoids overfitting. With the aid of the predictive means and variances given by Bayesian analysis, a learning function is proposed to sequentially select the most informative samples that are critical to build a PCE. This sequential learning scheme can highly enhance the computational efficiency of PCEs. Besides, importance sampling (IS) is incorporated into the sequential learning (SL)‐PCEs to deal with geotechnical problems with small failure probabilities. The proposed method of SL‐PCE‐IS is applied to three illustrative examples, which shows that the improved PCE method is more effective and efficient than the common PCEs method, leading to accurate estimations of small failure probabilities using fewer training samples.

中文翻译：

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使用贝叶斯回归的连续稀疏多项式混沌展开进行岩土工程可靠度估算

多项式混沌扩展（PCE）已被广泛用于估算岩土工程中的失效概率。但是，PCE存在两个缺陷：（a）PCE系数通过最小二乘最小化方法求解，这很容易导致过度拟合问题；（b）建立高阶PCE通常在计算上昂贵。为了克服上述缺点，采用贝叶斯回归技术评估PCE系数，这不仅提供了稀疏的解决方案，而且避免了过拟合。借助贝叶斯分析给出的预测性均值和方差，提出了一种学习功能，以依次选择对构建PCE至关重要的信息量最大的样本。这种顺序学习方案可以大大提高PCE的计算效率。除了，重要性采样（IS）被合并到顺序学习（SL）-PCE中，以处理故障概率较小的岩土问题。所提出的SL‐PCE‐IS方法应用于三个说明性示例，这表明改进的PCE方法比普通PCE方法更有效，从而可以使用较少的训练样本来准确估计较小的故障概率。