Estimating failure probability is a key task in the field of uncertainty quantification. In this domain, importance sampling has proven to be an effective estimation strategy; however, its efficiency heavily depends on the choice of the biasing distribution. An improperly selected biasing distribution can significantly increase estimation error. One approach to address this challenge is to leverage a less expensive, lower-fidelity surrogate. Building on the accessibility to such a model and its derivative on the random uncertain inputs, we introduce an importance sampling-based estimator, termed the Langevin bi-fidelity importance sampling (L-BF-IS), which uses score-function-based sampling algorithms to generate new samples and substantially reduces the mean square error (MSE) of failure probability estimation. The proposed method demonstrates lower estimation error, especially in high-dimensional input spaces and when limited high-fidelity evaluations are available. The L-BF-IS estimator's effectiveness is validated through experiments with two synthetic functions and two real-world applications governed by partial differential equations. These real-world applications involve a composite beam, which is represented using a simplified Euler-Bernoulli equation as a low-fidelity surrogate, and a steady-state stochastic heat equation, for which a pre-trained neural operator serves as the low-fidelity surrogate.

翻译：失效概率估计是不确定性量化领域的一项关键任务。在该领域中，重要性采样已被证明是一种有效的估计策略；然而，其效率在很大程度上取决于偏置分布的选择。选择不当的偏置分布会显著增加估计误差。应对这一挑战的一种方法是利用成本较低、保真度较低的代理模型。基于对此类模型及其对随机不确定输入导数的可访问性，我们引入了一种基于重要性采样的估计器，称为Langevin双保真度重要性采样（L-BF-IS）。该方法利用基于得分函数的采样算法生成新样本，并显著降低了失效概率估计的均方误差（MSE）。所提出的方法表现出更低的估计误差，特别是在高维输入空间以及可用的高保真度评估有限的情况下。L-BF-IS估计器的有效性通过两个合成函数和两个由偏微分方程控制的实际应用实验得到了验证。这些实际应用包括一个复合材料梁（使用简化的欧拉-伯努利方程作为低保真度代理模型进行表示）和一个稳态随机热方程（使用预训练的神经算子作为其低保真度代理模型）。