The estimation of rare event probabilities plays a pivotal role in diverse fields. Our aim is to determine the probability of a hazard or system failure occurring when a quantity of interest exceeds a critical value. In our approach, the distribution of the quantity of interest is represented by an energy density, characterized by a free energy function. To efficiently estimate the free energy, a bias potential is introduced. Using concepts from energy-based models (EBM), this bias potential is optimized such that the corresponding probability density function approximates a pre-defined distribution targeting the failure region of interest. Given the optimal bias potential, the free energy function and the rare event probability of interest can be determined. The approach is applicable not just in traditional rare event settings where the variable upon which the quantity of interest relies has a known distribution, but also in inversion settings where the variable follows a posterior distribution. By combining the EBM approach with a Stein discrepancy-based stopping criterion, we aim for a balanced accuracy-efficiency trade-off. Furthermore, we explore both parametric and non-parametric approaches for the bias potential, with the latter eliminating the need for choosing a particular parameterization, but depending strongly on the accuracy of the kernel density estimate used in the optimization process. Through three illustrative test cases encompassing both traditional and inversion settings, we show that the proposed EBM approach, when properly configured, (i) allows stable and efficient estimation of rare event probabilities and (ii) compares favorably against subset sampling approaches.

翻译：罕见事件概率的估计在多个领域中发挥着关键作用。我们的目标是确定当感兴趣的量超过临界值时，发生灾害或系统故障的概率。在我们的方法中，感兴趣量的分布由能量密度表示，并通过自由能函数进行表征。为了高效估计自由能，引入了偏置势。利用基于能量的模型的概念，该偏置势经过优化，使得相应的概率密度函数逼近针对感兴趣失效区域的预定义分布。给定最优偏置势后，可以确定自由能函数和感兴趣的罕见事件概率。该方法不仅适用于传统罕见事件场景（其中感兴趣量所依赖的变量具有已知分布），也适用于反演场景（其中变量遵循后验分布）。通过将基于能量的模型方法与基于斯坦因差异的停止准则相结合，我们旨在实现精度与效率的平衡。此外，我们探讨了偏置势的参数化和非参数化方法，后者无需选择特定的参数化形式，但强烈依赖于优化过程中所使用的核密度估计的精度。通过三个涵盖传统和反演场景的说明性测试案例，我们证明了所提出的基于能量的模型方法在适当配置下，（i）能够稳定高效地估计罕见事件概率，（ii）与子集采样方法相比具有优势。