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The estimation of rare event probabilities plays a pivotal role in diversefields. Our aim is to determine the probability of a hazard or system failureoccurring when a quantity of interest exceeds a critical value. In ourapproach, the distribution of the quantity of interest is represented by anenergy density, characterized by a free energy function. To efficientlyestimate the free energy, a bias potential is introduced. Using concepts fromenergy-based models (EBM), this bias potential is optimized such that thecorresponding probability density function approximates a pre-defineddistribution targeting the failure region of interest. Given the optimal biaspotential, the free energy function and the rare event probability of interestcan be determined. The approach is applicable not just in traditional rareevent settings where the variable upon which the quantity of interest relieshas a known distribution, but also in inversion settings where the variablefollows a posterior distribution. By combining the EBM approach with a Steindiscrepancy-based stopping criterion, we aim for a balanced accuracy-efficiencytrade-off. Furthermore, we explore both parametric and non-parametricapproaches for the bias potential, with the latter eliminating the need forchoosing a particular parameterization, but depending strongly on the accuracyof the kernel density estimate used in the optimization process. Through threeillustrative test cases encompassing both traditional and inversion settings,we show that the proposed EBM approach, when properly configured, (i) allowsstable and efficient estimation of rare event probabilities and (ii) comparesfavorably against subset sampling approaches.

An energy-based model approach to rare event probability estimation