Flow to Rare Events: An Application of Normalizing Flow in Temporal Importance Sampling for Automated Vehicle Validation*

Automated Vehicle (AV) validation based on simulated testing requires unbiased evaluation and high efficiency. One effective solution is to increase the exposure to risky rare events while reweighting the probability measure. However, characterizing the distribution of risky events is particularly challenging due to the paucity of samples and the temporality of continuous scenario variables. To solve it, we devise a method to represent, generate, and reweight the distribution of risky rare events. We decompose the temporal evolution of continuous variables into distribution components based on conditional probability. By introducing the Risk Indicator Function, the distribution of risky rare events is theoretically precipitated out of naturalistic driving distribution. This targeted distribution is practically generated via Normalizing Flow, which achieves exact and tractable probability evaluation of intricate distribution. The rare event distribution is then demonstrated as the advantageous Importance Sampling distribution. We also promote the technique of temporal Importance Sampling. The combined method, named as TrimFlow, is executed to estimate the collision rate of Car-following scenarios as a tentative practice. The results showed that sampling background vehicle maneuvers from rare event distribution could evolve testing scenarios to hazardous states. TrimFlow reduced 86.1% of tests compared to generating testing scenarios according to their exposure in the naturalistic driving environment. In addition, the TrimFlow method is not limited to one specific type of functional scenario.