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In this work, we develop an importance sampling estimator by coupling thereduced-order model and the generative model in a problem setting ofuncertainty quantification. The target is to estimate the probability that thequantity of interest (QoI) in a complex system is beyond a given threshold. Toavoid the prohibitive cost of sampling a large scale system, the reduced-ordermodel is usually considered for a trade-off between efficiency and accuracy.However, the Monte Carlo estimator given by the reduced-order model is biaseddue to the error from dimension reduction. To correct the bias, we still needto sample the fine model. An effective technique to reduce the variancereduction is importance sampling, where we employ the generative model toestimate the distribution of the data from the reduced-order model and use itfor the change of measure in the importance sampling estimator. To compensatethe approximation errors of the reduced-order model, more data that induce aslightly smaller QoI than the threshold need to be included into the trainingset. Although the amount of these data can be controlled by a posterior errorestimate, redundant data, which may outnumber the effective data, will be keptdue to the epistemic uncertainty. To deal with this issue, we introduce aweighted empirical distribution to process the data from the reduced-ordermodel. The generative model is then trained by minimizing the cross entropybetween it and the weighted empirical distribution. We also introduce a penaltyterm into the objective function to deal with the overfitting for morerobustness. Numerical results are presented to demonstrate the effectiveness ofthe proposed methodology.

Coupling the reduced-order model and the generative model for an importance sampling estimator