Surrogate modeling for probability distribution estimation: Uniform or adaptive design?

The active learning (AL) technique, one of the state-of-the-art methods for constructing surrogate models, has shown high accuracy and efficiency in forward uncertainty quantification (UQ) analysis. This paper provides a comprehensive study on AL-based global surrogates for computing the full distribution function, i.e., the cumulative distribution function (CDF) and the complementary CDF (CCDF). To this end, we investigate the three essential components for building surrogates, i.e., types of surrogate models, enrichment methods for experimental designs, and stopping criteria. For each component, we choose several representative methods and study their desirable configurations. In addition, we use a uniform design based on maximin-distance criteria as a baseline for measuring the improvement of using AL. Combining all the representative methods, a total of 1920 UQ analyses are carried out to solve 16 benchmark examples. The performance of the selected strategies is evaluated based on accuracy and efficiency. In the context of full distribution estimation, this study concludes that (i) The benefit of using AL is lower than expected and varies across different surrogate models, with three reasons for this performance variability analyzed in detail. (ii) Detailed recommendations are provided for the three surrogate components, depending on the features of the problems (especially the local nonlinearity), target accuracy, and computational budget.