On Active Learning for Gaussian Process-based Global Sensitivity Analysis

Abstract: This paper explores the application of active learning strategies to adaptively learn Sobol indices for global sensitivity analysis. We demonstrate that active learning for Sobol indices poses unique challenges due to the definition of the Sobol index as a ratio of variances estimated from Gaussian process surrogates. Consequently, learning strategies must either focus on convergence in the numerator or the denominator of this ratio. However, rapid convergence in either one does not guarantee convergence in the Sobol index. We propose a novel strategy for active learning that focuses on resolving the main effects of the Gaussian process (associated with the numerator of the Sobol index) and compare this with existing strategies based on convergence in the total variance (the denominator of the Sobol index). The new strategy, implemented through a new learning function termed the MUSIC (minimize uncertainty in Sobol index convergence), generally converges in Sobol index error more rapidly than the existing strategies based on the Expected Improvement for Global Fit (EIGF) and the Variance Improvement for Global Fit (VIGF). Both strategies are compared with simple sequential random sampling and the MUSIC learning function generally converges most rapidly for low-dimensional problems. However, for high-dimensional problems, the performance is comparable to random sampling. The new learning strategy is demonstrated for a practical case of adaptive experimental design for large-scale Boundary Layer Wind Tunnel experiments.