Integrating small data and shape prior knowledge with gradient-enhanced Kriging through adaptive knowledge sampling

Integrating prior knowledge into surrogate models constitutes an advanced approach to tackling the challenges posed by small data. The existing gradient-enhanced Kriging (GEK) method can enhance accuracy by incorporating gradient points of shape prior knowledge. Nonetheless, our findings indicate that the strategy employed for setting gradient points significantly impacts accuracy, even when the number of gradient points remains constant. To enhance the efficient utilization of gradient information, we propose a novel Adaptive Knowledge Sampling (AKS) strategy with an innovative acquisition function that balances local exploitation and global exploration, effectively extracting gradient points from shape prior knowledge. The acquisition function consists of three parts: (1) The Local Information Filling Criterion (LIFC), designed to mitigate model error; (2) The Global Information Filling Criterion (GIFC), aimed at reducing model uncertainty in data-sparse areas; and (3) The Minimum Distance Constraint (MDC), implemented to prevent excessive clustering of gradient points. To strike a balance between model accuracy and computational cost, an adaptive stopping condition is introduced to regulate the number of gradient points sampled. The impact of hyperparameters in the AKS strategy was analyzed in detail, and the method's performance was compared to existing approaches using six benchmark functions and a missile aerodynamic prediction case. The results demonstrate that AKS-GEK offers higher prediction accuracy and robustness when integrating small data with shape prior knowledge.