Supervised kernel principal component analysis-polynomial chaos-Kriging for high-dimensional surrogate modelling and optimization

Surrogate-based optimization (SBO) approach is becoming more and more popular in the expensive aerodynamic design of aircraft. However, with increasing number of design variables required for parameterizing a complex shape, SBO is suffering from the serious difficulty of the curse of dimensionality. To ameliorate this issue, a supervised nonlinear dimensionality-reduction surrogate modelling method was proposed. Such a method combines the supervised kernel principal component analysis (SKPCA) and polynomial chaos-Kriging (PCK) techniques into the jointly surrogate modelling process and adaptively establishes the accurate mapping from the high-dimensional inputs to the output of the system. This SKPCA-PCK method, which fully considers the effect of inputs on outputs and adaptively trains these hyper-parameters in the surrogate modelling process, escapes from the low prediction accuracy and instability of the surrogate model in conjunction with current linear or unsupervised dimensionality-reduction methods. Further, an efficient SKPCA-PCK-based global optimization method for high-dimensional aerodynamic design was developed. The performance of the proposed method is examined by investigating two numerical examples, the transonic RAE2822 airfoil and the wing of the NASA Common Research Model. Results demonstrate that the proposed SKPCA-PCK method significantly improves the modelling efficiency and accuracy compared to the unsupervised linear PCA-Kriging method. More importantly, the proposed SKPCA-PCK-based optimization method provides better performance and an appreciably higher optimization efficiency for expensive single-point and robust aerodynamic design involving high-dimensional design variables compared to the Kriging-based optimization method. These results provide further evidence that the proposed method provides a promising approach for mitigating the curse of dimensionality in SBO.