A PC-Kriging-HDMR integrated with an adaptive sequential sampling strategy for high-dimensional approximate modeling

TL;DR

This paper introduces a PC-Kriging-HDMR surrogate modeling method with an adaptive sampling strategy to efficiently handle high-dimensional nonlinear problems, significantly reducing computational costs while improving accuracy.

Contribution

It proposes a novel PC-Kriging-HDMR approach combined with a multi-stage adaptive sampling strategy for high-dimensional modeling, addressing the curse of dimensionality and variable coupling issues.

Findings

PC-Kriging-HDMR outperforms traditional single-surrogate models in high-dimensional nonlinear problems.

Sample requirements grow polynomially, not exponentially, with increasing dimensions, reducing computational costs.

The method shows superior accuracy and robustness compared to other Cut-HDMR approaches in numerical tests and practical applications.

Abstract

High-dimensional complex multi-parameter problems are prevalent in engineering, exceeding the capabilities of traditional surrogate models designed for low/medium-dimensional problems. These models face the curse of dimensionality, resulting in decreased modeling accuracy as the design parameter space expands. Furthermore, the lack of a parameter decoupling mechanism hinders the identification of couplings between design variables, particularly in highly nonlinear cases. To address these challenges and enhance prediction accuracy while reducing sample demand, this paper proposes a PC-Kriging-HDMR approximate modeling method within the framework of Cut-HDMR. The method leverages the precision of PC-Kriging and optimizes test point placement through a multi-stage adaptive sequential sampling strategy. This strategy encompasses a first-stage adaptive proportional sampling criterion and a second-stage central-based maximum entropy criterion. Numerical tests and a practical application involving a cantilever beam demonstrate the advantages of the proposed method. Key findings include: (1) The performance of traditional single-surrogate models, such as Kriging, significantly deteriorates in high-dimensional nonlinear problems compared to combined surrogate models under the Cut-HDMR framework (e.g., Kriging-HDMR, PCE-HDMR, SVR-HDMR, MLS-HDMR, and PC-Kriging-HDMR); (2) The number of samples required for PC-Kriging-HDMR modeling increases polynomially rather than exponentially as the parameter space expands, resulting in substantial computational cost reduction; (3) Among existing Cut-HDMR methods, no single approach outperforms the others in all aspects. However, PC-Kriging-HDMR exhibits improved modeling accuracy and efficiency within the desired improvement range compared to PCE-HDMR and Kriging-HDMR, demonstrating robustness.