  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.
