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.

翻译：工程中普遍存在高维复杂多参数问题，传统针对低/中维问题设计的代理模型难以应对此类场景。这些模型面临维数灾难，随着设计参数空间扩展，建模精度不断下降。此外，由于缺乏参数解耦机制，难以识别设计变量之间的耦合关系，尤其在高度非线性情况下更为突出。为解决上述挑战并提升预测精度、降低样本需求，本文在Cut-HDMR框架下提出一种PC-Kriging-HDMR近似建模方法。该方法利用PC-Kriging的精度优势，通过多阶段自适应序贯采样策略优化测试点布局。该策略包含第一阶段的自适应比例采样准则与第二阶段基于中心的最大熵准则。数值试验及悬臂梁实际应用案例验证了所提方法的优势。关键发现包括：(1) 在高维非线性问题中，传统单一代理模型（如Kriging）的性能相较于Cut-HDMR框架下的组合代理模型（如Kriging-HDMR、PCE-HDMR、SVR-HDMR、MLS-HDMR及PC-Kriging-HDMR）显著退化；(2) PC-Kriging-HDMR建模所需样本量随参数空间扩展呈多项式增长而非指数增长，大幅降低计算成本；(3) 现有Cut-HDMR方法中，尚无单一方法在所有方面均表现优异。然而，PC-Kriging-HDMR在目标改进范围内相比PCE-HDMR和Kriging-HDMR展现出更优的建模精度与效率，且具有良好的鲁棒性。