AI 中文总结
针对有限训练样本下不确定性量化问题，提出梯度增强样条维分解（GE-SDD）方法，利用函数值和偏导数训练代理，通过对角行加权矩阵等技术求解平衡系统，实验表明该方法比标准SDD和梯度增强克里金更准确、稳健。
AI 中文摘要
样条维分解（SDD）代理能有效表示不确定性量化（UQ）中具有局部特征和复杂非线性的高维工程响应。但有限训练数据会使基于函数值的系数估计严重病态。本文提出梯度增强SDD（GE-SDD），利用函数值和偏导数训练代理，用对角行加权矩阵平衡函数和导数块，通过概率加权Sobolev坐标下的岭回归求解平衡系统，用分组K折交叉验证选正则化参数。在二维连续指数函数、含三个不确定参数的线性动力系统和30维25杆桁架上评估，GE-SDD比标准SDD更准确，比梯度增强克里金更稳健地使用梯度。在非光滑基准上GE-SDD中位数NRMSE为1.022%，而克里金为8.731%。对于桁架，在中等及以上训练规模时，GE-SDD比克里金有更低NRMSE和更准确标准差估计。梯度增强的好处取决于输入维度、基分辨率、训练规模和目标UQ量。
英文摘要
A spline dimensional decomposition (SDD) surrogate effectively represents high-dimensional engineering responses with localized features and complex nonlinearities in uncertainty quantification (UQ). However, limited training data can make coefficient estimation from function values severely ill-conditioned. We propose gradient-enhanced SDD (GE-SDD), which trains the surrogate using function values and partial derivatives. A diagonal row-weight matrix balances the function and derivative blocks by their Frobenius norms. We solve the balanced system through ridge regression in probability-weighted Sobolev coordinates and select the regularization parameter using grouped K-fold cross-validation to prevent information leakage. Mapping the solution back to the L2-orthonormal SDD basis preserves closed-form mean and variance estimates. We evaluate the proposed GE-SDD on a two-dimensional continuous exponential function, a linear dynamical system with three uncertain parameters, and a 30-dimensional 25-bar truss. GE-SDD is more accurate than standard SDD and uses gradients more robustly than gradient-enhanced Kriging. GE-SDD achieves a median NRMSE of 1.022% on the nonsmooth benchmark, compared with 8.731% for Kriging. For the truss, GE-SDD yields lower NRMSE and more accurate standard-deviation estimates than Kriging at moderate training sizes and above. Overall, the benefits of gradient augmentation depend on input dimension, basis resolution, training size, and the target UQ quantity.
