Gradient-enhanced spline dimensional decomposition for uncertainty quantification with limited training samples

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.