A Surrogate-Informed Framework for Sparse Grid Interpolation

TL;DR

The paper tackles the cost of high-dimensional, expensive simulations by introducing a surrogate-informed refinement framework for sparse grid interpolation. It uses a zero-cost error indicator, based on the discrepancy between two nested surrogates at levels $w$ and $w-1$ evaluated on candidate points from level $w+1$, to rank refinement points and construct a final model that evaluates the true function only at high-impact points. The methodology includes two equivalent formulations—surrogate-informed addition and error-correction superposition—proven equivalent for nested polynomial sparse grids. Empirical tests on Ishigami, Sobol G, and an oscillatory function, plus a hydrogen-fueled burner case study, show that the approach achieves near $w+1$ accuracy with substantially fewer expensive evaluations, improving efficiency for uncertainty quantification and optimization in high-dimensional settings.

Abstract

Approximating complex, high-dimensional, and computationally expensive functions is a central problem in science and engineering. Standard sparse grids offer a powerful solution by mitigating the curse of dimensionality compared to full tensor grids. However, they treat all regions of the domain isotropically, which may not be efficient for functions with localized or anisotropic behavior. This work presents a surrogate-informed framework for constructing sparse grid interpolants, which is guided by an error indicator that serves as a zero-cost estimate for the hierarchical surplus. This indicator is calculated for all candidate points, defined as those in the next-level grid $w+1$ not already present in the base grid $w$. It quantifies the local approximation error by measuring the relative difference between the predictions of two consecutive interpolants of level $w$ and $w-1$. The candidates are then ranked by this metric to select the most impactful points for refinement up to a given budget or following another criterion, as, e.g., a given threshold in the error indicator. The final higher-order model is then constructed using a surrogate-informed approach: the objective function is evaluated only at the selected high-priority points, while for the remaining nodes of the $w+1$ grid, we assign the values predicted by the initial $w$-level surrogate. This strategy significantly reduces the required number of expensive evaluations, yielding a final model that closely approximates the accuracy of a fully-resolved $w+1$ grid at a fraction of the computational cost. The accuracy and efficiency of the proposed surrogate-informed refinement criterion is demonstrated for several analytic function and for a real engineering problem, i.e., the analysis of sensitivity to geometrical parameters of numerically predicted flashback phenomenon in hydrogen-fueled perforated burners.

Figures (22)

• Figure 1: Comparison of the error on 50 testing points for the $w=2$ and surrogate informed sparse grid fot the Ishigami's function
• Figure 2: Histogram of the error between the baseline and the informed model
• Figure 3: Response surface comparison for the Ishigami's function at $x_3=\pi$.
• Figure 4: Comparison between the baseline sparse grid, the surrogate-informed one and the true values for the Ishigami's function at $x_1=x_3=\pi$
• Figure 5: Convergence of maximum absolute error and RMSE for the Ishigami's function