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.

翻译：逼近复杂、高维且计算代价高昂的函数是科学与工程领域的核心问题。相比全张量网格，标准稀疏网格通过缓解维度灾难提供了强有力的解决方案。然而，其对各向同性的全域处理方法，对于具有局部化或各向异性特征的函数可能效率不足。本文提出一种基于代理模型信息的稀疏网格插值构造框架，该方法以误差指示器为指导，该指示器可作为分层盈余的零成本估计量。该指示器针对所有候选点（即定义在下一层网格$w+1$中但未出现在基础网格$w$中的点）进行计算，通过度量第$w$层与第$w-1$层两个连续插值预测值的相对差异来量化局部逼近误差。随后依据该指标对候选点进行排序，以根据给定预算或其他准则（如误差指示器的设定阈值）选取对网格细化最具影响的点。最终的高阶模型采用代理信息驱动方法构建：仅对选定的高优先级点进行目标函数求值，而对于$w+1$网格中剩余的节点，则赋予初始$w$层代理模型的预测值。该策略显著减少了昂贵函数求值的次数，最终获得的模型能以极小的计算成本逼近完全解析的$w+1$网格精度。本文通过多个解析函数实例及实际工程问题（即氢燃料多孔燃烧器中数值预测回火现象对几何参数的敏感性分析）验证了所提代理信息驱动细化准则的精确性与高效性。