Recent developments in surrogate construction predominantly focused on two strategies to improve surrogate accuracy. Firstly, component-wise domain scaling informed by cross-validation. Secondly, regression to construct response surfaces using additional information in the form of additional function-values sampled from multi-fidelity models and gradients. Component-wise domain scaling reliably improves the surrogate quality at low dimensions but has been shown to suffer from high computational costs for higher dimensional problems. The second strategy, adding gradients to train surrogates, typically results in regression surrogates. Counter-intuitively, these gradient-enhanced regression-based surrogates do not exhibit improved accuracy compared to surrogates only interpolating function values. This study empirically establishes three main findings. Firstly, constructing the surrogate in poorly scaled domains is the predominant cause of deteriorating response surfaces when regressing with additional gradient information. Secondly, surrogate accuracy improves if the surrogates are constructed in a fully transformed domain, by scaling and rotating the original domain, not just simply scaling the domain. The domain transformation scheme should be based on the local curvature of the approximation surface and not its global curvature. Thirdly, the main benefit of gradient information is to efficiently determine the (near) optimal domain in which to construct the surrogate. This study proposes a foundational transformation algorithm that performs near-optimal transformations for lower dimensional problems. The algorithm consistently outperforms cross-validation-based component-wise domain scaling for higher dimensional problems. A carefully selected test problem set that varies between 2 and 16-dimensional problems is used to clearly demonstrate the three main findings of this study.

翻译：近期代理模型构建的发展主要集中在两种提升代理精度的策略上：其一，基于交叉验证的分量式域缩放；其二，利用从多保真模型采样的附加函数值与梯度等额外信息构建响应面的回归方法。分量式域缩放能在低维问题中可靠提升代理质量，但在高维问题中计算成本显著升高。第二种策略通过添加梯度训练代理模型，通常生成回归型代理模型。出乎意料的是，此类梯度增强回归代理模型相较于仅插值函数值的代理模型并未展现出更高的精度。本研究通过实验验证三项核心发现：其一，在尺度不良的域中构建代理模型是导致结合额外梯度信息进行回归时响应面劣化的主要原因；其二，若在完全变换域（对原始域进行缩放与旋转，而非简单缩放）中构建代理模型，其精度将得到提升，且域变换方案应基于近似曲面的局部曲率而非全局曲率；其三，梯度信息的主要价值在于高效确定构建代理模型的（近）最优域。本文提出一种基础性变换算法，可针对低维问题执行近最优变换。对于高维问题，该算法始终优于基于交叉验证的分量式域缩放方法。通过精心选取的2至16维测试问题集，本研究清晰验证了上述三项核心发现。