Many problems in computational materials science and chemistry require the evaluation of expensive functions with locally rapid changes, such as the turn-over frequency of first principles kinetic Monte Carlo models for heterogeneous catalysis. Because of the high computational cost, it is often desirable to replace the original with a surrogate model, e.g., for use in coupled multiscale simulations. The construction of surrogates becomes particularly challenging in high-dimensions. Here, we present a novel version of the modified Shepard interpolation method which can overcome the curse of dimensionality for such functions to give faithful reconstructions even from very modest numbers of function evaluations. The introduction of local metrics allows us to take advantage of the fact that, on a local scale, rapid variation often occurs only across a small number of directions. Furthermore, we use local error estimates to weigh different local approximations, which helps avoid artificial oscillations. Finally, we test our approach on a number of challenging analytic functions as well as a realistic kinetic Monte Carlo model. Our method not only outperforms existing isotropic metric Shepard methods but also state-of-the-art Gaussian process regression.

中文翻译：

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基于局部度量误差的Shepard插值作为高维高度非线性材料模型的替代

计算材料科学和化学中的许多问题都需要评估具有局部快速变化的昂贵函数，例如用于非均相催化的第一原理动力学蒙特卡洛模型的转换频率。由于高计算成本，通常需要用替代模型代替原始模型，例如，用于耦合多尺度模拟。在高维中，替代物的构造变得特别具有挑战性。在这里，我们提出了一种改进的Shepard插值方法的新颖版本，它可以克服维数的诅咒这样的功能即使从数量很少的功能评估中也能忠实地进行重构。引入本地指标使我们可以利用以下事实：在本地范围内，快速变化通常仅在少数几个方向上发生。此外，我们使用局部误差估计来权衡不同的局部近似值，这有助于避免人为振荡。最后，我们在许多具有挑战性的分析函数以及现实的动力学蒙特卡洛模型上测试了我们的方法。我们的方法不仅优于现有的各向同性度量Shepard方法，而且还具有最新的高斯过程回归能力。