基于Voronoi基本区八元数框架的任意晶粒边界五自由度性质插值
Five Degree-of-Freedom Property Interpolation of Arbitrary Grain Boundaries via Voronoi Fundamental Zone Octonion Framework
摘要：
我们介绍了Voronoi基本区八元插值框架的晶界（GB）的结构-性能模型和替代品。VFZO框架提供了优于其他五个自由度为基础的属性插值方法，因为它被构造为一个点集在一个流形。这意味着直接计算的欧几里德距离近似于原始的八元数距离，并显著减少了计算运行时间（对于50000 x50000的成对距离矩阵，约7 CPU分钟vs 153 CPU天）。这种提高的效率通过使用显著更多的输入数据来促进更低的插值误差。我们展示了晶界能量插值结果的非光滑验证功能和模拟双晶体数据集的Fe和Ni使用四种插值方法：重心插值，高斯过程回归（GPR），逆距离加权，最近邻插值。针对50000个随机输入GB和10000个随机预测GB进行评估。最好的性能是实现与GPR，这导致在减少均方根误差（RMSE）的83.0%相对于一个恒定的，平均模型的RMSE。同样，在一个大的，嘈杂的，分子静态Fe模拟数据集上的插值将性能提高了34.4%，而在以前的工作中为21.2%。在小的、低噪声的MS Ni模拟数据集上的插值与原始八元数度量的插值结果相似（57.6%对56.4%）。我们的VFZO库（github.com/sgbaird-5dof/interp）中提供了一个矢量化、并行化的MATLAB插值函数（interp5DOF.m）和相关例程，可应用于其他晶体学点群。VFZO框架为计算GB之间的距离、估计任意GB的属性值以及对计算昂贵的5DOF函数和仿真的代理进行建模提供了优势。
英文摘要：
We introduce the Voronoi fundamental zone octonion interpolation framework for grain boundary (GB) structure-property models and surrogates. The VFZO framework offers an advantage over other five degree-of-freedom based property interpolation methods because it is constructed as a point set in a manifold. This means that directly computed Euclidean distances approximate the original octonion distance with significantly reduced computation runtime (~7 CPU minutes vs. 153 CPU days for a 50000x50000 pairwise-distance matrix). This increased efficiency facilitates lower interpolation error through the use of significantly more input data. We demonstrate grain boundary energy interpolation results for a non-smooth validation function and simulated bi-crystal datasets for Fe and Ni using four interpolation methods: barycentric interpolation, Gaussian process regression (GPR), inverse-distance weighting, and nearest-neighbor interpolation. These are evaluated for 50000 random input GBs and 10 000 random prediction GBs. The best performance was achieved with GPR, which resulted in a reduction of the root mean square error (RMSE) by 83.0% relative to RMSE of a constant, average model. Likewise, interpolation on a large, noisy, molecular statics Fe simulation dataset improves performance by 34.4% compared to 21.2% in prior work. Interpolation on a small, low-noise MS Ni simulation dataset is similar to interpolation results for the original octonion metric (57.6% vs. 56.4%). A vectorized, parallelized, MATLAB interpolation function (interp5DOF.m) and related routines are available in our VFZO repository (github.com/sgbaird-5dof/interp) which can be applied to other crystallographic point groups. The VFZO framework offers advantages for computing distances between GBs, estimating property values for arbitrary GBs, and modeling surrogates of computationally expensive 5DOF functions and simulations.
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