A FEM dataset of Ge film profiles and elastic energies for machine learning approximation of strain state and morphological evolution

Machine Learning (ML) can be conveniently applied to continuum materials simulations, allowing for the investigation of larger systems and longer timescales, pushing the limits of tractable systems. Here we provide a comprehensive dataset of strained Ge films on Si and their corresponding strain states, which can be used to train a ML model capable of such acceleration. Approximately 80k 2D cases are included, reporting the profiles h(x) and the corresponding elastic energy densities and strain fields. The profiles are conveniently sampled using Perlin-noise and pure-sine waves. A 100nm-large computational domain is considered. The mechanical equilibrium problem is solved using Finite Element Method (FEM). Ge is modeled as an isotropic material and an eigenstrain of 3.99% is used, as in Ge/Si(001).
The database has been exploited for training a (fully) Convolutional Neural Network (CNN) which maps the free surface profile h(x) to the corresponding energy density. If plugged into the proper time-dependent Partial Differential Equation, this term can be used to accelerate continuum simulations of the morphological evolution of strained films while retaining FEM-level accuracy. Tests of the reliability of such CNN model are also provided in the repository, together with the output of surface morphology minimization procedures and morphological evolution simulations during coarsening and growth. In the latter, evolution by surface diffusion has been considered as an important case, but applications to other mechanisms are possible. Generalization examples to larger computational cells with respect to those in the dataset are also available.

机器学习（ML）在连续介质材料模拟中得以便捷地应用，这不仅促进了对于更大系统规模和更长时间尺度的探究，亦拓展了可处理系统的边界。本数据集提供了在硅（Si）衬底上生长的应变锗（Ge）薄膜及其相应的应变状态的全面集锦，可用于训练具备此类加速功能的机器学习模型。数据集包含约80,000个二维案例，报告了h(x)轮廓、相应的弹性能量密度和应变场。轮廓采样采用Perlin噪声和纯正弦波进行。考虑的计算域尺度为100纳米。采用有限元方法（FEM）解决力学平衡问题。锗（Ge）被建模为各向同性材料，并使用了3.99%的本征应变，与Ge/Si(001)情况相吻合。该数据库已被用于训练一个（全）卷积神经网络（CNN），该网络将自由表面轮廓h(x)映射到相应的能量密度。若将此结果输入适当的时间相关偏微分方程，则可用于加速应变薄膜形态演化的连续介质模拟，同时保持有限元级别的精度。此外，还提供了该CNN模型可靠性的测试，包括表面形态最小化过程和形态演化模拟的输出，在后者中，表面扩散导致的演化被视为一个重要案例，但该应用亦适用于其他机制。针对数据集中更大计算单元的泛化示例也一并提供。