机器学习材料物理学：基于马氏体微观结构的弹性自由能数据训练的深度神经网络可以高精度地预测均匀化应力场
Machine learning materials physics: Deep neural networks trained on elastic free energy data from martensitic microstructures predict homogenized stress fields with high accuracy
摘要：
我们提出了一种方法来数值均匀化的弹性响应的微结构。我们的工作使用了深度神经网络表示，这些表示是在从马氏体相变的直接数值模拟（DNS）中获得的数据上训练的。导致微观结构的微观模型是基于非凸自由能量密度函数，产生马氏体变体，必须扩展到有限应变下的弹性梯度理论。这些应变梯度引入了界面能，并简化了模型，使大量的解决方案，每个具有精细的层压微观结构的入场。这些DNS解决方案的数值刚度和响应的精细尺度使得获得数据的成本很高，同时也促使人们为了工程设计的目的而寻找其响应的均匀化表示。通过使用标量自由能量密度数据在有效响应上训练深度神经网络（DNN）来降低问题的高维性。我们方法的新颖之处在于，经过训练的DNN还返回了衍生数据的高保真表示，特别是压力。这允许通过DNN表示再现连续弹性的经典超弹性响应。还包括详细的超参数优化研究，以及数据集大小的收敛性。
英文摘要：
We present an approach to numerical homogenization of the elastic response of microstructures. Our work uses deep neural network representations trained on data obtained from direct numerical simulation (DNS) of martensitic phase transformations. The microscopic model leading to the microstructures is based on non-convex free energy density functions that give rise to martensitic variants, and must be extended to gradient theories of elasticity at finite strain. These strain gradients introduce interfacial energies as well as coercify the model, enabling the admission of a large number of solutions, each having finely laminated microstructures. The numerical stiffness of these DNS solutions and the fine scales of response make the data expensive to obtain, while also motivating the search for homogenized representations of their response for the purpose of engineering design. The high-dimensionality of the problem is reduced by training deep neural networks (DNNs) on the effective response by using the scalar free energy density data. The novelty in our approach is that the trained DNNs also return high-fidelity representations of derivative data, specifically the stresses. This allows the recapitulation of the classic hyperelastic response of continuum elasticity via the DNN representation. Also included are detailed optimization studies over hyperparameters, and convergence with size of datasets.
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