SO(3)-invariance of informed-graph-based deep neural network for anisotropic elastoplastic materials
SO(3)-invariance of informed-graph-based deep neural network for anisotropic elastoplastic materials
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DOI：
10.1016/j.cma.2020.112875
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发表时间：
2020-05-01
影响因子：
7.2
通讯作者：
Sun, WaiChing
中科院分区：
文献类型：
--
作者：
Heider, Yousef;Wang, Kun;Sun, WaiChing
本文研究了经典多层和基于信息图的神经网络的监督机器学习所生成的模拟各向异性弹塑性响应中表现出的框架不变性（及其缺乏），并提出了不同的补救措施来解决这一缺陷。在弹塑性模型中的物理量和状态变量之间的固有的层次关系首先表示为通知，有向图，其中三个变化的图形进行了测试。虽然前馈神经网络用于训练与路径无关的本构关系（例如，弹性），使用递归神经网络来复制依赖于变形历史（即路径依赖）的响应。在处理客观性不足时，我们使用谱形式来表示张量，随后，三个度量，欧拉角之间的欧氏距离，单位矩阵的距离和李代数中单位球面上的测地线，可以用来构成监督机器学习的目标函数。在此，目的是最小化真实和预测的3D旋转实体之间的测量距离。在此之后，我们进行了数值实验，研究这些理论上等价的指标如何导致监督机器学习的效率以及所得模型的准确性和鲁棒性的差异。神经网络模型训练张量表示在组件的形式为一个给定的笛卡尔坐标系被用作基准。我们的数值试验表明，即使在相同的信息量和数据，质量的各向异性弹塑性模型是高度敏感的张量的方式表示和测量。结果表明，使用损失函数的基础上测地线上的单位球面李代数与一个知情的，有向图产生显着更准确的旋转预测比其他测试方法。(C)2020爱思唯尔B.V.保留所有权利。
This paper examines the frame-invariance (and the lack thereof) exhibited in simulated anisotropic elasto-plastic responses generated from supervised machine learning of classical multi-layer and informed-graph-based neural networks, and proposes different remedies to fix this drawback. The inherent hierarchical relations among physical quantities and state variables in an elasto-plasticity model are first represented as informed, directed graphs, where three variations of the graph are tested. While feed-forward neural networks are used to train path-independent constitutive relations (e.g., elasticity), recurrent neural networks are used to replicate responses that depends on the deformation history, i.e. or path dependent. In dealing with the objectivity deficiency, we use the spectral form to represent tensors and, subsequently, three metrics, the Euclidean distance between the Euler Angles, the distance from the identity matrix, and geodesic on the unit sphere in Lie algebra, can be employed to constitute objective functions for the supervised machine learning. In this, the aim is to minimize the measured distance between the true and the predicted 3D rotation entities. Following this, we conduct numerical experiments on how these metrics, which are theoretically equivalent, may lead to differences in the efficiency of the supervised machine learning as well as the accuracy and robustness of the resultant models. Neural network models trained with tensors represented in component form for a given Cartesian coordinate system are used as a benchmark. Our numerical tests show that, even given the same amount of information and data, the quality of the anisotropic elasto-plasticity model is highly sensitive to the way tensors are represented and measured. The results reveal that using a loss function based on geodesic on the unit sphere in Lie algebra together with an informed, directed graph yield significantly more accurate rotation prediction than the other tested approaches. (C) 2020 Elsevier B.V. All rights reserved.