Abstract We present a machine learning approach that integrates geometric deep learning and Sobolev training to generate a family of finite strain anisotropic hyperelastic models that predict the homogenized responses of polycrystals previously unseen during the training. While hand-crafted hyperelasticity models often incorporate homogenized measures of microstructural attributes, such as the porosity or the averaged orientation of constituents, these measures may not adequately represent the topological structures of the attributes. We fill this knowledge gap by introducing the concept of the weighted graph as a new high-dimensional descriptor that represents topological information, such as the connectivity of anisotropic grains in an assemble. By leveraging a graph convolutional deep neural network in a hybrid machine learning architecture previously used in Frankel et al. (2019), the artificial intelligence extracts low-dimensional features from the weighted graphs and subsequently learns the influence of these low-dimensional features on the resultant stored elastic energy functionals. To ensure smoothness and prevent unintentionally generating a non-convex stored energy functional, we adopt the Sobolev training method for neural networks such that a stress measure is obtained implicitly by taking directional derivatives of the trained energy functional. Results from numerical experiments suggest that Sobolev training is capable of generating a hyperelastic energy functional that predicts both the elastic energy and stress measures more accurately than the classical training that minimizes L 2 norms. Verification exercises against unseen benchmark FFT simulations and phase-field fracture simulations that employ the geometric learning generated elastic energy functional are conducted to demonstrate the quality of the predictions.

中文翻译：

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计算力学的几何深度学习第一部分：各向异性超弹性

摘要 我们提出了一种机器学习方法，该方法将几何深度学习和 Sobolev 训练相结合，以生成一系列有限应变各向异性超弹性模型，该模型可预测以前在训练过程中未见过的多晶的均匀响应。虽然手工制作的超弹性模型通常包含微观结构属性的均匀测量，例如孔隙率或成分的平均方向，但这些测量可能无法充分代表属性的拓扑结构。我们通过引入加权图的概念作为表示拓扑信息的新高维描述符来填补这一知识空白，例如组装中各向异性晶粒的连通性。通过在之前 Frankel 等人使用的混合机器学习架构中利用图卷积深度神经网络。(2019)，人工智能从加权图中提取低维特征，然后学习这些低维特征对结果存储弹性能量泛函的影响。为了确保平滑并防止无意中生成非凸存储能量泛函，我们采用 Sobolev 神经网络训练方法，通过对受训能量泛函取方向导数隐式地获得应力度量。数值实验的结果表明，与最小化 L 2 范数的经典训练相比，Sobolev 训练能够生成超弹性能量函数，该函数可以更准确地预测弹性能量和应力测量值。针对看不见的基准 FFT 模拟和使用几何学习生成的弹性能量函数的相场断裂模拟进行验证练习，以证明预测的质量。