We develop a new neural network architecture that strictly enforces constitutive constraints such as polyconvexity, frame-indifference, zero strain energy with zero deformations, and the symmetry of the stress and material stiffness. Additionally, we show that for this neural network, the accuracy is significantly improved by using a Sobolev minimization strategy that includes derivative terms. Using our network and Sobolev minimization, we obtain a NMSE of 0.15% for the energy, 0.815% averaged across the components of the stress, and 5.4% averaged across the components of the stiffness. This machine learned constitutive model was deployed in a finite element simulation of a facet capsular ligament. The displacement fields and stress-strain curves where compared to a multiscale simulation that required running on a GPU based supercomputer. At 70% strain, the model using the neural network had less than 10% relative error in the mean stress value.

翻译：我们开发了一种新型神经网络架构，该架构严格强制执行本构约束条件，包括多凸性、框架无关性、零应变与零变形下的能量为零，以及应力与材料刚度的对称性。此外，我们证明对该神经网络采用包含导数项的Sobolev最小化策略可显著提高精度。运用该网络与Sobolev最小化方法，我们获得了0.15%的能量归一化均方误差（NMSE）、应力分量平均0.815%的误差，以及刚度分量平均5.4%的误差。该机器学习本构模型被部署在关节囊韧带的有限元仿真中，并将位移场与应力-应变曲线与基于GPU超级计算机运行的多尺度仿真结果进行对比。在70%应变条件下，采用神经网络的模型在平均应力值上的相对误差低于10%。