We develop a new neural network architecture that strictly enforces constitutive constraints such as polyconvexity, frame-indifference, and the symmetry of the stress and material stiffness. Additionally, we show that the accuracy of the stress and material stiffness predictions is significantly improved for this neural network by using a Sobolev minimization strategy that includes derivative terms. Using our neural network, we model the constitutive behavior of fibrous-type discrete network material. With Sobolev minimization, we obtain a normalized mean square error of 0.15% for the strain energy density, 0.815% averaged across the components of the stress, and 5.4% averaged across the components of the stiffness tensor. This machine-learned constitutive model was deployed in a finite element simulation of a facet capsular ligament. The displacement fields and stress-strain curves were compared to a multiscale simulation that required running on a GPU-based supercomputer. The new approach maintained upward of 85% accuracy in stress up to 70% strain while reducing the computation cost by orders of magnitude.

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，如多凸性、框架无关性以及应力和材料刚度的对称性。此外，我们证明，通过采用包含导数项的Sobolev最小化策略，该神经网络的应力和材料刚度预测精度显著提高。利用我们的神经网络，我们对纤维型离散网络材料的本构行为进行了建模。采用Sobolev最小化方法，我们获得了应变能密度归一化均方误差为0.15%，应力分量平均归一化均方误差为0.815%，刚度张量分量平均归一化均方误差为5.4%。这种机器学习得到的本构模型被应用于关节囊韧带的有限元模拟中。其位移场和应力-应变曲线与需要在GPU超级计算机上运行的多尺度模拟结果进行了比较。新方法在高达70%的应变下保持了85%以上的应力预测精度，同时将计算成本降低了数个数量级。