We present a new neural-network architecture, called the Cholesky-factored symmetric positive definite neural network (SPD-NN), for modeling constitutive relations in computational mechanics. Instead of directly predicting the stress of the material, the SPD-NN trains a neural network to predict the Cholesky factor of the tangent stiffness matrix, based on which the stress is calculated in incremental form. As a result of this special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies the second order work criterion (Hill's criterion) and time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations. Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs. We demonstrate the effectiveness of SPD-NN on hyperelastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of SPD-NN make it a promising tool for a wide range of constitutive modeling applications.

我们提出了一种新的神经网络架构，称为Cholesky分解对称正定神经网络（SPD-NN），用于对计算力学中的本构关系进行建模。SPD-NN代替直接预测材料的应力，而是训练神经网络来预测切线刚度矩阵的Cholesky因子，并以此为基础以增量形式计算应力。由于这种特殊的结构，SPD-NN对应变能函数弱加了凸度，满足了路径依赖材料的二阶工作准则（希尔准则）和时间一致性，因此提高了数值稳定性，特别是当SPD- NN用于有限元模拟。根据可用数据的类型，我们提出两种训练方法，即直接训练应变和应力对，间接训练负荷和位移对。我们从固体力学论证了SPD-NN在超弹性，弹塑性和多尺度纤维增强板问题上的有效性。SPD-NN的通用性和鲁棒性使其成为各种本构模型应用程序的有前途的工具。