We present a data-driven framework for the multiscale modeling of anisotropic finite strain elasticity based on physics-augmented neural networks (PANNs). Our approach allows the efficient simulation of materials with complex underlying microstructures which reveal an overall anisotropic and nonlinear behavior on the macroscale. By using a set of invariants as input, an energy-type output and by adding several correction terms to the overall energy density functional, the model fulfills multiple physical principles by construction. The invariants are formed from the right Cauchy-Green deformation tensor and fully symmetric 2nd, 4th or 6th order structure tensors which enables to describe a wide range of symmetry groups. Besides the network parameters, the structure tensors are simultaneously calibrated during training so that the underlying anisotropy of the material is reproduced most accurately. In addition, sparsity of the model with respect to the number of invariants is enforced by adding a trainable gate layer and using lp regularization. Our approach works for data containing tuples of deformation, stress and material tangent, but also for data consisting only of tuples of deformation and stress, as is the case in real experiments. The developed approach is exemplarily applied to several representative examples, where necessary data for the training of the PANN surrogate model are collected via computational homogenization. We show that the proposed model achieves excellent interpolation and extrapolation behaviors. In addition, the approach is benchmarked against an NN model based on the components of the right Cauchy-Green deformation tensor.

翻译：本文提出了一种基于物理增强神经网络的数据驱动多尺度建模框架，用于各向异性有限应变弹性力学分析。该方法能够高效模拟具有复杂微观结构的材料，这些材料在宏观尺度上表现出整体各向异性和非线性行为。通过采用一组不变量作为输入、以能量型函数作为输出，并在整体能量密度泛函中添加若干修正项，该模型从构造上满足了多重物理原理。这些不变量由右柯西-格林变形张量与完全对称的二阶、四阶或六阶结构张量构成，从而能够描述广泛的对称群类别。除网络参数外，结构张量在训练过程中同步校准，以最精确地复现材料的本征各向异性特性。此外，通过引入可训练门控层并采用lp正则化方法，增强了模型在不变量数量方面的稀疏性。本方法不仅适用于包含变形、应力和材料切线模量数据元组的训练数据，也能处理仅含变形与应力数据元组的实际情况（如真实实验场景）。我们将所提出的方法应用于多个代表性案例，其中物理增强神经网络代理模型训练所需数据通过计算均匀化方法获取。研究表明，该模型展现出优异的插值和外推性能。此外，通过与基于右柯西-格林变形张量分量的神经网络模型进行对比，验证了本方法的优越性。