In the present work, advanced spatial and temporal discretization techniques are tailored to hyperelastic physics-augmented neural networks, i.e., neural network based constitutive models which fulfill all relevant mechanical conditions of hyperelasticity by construction. The framework takes into account the structure of neural network-based constitutive models, in particular, that their derivatives are more complex compared to analytical models. The proposed framework allows for convenient mixed Hu-Washizu like finite element formulations applicable to nearly incompressible material behavior. The key feature of this work is a tailored energy-momentum scheme for time discretization, which allows for energy and momentum preserving dynamical simulations. Both the mixed formulation and the energy-momentum discretization are applied in finite element analysis. For this, a hyperelastic physics-augmented neural network model is calibrated to data generated with an analytical potential. In all finite element simulations, the proposed discretization techniques show excellent performance. All of this demonstrates that, from a formal point of view, neural networks are essentially mathematical functions. As such, they can be applied in numerical methods as straightforwardly as analytical constitutive models. Nevertheless, their special structure suggests to tailor advanced discretization methods, to arrive at compact mathematical formulations and convenient implementations.

翻译：本文针对超弹性物理增强神经网络（即通过构造满足超弹性所有相关力学条件的神经网络本构模型），定制了先进的时空离散化技术。该框架充分考虑了基于神经网络的本构模型的结构特点，特别是其导数比解析模型更为复杂。所提出的框架允许采用便捷的混合Hu-Washizu类有限元公式，适用于近似不可压缩材料行为。本文的主要特色是一种定制的能量-动量方案用于时间离散化，该方案能够实现能量和动量守恒的动态模拟。混合公式和能量-动量离散化均应用于有限元分析中。为此，我们使用解析势函数生成的数据对超弹性物理增强神经网络模型进行了校准。在所有有限元模拟中，所提出的离散化技术均表现出卓越的性能。这表明，从形式角度来看，神经网络本质上是数学函数。因此，它们可以像解析本构模型一样直接应用于数值方法中。尽管如此，其特殊结构提示我们定制先进的离散化方法，以形成紧凑的数学公式和便捷的实现方案。