  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.
