We propose a physics-augmented neural network (PANN) framework for finite strain incompressible viscoelasticity within the generalized standard materials theory. The formulation is based on the multiplicative decomposition of the deformation gradient and enforces unimodularity of the inelastic deformation part throughout the evolution. Invariant-based representations of the free energy and the dual dissipation potential by (partially) monotonic and fully input-convex neural networks ensure thermodynamic consistency, objectivity, and material symmetry by construction. The evolution of the internal variables during training is handled by solving the evolution equations using an implicit exponential time integrator. In addition, a trainable gate layer combined with ℓ_p regularization automatically identifies the required number of internal variables during training. The PANN is calibrated with synthetic and experimental data, showing excellent agreement for a wide range of deformation rates and different load paths. We also show that the proposed model achieves excellent interpolation as well as plausible and accurate extrapolation behaviors. In addition, we demonstrate consistency of the PANN with linear viscoelasticity by linearization of the full model.