Accurately modeling the nonlinear viscoelastic response of filled elastomers under large deformations remains a challenging task, especially when phenomena such as the Payne effect dominate the mechanical behavior. Generalized Maxwell models employing phenomenological viscosity laws often fail to reproduce the pronounced amplitude dependence observed in filled elastomers. In this work, we propose a hybrid constitutive framework that extends generalized Maxwell rheological models by introducing a Deep Rheological Element (DRE), in which a deep neural network (DNN) provides a direct and thermodynamically consistent representation of the dashpot viscosity. The DRE is formulated within a finite-strain framework based on the multiplicative decomposition of the deformation gradient, under the assumptions of material isotropy and incompressibility, allowing the constitutive model to reproduce the strong nonlinear viscoelastic behavior characteristic of filled elastomers. Thermodynamic consistency is preserved by enforcing a positivity constraint on the DNN output, ensuring non-negative energy dissipation. The network is trained in two stages: it is first initialized using synthetic data generated by a calibrated phenomenological model and then refined with time-domain stress-strain data obtained from dynamic mechanical analysis (DMA) tests. Comparative numerical results confirm that the DRE outperforms phenomenological viscosity formulations in reproducing the amplitude-dependent storage and loss moduli of filled rubbers. To further assess the computational performance of the proposed approach, the Deep Rheological Element was implemented in FEniCSx. The flexibility of the methodology paves the way for future extensions towards anisotropic and multiphysics viscoelasticity.