Modeling the time-dependent behavior of elastomers under finite strains often relies on rheological networks, such as generalized Maxwell models, to represent both elastic and inelastic features. However, phenomenological viscosity functions can prove insufficient in capturing the strong amplitude-dependent complex moduli observed in filled rubbers, commonly referred to as the Payne effect. These limitations become critical when designing or simulating devices that operate under large cyclic or transient deformations. In this work, we propose a hybrid approach that augments the classical rheological model by incorporating a Deep Rheological Element (DRE). In this element, a deep neural network (DNN) directly supplies the dashpot viscosity, enabling the constitutive law to handle pronounced nonlinearities with reduced reliance on extensive experimental datasets. Thermodynamic consistency is preserved through a simple positivity constraint on the network output, guaranteeing that dissipation remains non-negative. The constitutive framework is built upon the multiplicative decomposition of the deformation gradient. To mitigate the scarcity of high-density data, the DNN undergoes a two-step training process. First, synthetic viscosity data are generated using a phenomenological model available in the literature and used to pre-train the network. Next, time-domain stress-strain histories derived from dynamic mechanical analysis (DMA) measurements refine its parameters, significantly reducing experimental requirements. Numerical comparisons show that this DNN-based approach reproduces the amplitude dependence of both storage and loss moduli in filled rubbers more accurately than phenomenological viscosity laws. The methodology can be readily extended to incorporate any hyperelastic springs or additional inelastic mechanisms, offering a flexible yet robust framework to tackle a wide range of viscoelastic solids.