This work presents a physics-informed neural network (PINN) based framework to model the strain-rate and temperature dependence of the deformation fields in elastic-viscoplastic solids. To avoid unbalanced back-propagated gradients during training, the proposed framework uses a simple strategy with no added computational complexity for selecting scalar weights that balance the interplay between different terms in the physics-based loss function. In addition, we highlight a fundamental challenge involving the selection of appropriate model outputs so that the mechanical problem can be faithfully solved using a PINN-based approach. We demonstrate the effectiveness of this approach by studying two test problems modeling the elastic-viscoplastic deformation in solids at different strain rates and temperatures, respectively. Our results show that the proposed PINN-based approach can accurately predict the spatio-temporal evolution of deformation in elastic-viscoplastic materials.