We introduce a continuum model for polymer melts filled with nanoparticles capable of describing in a unified and self-consistent way their microstructure, phase behavior, and rheology in both the linear and nonlinear regimes. It is based on the Hamiltonian formulation of transport phenomena for fluids with a complex microstructure with the final dynamic equations derived by means of a generalized (Poisson plus dissipative) bracket. The model describes the polymer nanocomposite melt at a mesoscopic level by using three fields (state variables): a vectorial (the momentum density) and two tensorial ones (the conformation tensor for polymer chains and the orientation tensor for nanoparticles). The dynamic equations are developed for nanoparticles with an arbitrary shape but then they are specified to the case of spherical ones. Restrictions on the parameters of the model are provided by analyzing its thermodynamic admissibility. A key ingredient of the model is the expression for the Helmholtz free energy A of the polymer nanocomposite. At equilibrium this reduces to the form introduced by Mackay et al. (Science 2006, 311, 1740-1743) to explain the phase behavior of polystyrene melts filled with silica nanopartides. Beyond equilibrium, A contains additional terms that account for the coupling between microstructure and flow. In the absence of chain elasticity, the proposed evolution equations capture known models for the hydrodynamics of a Newtonian suspension of particles. A thorough comparison against several sets of experimental and simulation data demonstrates the unique capability of the model to accurately describe chain conformation and swelling in polymer melt nanocomposites and to reliably fit measured rheological data for their shear and complex viscosity over large ranges of volume fractions and deformation rates.