We present a deep learning framework that leverages computational homogenization expertise to predict the local stress field and homogenized moduli of heterogeneous materials with two- and three-dimensional periodicity, which is named physics-informed Deep Homogenization Networks (DHN). To this end, the displacement field of a repeating unit cell is expressed as two-scale expansion in terms of averaging and fluctuating contributions dependent on the global and local coordinates, respectively, under arbitrary multi-axial loading conditions. The latter is regarded as a mesh-free periodic domain estimated using fully connected neural network layers by minimizing residuals of Navier's displacement equations of anisotropic microstructured materials for specified macroscopic strains with the help of automatic differentiation. Enabled by the novel use of a periodic layer, the boundary conditions are encoded directly in the DHN architecture which ensures exact satisfaction of the periodicity conditions of displacements and tractions without introducing additional penalty terms. To verify the proposed model, the local field variables and homogenized moduli were examined for various composites against the finite-element technique. We also demonstrate the feasibility of the proposed framework for simulating unit cells with locally irregular fibers via transfer learning and find a significant enhancement in the accuracy of stress field recovery during neural network retraining.