We present a novel elasticity-inspired data-driven Fourier homogenization network (FHN) theory for periodic heterogeneous microstructures with square or hexagonal arrays of cylindrical fibers. Towards this end, two custom-tailored networks are harnessed to construct microscopic displacement functions in each phase of composite materials, based on the exact Fourier series solutions of Navier’s displacement differential equations. The fiber and matrix networks are seamlessly connected through a common loss function by enforcing the continuity conditions, in conjunction with periodicity boundary conditions, of both tractions and displacements. These conditions are evaluated on a set of weighted collocation points located on the fiber/matrix interface and the exterior faces of the unit cell, respectively. The partial derivatives of displacements are computed effortlessly through the automatic differentiation functionality. During the training of the FHN model, the total loss function is minimized with respect to the Fourier series parameters using gradient descent and concurrently maximized with respect to the adaptive weights using gradient ascent. The transfer learning technique is employed to speed up the training of new geometries by leveraging a pre-trained model. Comparison with finite-element/volume-based unit cell solutions under various loading scenarios showcases the computational capability of the proposed method. The utility of the proposed technique is further demonstrated by capturing the interfacial debonding in unidirectional composites via a cohesive interface model.