The presence of heterogeneities and significant property mismatches in soft composites lead to complex behaviors that are challenging to model with conventional analytical or numerical homogenization techniques. The present work introduces a new physics-informed deep neural network framework to characterize microscopic displacements and stress fields in soft composites with periodic microstructures undergoing finite deformation. The main obstacle we address is the construction of specific loss functions incorporating intricate knowledge of finite strain homogenization theory, which is valid for arbitrary macroscopic deformation gradients [1]. Notably, a multi-network model is utilized to capture the discontinuities in material properties and solution fields within the composites. These neural networks communicate with each other through interface traction and displacement continuity conditions within the loss function. In addition, to exactly impose the periodicity boundary in hexagonal and square unit cells, the neural network architectures are modified by incorporating a set of trainable sinusoidal functions. A significant advantage of the current framework is that it allows for a straightforward solution of the governing partial differential equations expressed in terms of the first Piola-Kirchhoff stresses, eliminating the need for iterative formulations of the residual vector and tangent matrix required by the physics-based modelling. We extensively assess the accuracy of the proposed theory by comparison with isogeometric analysis to determine the displacement and Cauchy stress fields in square and hexagonal arrays of fibers/porosities, demonstrating neural networks as a powerful alternative to the conventional numerical approaches in finite deformation analysis of microstructural materials.