This contribution presents a new physics-informed deep homogenization neural network model for identifying local displacement and stress fields, as well as homogenized moduli, of nanocomposites with periodic arrays of porosities under general loading conditions. Notably, it accounts for the surface elasticity effect, utilizing the Gurtin-Murdoch interface theory. First of all, a fully connected neural network model is established that maps the spatial coordinates, passing first through several sinusoidal functions, to the microscopic displacements. The loss function is formulated as the weighted sum of residuals of Navier-Cauchy equations in the bulk domains and the Young-Laplace equations on the energetic surfaces, evaluated on separate sets of collocation points. To more effectively predict stress concentrations inside the microstructures, we introduce fully trainable weights to each collocation point. The capacity and effectiveness of the new homogenization technique for capturing the size-dependent local and global response of nanocomposites with distinct pore sizes and shapes are verified upon extensive comparisons with the finite-element benchmark results, under various loading conditions. New results showcase the proposed theory’s ability to model random distributions of nano-porosities with a high degree of accuracy, a task not easily achievable with alternative techniques except for the specialized finite-element method.