In this talk, we present physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating multiscale PDEs. From a continuous perspective, our formulation corresponds to a non-standard PDE-constrained optimization problem with a PINN-type objective. From a discrete standpoint, the formulation represents a hybrid numerical solver that utilizes both neural networks and finite elements. In our setting, the neural network approximates a fine-scale problem, and a coarse-scale problem constrains the learning process. We show that incorporating coarse-scale information into the neural network training process through our modelling framework significantly enhances the overall learning process and results in improved convergence properties and accuracy of the PINN method. The mathematical framework for such a formulation, and the relevance of the hybrid solver to numerical homogenization are discussed.