Scientific Machine Learning and, in particular, hybrid methods combining classical numerical techniques for solving Partial Differential Equations with deep neural networks, have recently become increasingly popular. Thanks to their flexibility and nonlinearity, neural networks can be used to alleviate the bottlenecks of classical methods, improve their accuracy, or exploit external data in a more effective way. However, despite their growing popularity, the theoretical foundations of such hybrid methods are still in their infancy and require new techniques and ideas. In this talk, we focus on the theoretical background of a specific hybrid method: the Neural Approximated Virtual Element Method (NAVEM) [1, 2]. This method combines the Virtual Element Method (VEM) with neural networks to derive a polygonal method that does not require any stabilization or projection operator. After a short introduction to the method and its possible generalizations, an a priori error estimate with respect to mesh refinement is derived and analysed. Particular attention is given to the generalizability of the main ideas of the proof, with the aim of suggesting approaches that may be extended to other mesh-based hybrid methods. The obtained a priori error estimate can be seen as a generalization of the classical VEM one, incorporating the involved neural networks and their accuracy, which is independent of the mesh size.