The Neural Approximated Virtual Element Method (NAVEM) is a recent neural-based discretization technique for partial differential equations on polygonal meshes, in which the basis functions of the local Virtual Element Method (VEM) space are approximated by pre-trained neural networks and subsequently employed within a standard finite element framework. By exploiting an efficient offline–online decomposition, NAVEM approximately solves the elemental Laplace problems that define VEM basis functions, thereby avoiding the VEM projection and stabilization operators, while preserving its geometric flexibility. In the standard NAVEM formulation, the basis functions are exactly harmonic, but the linear boundary Dirichlet data are only approximately enforced through neural networks. Moreover, the independently constructed elemental approximations lead to NAVEM basis functions that are not continuous across adjacent elements. In this contribution, we introduce two continuous variants of NAVEM, termed B-NAVEM and P-NAVEM, which restore exact inter-element continuity of the basis functions. Both approaches rely on fully connected feed-forward neural networks and enforce boundary Dirichlet conditions exactly by construction. The B-NAVEM method employs a Physics-Informed Neural Network formulation to approximate the local PDE problems characterizing virtual element basis functions, yielding functions that are polynomial on element boundaries and approximately harmonic in the interior. In contrast, the P-NAVEM approach does not aim to approximate the VEM space itself, but it directly constructs a local approximation space by training the neural network to minimize a polynomial reproducibility error. Indeed, polynomial reproducibility is the key property for achieving optimal convergence rates. Numerical experiments on linear and nonlinear model problems compare the performance of the different neural-based models and show their advantages over the standard VEM method.