The rapid evolution of Physics Informed Neural Networks (PINNs) and Variational PINNs (VPINNs) signifies a transformative shift in the field of computational mathematics, particularly in solving complex non-linear partial differential equations (PDEs). These methodologies efficiently encode physical laws into the architecture of neural networks, allowing the numerical approximation of PDEs. In this context, we propose to combine PINNs and the traditional finite volume methods (FVM), which are widely used for numerical approximations of hyperbolic conservation laws. First, we consider the integral form of the hyperbolic conservation law and a suitable partition of our computational mesh into cells. Next, we will use a neural networks, with as many input neurons as cells, as a reconstruction operator in the FVM framework. The role of the neural network is also involved in the calculation of the solution averages in the cells for input into a loss function related to the finite volume schemes. The advantage of using neural networks is to be able to consider implicit methods without increasing the complexity, which allows us to increase stability and to be able to use larger time steps. The neural network performs the crucial task of reconstructing intercell fluxes. The way the method is assembled will allow us to construct in the future well-balanced methods in this framework, or to make use of entropy-conservative numerical fluxes. Moreover, the efficiency of the proposed method will be demonstrated through its application to a variety of hyperbolic conservation laws, including transport equations, Burgers' equations and Shallow Water equations, both in 1D and 2D.