The work aims to investigate the utilization of neural networks in classical computational methods used in CFD. This study focuses on a hybrid finite volume method that employs a neural network for computing numerical fluxes. The proposed method effectively combines a classical computational approach, ensuring various mathematical properties, such as conservativity, with a neural network to approximate complex numerical flux relationships. This arrangement is advantageous compared to a purely graph-based neural network method, which computes a new time level in two steps but lacks, for example, conservativity and stability. Another advantage of the investigated method is its universal applicability, as it can be easily implemented using, for instance, the TensorFlow library and can be evaluated using GPU. The hybrid finite volume neural network method is applied to the flow problems of inviscid and viscous compressible fluids in a blade cascade geometries.