The purpose of this work is to develop new Neural Network architectures for flux regression problems in Discrete Fracture Networks (DFNs) taking advantage of the graphs representing the DFNs. We build a “Graph Informed NN” (GI-NN), with layers characterized by the adjacency matrix, introducing a novel typology of Graph Convolutional Neural Network. After an introduction to NN applications for flow simulation problems in DFNs, the creation of GI-NNs is described and the flux regression performances are analyzed.