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In this work, we extend the formulation of the spatial-based graph convolutional networks with a new architecture, called the graph-informed neural network (GINN). This new architecture is specifically designed for regression tasks on graph-structured data that are not suitable for the well-known graph neural networks, such as the regression of functions with the domain and codomain defined on two sets of values for the vertices of a graph. In particular, we formulate a new graph-informed (GI) layer that exploits the adjacent matrix of a given graph to define the unit connections in the neural network architecture, describing a new convolution operation for inputs associated with the vertices of the graph. We study the new GINN models with respect to two maximum-flow test problems of stochastic flow networks. GINNs show very good regression abilities and interesting potentialities. Moreover, we conclude by describing a real-world application of the GINNs to a flux regression problem in underground networks of fractures.