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.

In several applications concerning underground flow simulations in fractured media, the fractured rock matrix is modeled by means of the Discrete Fracture Network (DFN) model. The fractures are typically described through stochastic parameters sampled from known distributions. In this framework, it is worth considering the application of suitable complexity reduction techniques, also in view of possible uncertainty quantification analyses or other applications requiring a fast approximation of the flow through the network. Herein, we propose the application of Neural Networks to flux regression problems in a DFN characterized by stochastic trasmissivities as an approach to predict fluxes.

In global optimization problems, diversification approaches are often necessary to overcome the convergence toward local optima. One approach is the multi-start method, where a set of different starting configurations are taken into account to designate the best local minimum returned by the multiple optimization procedures as the (possible) global optimum. Therefore, parallelization is crucial for multi-start. In this work, we present a new multi-start approach for gradient-based optimization methods that exploits the reverse Automatic Differentiation to perform efficiently. In particular, for each step, this Automatic Differentiation-based method is able to compute the N gradients of N optimization procedures extremely quickly, exploiting the implicit parallelization guaranteed by the computational graph representation of the multi-start problem. The practical advantages of the proposed method are illustrated by analyzing the time complexity from a theoretical point of view and showing numerical examples where the speed-up is between $\times 40$ and $\times 100$, with respect to classic parallelization methods. Moreover, we show that our AD-based multi-start approach can be implemented by using tailored shallow Neural Networks, taking advantage of the built-in optimization procedures of the Deep Learning frameworks.