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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 this work, we investigate the sensitivity of a family of multi-task Deep Neural Networks (DNN) trained to predict fluxes through given Discrete Fracture Networks (DFNs), stochastically varying the fracture transmissivities. In particular, detailed performance and reliability analyses of more than two hundred Neural Networks (NN) are performed, training the models on sets of an increasing number of numerical simulations made on several DFNs with two fixed geometries (158 fractures and 385 fractures) and different transmissibility configurations. A quantitative evaluation of the trained NN predictions is proposed, and rules fitting the observed behavior are provided to predict the number of training simulations that are required for a given accuracy with respect to the variability in the stochastic distribution of the fracture transmissivities. A rule for estimating the cardinality of the training dataset for different configurations is proposed. From the analysis performed, an interesting regularity of the NN behaviors is observed, despite the stochasticity that imbues the whole training process. The proposed approach can be relevant for the use of deep learning models as model reduction methods in the framework of uncertainty quantification analysis for fracture networks and can be extended to similar geological problems (for example, to the more complex discrete fracture matrix models). The results of this study have the potential to grant concrete advantages to real underground flow characterization problems, making computational costs less expensive through the use of NNs.

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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.

In this paper, a rectangular aerostatic bearing with multiple supply holes is optimised with a multiobjective optimisation approach. The design variables taken into account are the supply holes position, their number and diameter, the supply pressure, while the objective functions are the load capacity, the air consumption and the stiffness and damping coefficients. A genetic algorithm is applied in order to find the Pareto set of solutions. The novelty with respect to other optimisations which can be found in literature is that number and location of the supply holes is completely free and not associated to a pre-defined scheme. A vector x associated with the supply holes location is introduced in the design parameters and given in input to the optimizer.