Discrete Fracture Network (DFN) flow simulations are commonly used to determine the outflow in fractured media for critical applications. Here, we extend the formulation of spatial graph neural networks with a new architecture, called Graph Informed Neural Network (GINN), to speed up the Uncertainty Quantification analyses for DFNs. We show that the GINN model allows better Monte Carlo estimates of the mean and standard deviation of the outflow of a test case DFN.