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We introduce a multi-tasking graph convolutional neural network, HydraGNN, to simultaneously predict both global and atomic physical properties and demonstrate with ferromagnetic materials. We train HydraGNN on an open-source ab initio density functional theory (DFT) dataset for iron-platinum (FePt) with a fixed body centered tetragonal (BCT) lattice structure and fixed volume to simultaneously predict the mixing enthalpy (a global feature of the system), the atomic charge transfer, and the atomic magnetic moment across configurations that span the entire compositional range. By taking advantage of underlying physical correlations between material properties, multi-task learning (MTL) with HydraGNN provides effective training even with modest amounts of data. Moreover, this is achieved with just one architecture instead of three, as required by single-task learning (STL). The first convolutional layers of the HydraGNN architecture are shared by all learning tasks and extract features common to all material properties. The following layers discriminate the features of the different properties, the results of which are fed to the separate heads of the final layer to produce predictions. Numerical results show that HydraGNN effectively captures the relation between the configurational entropy and the material properties over the entire compositional range. Overall, the accuracy of simultaneous MTL predictions is comparable to the accuracy of the STL predictions. In addition, the computational cost of training HydraGNN for MTL is much lower than the original DFT calculations and also lower than training separate STL models for each property.