Data-Driven Computational Mechanics has emerged over the last decade as a novel paradigm for the numerical solution of governing equations in solid and fluid mechanics. Rather than identifying constitutive material laws through fitted models, the data-driven approach assigns to each material point the closest admissible state from a prescribed material dataset, while simultaneously satisfying essential physical constraints such as conservation laws and compatibility conditions. In this context, the solution of a partial differential equation is recast as a discrete–continuous constrained optimization problem, in which the optimal distribution of material data over the computational domain is sought simultaneously with the corresponding continuous primal fields, including pressures, gradients, and fluxes. A promising strategy for solving such problems is based on alternating direction methods. Within this framework, the constrained optimization problem is addressed through the solution of the optimality conditions associated with an augmented Lagrangian formulation, which incorporates proximity measures together with conservation law and compatibility constraints. For a prescribed distribution of discrete fields, these conditions are first solved to determine the corresponding continuous primal fields. Subsequently, these fields are projected, in a metric sense, onto the material dataset, thereby updating the discrete assignment. This two-step process is repeated until convergence is achieved. Recent works in the literature have mainly focused on the numerical approximation of the first step, that is, on efficient and accurate discretizations of the resulting optimality conditions. In this work, as a first step toward solving the full optimization problem including the projection stage, we consider the problem of incompressible fluid flow in hydraulic networks consisting of a set of one-dimensional elements represented as a graph. Pressure is defined at each node, while pressure gradients and fluid fluxes are defined on each element, leading to an element-wise representation. Different solution strategies are investigated, including a brute-force approach leveraging GPU processing and alternating direction methods with different initialization strategies. The accuracy and robustness of the proposed approaches are assessed through numerical examples involving different network configurations and noisy material datasets.