A challenging problem in complex networks is the network reconstruction problem from data. This work deals with a class of networks denoted as conserved networks, in which a flow associated with every edge and the flows are conserved at all non-source and non-sink nodes. We propose a novel polynomial time algorithm to reconstruct conserved networks from flow data by exploiting graph theoretic properties of conserved networks combined with learning techniques. We prove that exact network reconstruction is possible for arborescence networks. We also extend the methodology for reconstructing networks from noisy data and explore the reconstruction performance on arborescence networks with different structural characteristics.