Predicting network structure and flows on edges is a precondition to effective planning and disaster response in critical infrastructure networks. Accordingly, we propose to develop algorithms inspired by network science, physical modeling, and machine learning to determine this information from incomplete observations. Using both physical domain-specific and data-driven approaches, these algorithms will address diverse problems including: the reconstruction of network topology and parameters, the estimation of network flows, the optimal selection and scheduling of multiple types of sensors, and the modeling of interdependencies in multilayer networks. Together these methods will develop a rigorous theory for solving the inverse problem of edge and flow determination from observations of critical infrastructure networks.