Network flow models are used in both continuous and discrete optimization to describe fluid-like transitions. To close the gap between these two perspectives, the focus of this project is to extend the theory of flows over time by considering a) sparse representations of solutions and b) the sparsification of network structures. By using temporally repeated flows, a sparse representation of (approximate) solutions can be achieved. To incorporate non-constant transit times and state-dependent network cost, dynamic programming algorithms on tree-like networks are derived. Extensions to nonlinear dynamics are investigated.