Inferring direct interactions in complex networked systems from time series data constitutes a challenging open problem of current research. Major obstacles include the often limited number of time points accessible, unknown or inaccurate dynamical systems models in many practical applications, the impossibility to infer topological information from invariant collective dynamics such as synchronized states, and the required computational effort. Here, we propose and analyze a mathematical scheme that transforms observed transient dynamics towards invariant states in to accelerated reference frames to reveal network interactions. The transformation yields simple linear constraints relating a number of short observed time series (of only a few data points) of the dynamics to estimates the absence, presence and strength of direct physical interactions in a computationally efficient way. As we illustrate numerically, the scheme applies across transient dynamics towards periodic and chaotic, phase-locked and other synchronized states. Reconstruction robustly reveals the entire connectivity of network dynamical systems with increased reconstruction quality for large and for sparse networks.