Learning complex network dynamics is fundamental for understanding, modeling, and controlling real-world complex systems. Though great efforts have been made to predict the future states of nodes on networks, the capability of capturing long-term dynamics remains largely limited. This is because they overlook the fact that long-term dynamics in complex network are predominantly governed by their inherent low-dimensional manifolds, i.e., skeletons. Therefore, we propose the Dynamics-Invariant Skeleton Neural Net}work (DiskNet), which identifies skeletons of complex networks based on the renormalization group structure in hyperbolic space to preserve both topological and dynamics properties. Specifically, we first condense complex networks with various dynamics into simple skeletons through physics-informed hyperbolic embeddings. Further, we design graph neural ordinary differential equations to capture the condensed dynamics on the skeletons. Finally, we recover the skeleton networks and dynamics to the original ones using a degree-based super-resolution module. Extensive experiments across three representative dynamics as well as five real-world and two synthetic networks demonstrate the superior performances of the proposed DiskNet, which outperforms the state-of-the-art baselines by an average of 10.18\% in terms of long-term prediction accuracy. Code for reproduction is available at: https://github.com/tsinghua-fib-lab/DiskNet.