Network topology plays an important role in governing the collective dynamics. Partial synchronization (PaS) on regular networks with a few non-local links is found and explored. Different PaS patterns are observed for different ways of non-local couplings. The criterion for the existence and stability of the PaS state is studied. The emergence of PaS is related to the loss of degeneration in Lyapunov exponent spectrum. Theoretical and numerical analysis indicate that non-local coupling may drastically change the dynamical feature of the network, emphasizing the important topological dependence of collective dynamics on networks.