A non-intrusive reduced order model (ROM) that combines a proper orthogonal decomposition (POD) and an artificial neural network (ANN) is primarily studied to investigate the applicability of the proposed ROM in recovering the solutions with shocks and strong gradients accurately and resolving fine-scale structures efficiently for hyperbolic conservation laws. Its accuracy is demonstrated by solving a high-dimensional parametrized ODE and the one-dimensional viscous Burgers? equation with a parameterized diffusion coefficient. The two-dimensional singlemode Rayleigh-Taylor instability (RTI), where the amplitude of the small perturbation and time are considered as free parameters, is also simulated. An adaptive sampling method in time during the linear regime of the RTI is designed to reduce the number of snapshots required for POD and the training of ANN. The extensive numerical results show that the ROM can achieve an acceptable accuracy with improved efficiency in comparison with the standard full order method.