In this talk, a non-intrusive reduced order modeling of convection dominated flows based on the proper orthogonal decomposition (POD) and the artificial neural network (ANN) is designed to simulate the linear and quasi-nonlinear regimes of Rayleigh-Taylor instability (RTI) phenomenon. The temporal and spatial accuracy of the proposed method is demonstrated by solving a high-dimensional parametrized ODE and one-dimensional viscous Burgers’ equation with a parameterized diffusion coefficient. The results illustrate that the proposed ROM can achieve a good agreement with the corresponding full-order solutions. Particularly, RTI is simulated by the proposed method, where the amplitude of the small perturbation and time are considered as free parameters. Furthermore, an adaptive sampling method in time is proposed to decrease the number of samples in the parameter space during the linear regime of RTI, which results in the number of snapshots of the full-order solutions required for POD and the training of the ANN is correspondingly reduced. The numerical results show that the adaptive sampling method can achieve an equivalent accuracy and an improved efficiency.