This study presents a neural network-based predictive control (abbreviated as NNMPC in subsequent content) approach for quadrotor tracking. By learning dynamical behaviors from experimental flight data, a neural ordinary differential model (NODM) is built first, which is particularly suitable for situations where aerodynamic conditions are complex and difficult to model formulaically. Subsequently, the NODM is used for designing the predictive control by considering the constraint conditions. Here, the proposed method uses a linearized model of the NODM established to effectively handle the neural network system for predicting the states, reducing the computation time. Finally, simulation results show that the proposed method can achieve a good tracking performance.