Model Predictive Control (MPC) is widely applied to address control challenges in complex dynamic systems. However, traditional MPC methods rely on precise system models which are often difficult to obtain in practical application, and the computational amount of complex nonlinear systems is difficult to meet the real-time requirements. In this paper, a model predictive control method based on locally approximated neural networks (LANN-MPC) is proposed, which uses neural network to fit the nonlinear dynamics model of UAV and performs local approximation in MPC prediction. This approach overcomes the limitation of traditional control methods requiring accurate system mechanism model and effectively reduces the computational complexity of control instruction optimization, thereby improving the control efficiency. Simulation results show that the locally approximated neural network MPC method achieves excellent real-time performance while maintaining control accuracy close to that of a fully neural network-based MPC approach. This makes it ideal for the efficient real-time control scenarios of fixed wing UAVs.