A direct trajectory optimization method that uses neural network approximations is presented. Neural networks are trained to model objective functions, vehicle dynamics, and non-linear constraints. The neural network method reduces computational requirements by removing the need for collocation and providing fast analytical computation of gradients. The method was shown to significantly reduce computational costs while resulting in trajectories comparable to direct collocation and pseudospectral methods. The method is applied to a multi-aircraft surveillance mission to demonstrate the scalability of the method when adding additional aircraft or objective functions to the optimization problem. Cases are presented in which neural networks can be reused for different purposes without additional training.