A neural network approximation to direct trajectory optimization methods is presented. The method uses neural networks to approximate the dynamics and objective equations integrated over a given time interval. The trajectory is then built recursively and treated as a nonlinear programming problem. The method is compared to a direct collocation method as well as more recent pseudospectral methods and shows competitive results while being computationally faster. In addition, a neural network provides a continuously differentiable function approximation which may be advantageous when a discontinuous objective function is used in a nonlinear solver. A surveillance trajectory planning problem for an unmanned aerial vehicle is given as an example application and results are presented for all three methods.