A neural network learns trajectory of motion from the least action principle

This paper considers the problem of training layered neural networks to generate sequences of states. Aiming at application for situations when an integral characteristic of the process is known rather than the specific sequence of states we put forward a method in which underlying general principle is used as a foundation for the learning procedure. To illustrate the ability of a network to learn a task and to generalize algorithm we consider an example where a network generates sequences of states referred to as trajectories of motion of a particle under an external field. Training is grounded on the employment of the least action principle. In the course of training at restricted sections of the path the network elaborates a recurrent rule for the trajectory generation. The rule proves to be equivalent to the correct equation of motion for the whole trajectory.