  Extracting physical laws from observation data is a central challenge in many
diverse areas of science and engineering. We propose Optimal Control Neural
Networks (OCN) to learn the laws of vector fields in dynamical systems, with no
assumption on their analytical form, given data consisting of sampled
trajectories. The OCN framework consists of a neural network representation and
an optimal control formulation. We provide error bounds for both the solution
and the vector field. The bounds are shown to depend on both the training error
and the time step between the observation data. We also demonstrate the
effectiveness of OCN, as well as its generalization ability, by testing on
several canonical systems, including the chaotic Lorenz system.
