  We review several of the most widely used techniques for training recurrent
neural networks to approximate dynamical systems, then describe a novel
algorithm for this task. The algorithm is based on an earlier theoretical
result that guarantees the quality of the network approximation. We show that a
feedforward neural network can be trained on the vector field representation of
a given dynamical system using backpropagation, then recast, using matrix
manipulations, as a recurrent network that replicates the original system's
dynamics. After detailing this algorithm and its relation to earlier
approaches, we present numerical examples that demonstrate its capabilities.
One of the distinguishing features of our approach is that both the original
dynamical systems and the recurrent networks that simulate them operate in
continuous time.
