  We describe a recurrent neural network (RNN) based architecture to learn the
flow function of a causal, time-invariant and continuous-time control system
from trajectory data. By restricting the class of control inputs to piecewise
constant functions, we show that learning the flow function is equivalent to
learning the input-to-state map of a discrete-time dynamical system. This
motivates the use of an RNN together with encoder and decoder networks which
map the state of the system to the hidden state of the RNN and back. We show
that the proposed architecture is able to approximate the flow function by
exploiting the system's causality and time-invariance. The output of the
learned flow function model can be queried at any time instant. We
experimentally validate the proposed method using models of the Van der Pol and
FitzHugh Nagumo oscillators. In both cases, the results demonstrate that the
architecture is able to closely reproduce the trajectories of these two
systems. For the Van der Pol oscillator, we further show that the trained model
generalises to the system's response with a prolonged prediction time horizon
as well as control inputs outside the training distribution. For the
FitzHugh-Nagumo oscillator, we show that the model accurately captures the
input-dependent phenomena of excitability.
