  We consider the general class of time-homogeneous stochastic dynamical
systems, both discrete and continuous, and study the problem of learning a
representation of the state that faithfully captures its dynamics. This is
instrumental to learning the transfer operator or the generator of the system,
which in turn can be used for numerous tasks, such as forecasting and
interpreting the system dynamics. We show that the search for a good
representation can be cast as an optimization problem over neural networks. Our
approach is supported by recent results in statistical learning theory,
highlighting the role of approximation error and metric distortion in the
learning problem. The objective function we propose is associated with
projection operators from the representation space to the data space, overcomes
metric distortion, and can be empirically estimated from data. In the
discrete-time setting, we further derive a relaxed objective function that is
differentiable and numerically well-conditioned. We compare our method against
state-of-the-art approaches on different datasets, showing better performance
across the board.
