  Chaotic systems are notoriously challenging to predict because of their
sensitivity to perturbations and errors due to time stepping. Despite this
unpredictable behavior, for many dissipative systems the statistics of the long
term trajectories are governed by an invariant measure supported on a set,
known as the global attractor; for many problems this set is finite
dimensional, even if the state space is infinite dimensional. For Markovian
systems, the statistical properties of long-term trajectories are uniquely
determined by the solution operator that maps the evolution of the system over
arbitrary positive time increments. In this work, we propose a machine learning
framework to learn the underlying solution operator for dissipative chaotic
systems, showing that the resulting learned operator accurately captures
short-time trajectories and long-time statistical behavior. Using this
framework, we are able to predict various statistics of the invariant measure
for the turbulent Kolmogorov Flow dynamics with Reynolds numbers up to 5000.
