  Temporal averages in chaotic systems can often be approximated using
collections of periodic orbits--unstable time-periodic solutions of the
governing equations--embedded within the chaotic set. This connection is
formalized by periodic orbit theory, with a large number of such solutions
required to obtain a good approximation. Here, we describe an alternative,
data-driven approach that allows for an accurate approximation of temporal
averages even when the number of time-periodic solutions used is quite small.
Moreover, this approach yields convergence to temporal averages, in terms of
the number of solutions used, that far outperforms periodic orbit theory.
