Computing chaotic time-averages from few periodic or non-periodic orbits

For appropriately chosen weights, temporal averages in chaotic systems can be approximated as a weighted sum of averages over reference states, such as unstable periodic orbits. Under strict assumptions, such as completeness of the orbit library, these weights can be formally derived using periodic orbit theory. When these assumptions are violated, weights can be obtained empirically using a Markov partition of the chaotic set. Here, we describe an alternative, data-driven approach to computing weights that allows for an accurate approximation of temporal averages from a variety of reference states, including both periodic orbits and non-periodic trajectory segments embedded within the chaotic set. For a broad class of observables, we demonstrate that the resulting reduced-order statistical description significantly outperforms those based on periodic orbit theory or Markov models, achieving superior accuracy while requiring far fewer reference states-two critical properties for applications to high-dimensional chaotic systems.