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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\u2014two critical properties for applications to high-dimensional chaotic systems.<\/jats:p>","DOI":"10.1063\/5.0264212","type":"journal-article","created":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T12:32:08Z","timestamp":1750249928000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Computing chaotic time-averages from few periodic or non-periodic orbits"],"prefix":"10.1063","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0208-2439","authenticated-orcid":false,"given":"Joshua L.","family":"Pughe-Sanford","sequence":"first","affiliation":[{"name":"School of Physics, Georgia Institute of Technology , 837 State St NW, Atlanta, Georgia 30332,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5043-3934","authenticated-orcid":false,"given":"Sam","family":"Quinn","sequence":"additional","affiliation":[{"name":"School of Physics, Georgia Institute of Technology , 837 State St NW, Atlanta, Georgia 30332,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-9947-1867","authenticated-orcid":false,"given":"Teodor","family":"Balabanski","sequence":"additional","affiliation":[{"name":"School of Physics, Georgia Institute of Technology , 837 State St NW, Atlanta, Georgia 30332,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6220-4701","authenticated-orcid":false,"given":"Roman O.","family":"Grigoriev","sequence":"additional","affiliation":[{"name":"School of Physics, Georgia Institute of Technology , 837 State St NW, Atlanta, Georgia 30332,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2025,6,18]]},"reference":[{"key":"2025061812320462800_c1","doi-asserted-by":"publisher","first-page":"744208","DOI":"10.3389\/fmars.2021.744208","article-title":"A cross disciplinary framework for cost-benefit optimization of marine litter cleanup at regional scale","volume":"8","year":"2021","journal-title":"Front. 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