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Using a new numerical method for converging unstable invariant two-tori in a chaotic partial differential equation (PDE), and exploiting symmetry breaking of relative periodic orbits to detect those tori, we identify many quasiperiodic, unstable, invariant two-torus solutions of a modified Kuramoto\u2013Sivashinsky equation. The set of tori covers significant parts of the chaotic attractor and weighted averages of the properties of the tori\u2014with weights computed based on their respective stability eigenvalues\u2014approximate average quantities for the chaotic dynamics. These results are a step toward exploiting higher-dimensional invariant sets to describe general hyperchaotic systems, including dissipative spatiotemporally chaotic PDEs.<\/jats:p>","DOI":"10.1063\/5.0143689","type":"journal-article","created":{"date-parts":[[2023,8,3]],"date-time":"2023-08-03T13:36:14Z","timestamp":1691069774000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":9,"title":["Predicting chaotic statistics with unstable invariant tori"],"prefix":"10.1063","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2066-072X","authenticated-orcid":false,"given":"Jeremy P.","family":"Parker","sequence":"first","affiliation":[{"name":"Emergent Complexity in Physical Systems Laboratory (ECPS), \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne , CH-1015 Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5684-4681","authenticated-orcid":false,"given":"Omid","family":"Ashtari","sequence":"additional","affiliation":[{"name":"Emergent Complexity in Physical Systems Laboratory (ECPS), \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne , CH-1015 Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8617-8998","authenticated-orcid":false,"given":"Tobias M.","family":"Schneider","sequence":"additional","affiliation":[{"name":"Emergent Complexity in Physical Systems Laboratory (ECPS), \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne , CH-1015 Lausanne, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2023,8,3]]},"reference":[{"key":"2023080313360498700_c1","doi-asserted-by":"publisher","first-page":"1010","DOI":"10.2307\/2373414","article-title":"An improved closing lemma and a general density theorem","volume":"89","year":"1967","journal-title":"Am. 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