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U.S.A."],"published-print":{"date-parts":[[2026,10,20]]},"abstract":"<jats:p>Chaotic dynamics, commonly seen in weather systems and fluid turbulence, are characterized by their sensitivity to initial conditions, which makes accurate prediction challenging. Recent approaches have focused on developing data-driven models that attempt to preserve invariant statistics over long horizons since many chaotic systems exhibit dissipative behaviors and ergodicity. Despite the recent progress in such models, they are still often prone to generating unbounded trajectories, leading to invalid statistics evaluation. To address this fundamental challenge, we introduce a modular framework that provides formal guarantees of trajectory boundedness for neural network chaotic dynamics models. Our core contribution is a dissipative projection layer that leverages control-theoretic principles to ensure the learned system is dissipative. Specifically, our framework simultaneously learns a dynamics emulator and an energy-like function, where the latter is used to construct an algebraic dissipative constraint within the projection layer. Furthermore, the learned invariant level set provides an outer estimate for the system\u2019s strange attractor, which is known to be difficult to characterize due to its complex geometry. We demonstrate our model\u2019s ability to produce bounded long-horizon forecasts that preserve invariant statistics for chaotic dynamical systems, including Lorenz 96 and a reduced-order model of the Kuramoto\u2013Sivashinsky equation.<\/jats:p>","DOI":"10.1073\/pnas.2533077123","type":"journal-article","created":{"date-parts":[[2026,10,12]],"date-time":"2026-10-12T19:03:09Z","timestamp":1791831789000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["Learning dissipative chaotic dynamics with boundedness guarantees"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0009-0005-0198-6929","authenticated-orcid":false,"given":"Sunbochen","family":"Tang","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology","place":["Cambridge"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0302-0691","authenticated-orcid":false,"given":"Themistoklis P.","family":"Sapsis","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology","place":["Cambridge"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4299-2963","authenticated-orcid":false,"given":"Navid","family":"Azizan","sequence":"additional","affiliation":[{"id":[{"id":"https:\/\/ror.org\/042nb2s44","id-type":"ROR","asserted-by":"publisher"}],"name":"Massachusetts Institute of Technology","place":["Cambridge"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,10,12]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"publisher","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2"},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1143\/PTPS.64.346"},{"key":"e_1_3_4_3_2","doi-asserted-by":"crossref","unstructured":"G. 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