{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,11,6]],"date-time":"2026-11-06T19:16:16Z","timestamp":1793992576829,"version":"4.8.0"},"reference-count":26,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T00:00:00Z","timestamp":1775779200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,5,25]],"date-time":"2026-05-25T00:00:00Z","timestamp":1779667200000},"content-version":"vor","delay-in-days":45,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"funder":[{"name":"SKF Magnetic Mechatronics"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Sci Rep"],"DOI":"10.1038\/s41598-026-47609-0","type":"journal-article","created":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T01:35:54Z","timestamp":1775784954000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["RRAEDy: adaptive latent linearization of nonlinear dynamical systems"],"prefix":"10.1038","volume":"16","author":[{"given":"Jad","family":"Mounayer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sebastian","family":"Rodriguez","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jerome","family":"Tomezyk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chady","family":"Ghnatios","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Francisco","family":"Chinesta","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,4,10]]},"reference":[{"key":"47609_CR1","unstructured":"Tu, J.\u00a0H., Rowley, C.\u00a0W., Luchtenburg, D.\u00a0M., Brunton, S.\u00a0L. & Kutz, J.\u00a0N. On dynamic mode decomposition: Theory and applications. arXiv preprint arXiv:1312.0041 (2013)."},{"key":"47609_CR2","doi-asserted-by":"publisher","first-page":"3932","DOI":"10.1073\/pnas.1517384113","volume":"113","author":"SL Brunton","year":"2016","unstructured":"Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. 113, 3932\u20133937. https:\/\/doi.org\/10.1073\/pnas.1517384113 (2016).","journal-title":"Proc. Natl. Acad. Sci."},{"key":"47609_CR3","unstructured":"Chen, R. T.\u00a0Q., Rubanova, Y., Bettencourt, J. & Duvenaud, D. Neural ordinary differential equations. arXiv:1806.07366 (2019)."},{"key":"47609_CR4","unstructured":"Kidger, P., Morrill, J., Foster, J. & Lyons, T. Neural controlled differential equations for irregular time series. arXiv:2005.08926 (2020)."},{"key":"47609_CR5","unstructured":"Chung, J., Gulcehre, C., Cho, K. & Bengio, Y. Empirical evaluation of gated recurrent neural networks on sequence modeling. arXiv:1412.3555 (2014)."},{"key":"47609_CR6","doi-asserted-by":"publisher","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","volume":"9","author":"S Hochreiter","year":"1997","unstructured":"Hochreiter, S. & Schmidhuber, J. Long short-term memory. Neural Comput. 9, 1735\u20131780. https:\/\/doi.org\/10.1162\/neco.1997.9.8.1735 (1997).","journal-title":"Neural Comput."},{"key":"47609_CR7","unstructured":"Schmidt, R.\u00a0M. Recurrent neural networks (rnns): A gentle introduction and overview. arXiv:1912.05911 (2019)."},{"key":"47609_CR8","doi-asserted-by":"crossref","unstructured":"Lea, C., Vidal, R., Reiter, A. & Hager, G.\u00a0D. Temporal convolutional networks: A unified approach to action segmentation. arXiv:1608.08242 (2016).","DOI":"10.1007\/978-3-319-49409-8_7"},{"key":"47609_CR9","doi-asserted-by":"crossref","unstructured":"Fronk, C. & Petzold, L. The vanishing gradient problem for stiff neural differential equations. arXiv:2508.01519 (2025).","DOI":"10.1063\/5.0294317"},{"key":"47609_CR10","unstructured":"Ghazal, E.\u00a0A. et al. Application of reduced-order models for temporal multiscale representations in the prediction of dynamical systems. arXiv:2510.18925 (2025)."},{"key":"47609_CR11","doi-asserted-by":"crossref","unstructured":"Thummerer, T. & Mikelsons, L. Eigen-informed neuralodes: Dealing with stability and convergence issues of neuralodes. arXiv:2302.10892 (2023).","DOI":"10.2139\/ssrn.4819144"},{"key":"47609_CR12","unstructured":"Gal, Y. & Ghahramani, Z. A theoretically grounded application of dropout in recurrent neural networks. arXiv:1512.05287 (2016)."},{"key":"47609_CR13","unstructured":"Rubanova, Y., Chen, R. T.\u00a0Q. & Duvenaud, D. Latent odes for irregularly-sampled time series. arXiv:1907.03907 (2019)."},{"key":"47609_CR14","doi-asserted-by":"crossref","unstructured":"Susik, R. Recurrent Autoencoder with Sequence-Aware Encoding. 47\u201357 (Springer, 2021).","DOI":"10.1007\/978-3-030-77964-1_4"},{"key":"47609_CR15","doi-asserted-by":"publisher","first-page":"2096","DOI":"10.1137\/17M1125236","volume":"16","author":"H Arbabi","year":"2017","unstructured":"Arbabi, H. & Mezi\u0107, I. Ergodic theory, dynamic mode decomposition, and computation of spectral properties of the Koopman operator. SIAM J. Appl. Dyn. Syst. 16, 2096\u20132126. https:\/\/doi.org\/10.1137\/17M1125236 (2017).","journal-title":"SIAM J. Appl. Dyn. Syst."},{"key":"47609_CR16","doi-asserted-by":"crossref","unstructured":"Brunton, S.\u00a0L., Budi\u0161i\u0107, M., Kaiser, E. & Kutz, J.\u00a0N. Modern Koopman theory for dynamical systems. arXiv:2102.12086 (2021).","DOI":"10.1137\/21M1401243"},{"key":"47609_CR17","doi-asserted-by":"publisher","DOI":"10.1063\/5.0073893","author":"DJ Alford-Lago","year":"2022","unstructured":"Alford-Lago, D. J., Curtis, C. W., Ihler, A. T. & Issan, O. Deep learning enhanced dynamic mode decomposition. Chaos Interdiscip. J. Nonlinear Sci. https:\/\/doi.org\/10.1063\/5.0073893 (2022).","journal-title":"Chaos Interdiscip. J. Nonlinear Sci."},{"key":"47609_CR18","doi-asserted-by":"crossref","unstructured":"Saha, P. & Mukhopadhyay, S. Bridging autoencoders and dynamic mode decomposition for reduced-order modeling and control of PDEs. arXiv:2409.06101 (2024).","DOI":"10.1109\/CDC56724.2024.10886639"},{"key":"47609_CR19","doi-asserted-by":"publisher","DOI":"10.1038\/s41467-018-07210-0","author":"B Lusch","year":"2018","unstructured":"Lusch, B., Kutz, J. N. & Brunton, S. L. Deep learning for universal linear embeddings of nonlinear dynamics. Nat. Commun. https:\/\/doi.org\/10.1038\/s41467-018-07210-0 (2018).","journal-title":"Nat. Commun."},{"key":"47609_CR20","doi-asserted-by":"crossref","unstructured":"Mounayer, J., Rodriguez, S., Ghnatios, C., Farhat, C. & Chinesta, F. Rank reduction autoencoders. arXiv:2405.13980 (2025).","DOI":"10.2139\/ssrn.5718813"},{"key":"47609_CR21","doi-asserted-by":"crossref","unstructured":"Idrissi, M. E.\u00a0F. et al. A new framework for generative design, real-time prediction, and inverse design optimization: Application to microstructure (2025).","DOI":"10.21203\/rs.3.rs-7478922\/v1"},{"key":"47609_CR22","doi-asserted-by":"crossref","unstructured":"Mounayer, J., Tierz, A., Tomezyk, J., Ghnatios, C. & Chinesta, F. Variational rank reduction autoencoders. arXiv:2505.09458 (2025).","DOI":"10.2139\/ssrn.5718813"},{"key":"47609_CR23","unstructured":"Kidger, P. & Garcia, C. Equinox: Neural networks in JAX via callable PyTrees and filtered transformations. In Differentiable Programming Workshop at Neural Information Processing Systems 2021 (2021)."},{"key":"47609_CR24","unstructured":"Bradbury, J. et al. JAX: Composable transformations of Python+NumPy programs (2018)."},{"key":"47609_CR25","unstructured":"Zhuang, J. et al. Adabelief optimizer: Adapting stepsizes by the belief in observed gradients. In Advances in Neural Information Processing Systems (NeurIPS) . Vol. 33 (2020)."},{"key":"47609_CR26","unstructured":"Proctor, J.\u00a0L., Brunton, S.\u00a0L. & Kutz, J.\u00a0N. Dynamic mode decomposition with control. arXiv:1409.6358 (2014)."}],"container-title":["Scientific Reports"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-47609-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-47609-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-47609-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,24]],"date-time":"2026-05-24T23:33:53Z","timestamp":1779665633000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.nature.com\/articles\/s41598-026-47609-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,4,10]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,12]]}},"alternative-id":["47609"],"URL":"https:\/\/doi.org\/10.1038\/s41598-026-47609-0","relation":{},"ISSN":["2045-2322"],"issn-type":[{"value":"2045-2322","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,4,10]]},"assertion":[{"value":"20 January 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 April 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 April 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"16103"}}