{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/deep-learning-for-universal-linear-embeddings","title":"Deep learning for universal linear embeddings of nonlinear dynamics","arxiv_id":"1712.09707","date":"2017-12-27","proceeding":null,"authors":["Bethany Lusch","J. Nathan Kutz","Steven L. Brunton"],"abstract":"Identifying coordinate transformations that make strongly nonlinear dynamics\napproximately linear is a central challenge in modern dynamical systems. These\ntransformations have the potential to enable prediction, estimation, and\ncontrol of nonlinear systems using standard linear theory. The Koopman operator\nhas emerged as a leading data-driven embedding, as eigenfunctions of this\noperator provide intrinsic coordinates that globally linearize the dynamics.\nHowever, identifying and representing these eigenfunctions has proven to be\nmathematically and computationally challenging. This work leverages the power\nof deep learning to discover representations of Koopman eigenfunctions from\ntrajectory data of dynamical systems. Our network is parsimonious and\ninterpretable by construction, embedding the dynamics on a low-dimensional\nmanifold that is of the intrinsic rank of the dynamics and parameterized by the\nKoopman eigenfunctions. In particular, we identify nonlinear coordinates on\nwhich the dynamics are globally linear using a modified auto-encoder. We also\ngeneralize Koopman representations to include a ubiquitous class of systems\nthat exhibit continuous spectra, ranging from the simple pendulum to nonlinear\noptics and broadband turbulence. Our framework parametrizes the continuous\nfrequency using an auxiliary network, enabling a compact and efficient\nembedding at the intrinsic rank, while connecting our models to half a century\nof asymptotics. In this way, we benefit from the power and generality of deep\nlearning, while retaining the physical interpretability of Koopman embeddings.","url_abs":"http://arxiv.org/abs/1712.09707v2","url_pdf":"http://arxiv.org/pdf/1712.09707v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"deep-learning-for-universal-linear-embeddings","repo_url":"https://github.com/BethanyL/DeepKoopman","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":{"status":"unanswered"}},{"paper_slug":"deep-learning-for-universal-linear-embeddings","repo_url":"https://github.com/opaliss/dmd_autoencoder","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"unanswered"}}],"tasks":[{"task_slug":"deep-learning","task_name":"Deep Learning"}],"methods":[{"method_slug":"interpretability","method_name":"Interpretability"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1712.09707","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}