{"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/learning-normal-form-autoencoders-for-data","title":"Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equations","arxiv_id":"2106.05102","date":"2021-06-09","proceeding":null,"authors":["Manu Kalia","Steven L. Brunton","Hil G. E. Meijer","Christoph Brune","J. Nathan Kutz"],"abstract":"Complex systems manifest a small number of instabilities and bifurcations that are canonical in nature, resulting in universal pattern forming characteristics as a function of some parametric dependence. Such parametric instabilities are mathematically characterized by their universal un-foldings, or normal form dynamics, whereby a parsimonious model can be used to represent the dynamics. Although center manifold theory guarantees the existence of such low-dimensional normal forms, finding them has remained a long standing challenge. In this work, we introduce deep learning autoencoders to discover coordinate transformations that capture the underlying parametric dependence of a dynamical system in terms of its canonical normal form, allowing for a simple representation of the parametric dependence and bifurcation structure. The autoencoder constrains the latent variable to adhere to a given normal form, thus allowing it to learn the appropriate coordinate transformation. We demonstrate the method on a number of example problems, showing that it can capture a diverse set of normal forms associated with Hopf, pitchfork, transcritical and/or saddle node bifurcations. This method shows how normal forms can be leveraged as canonical and universal building blocks in deep learning approaches for model discovery and reduced-order modeling.","url_abs":"https://arxiv.org/abs/2106.05102v1","url_pdf":"https://arxiv.org/pdf/2106.05102v1.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":"learning-normal-form-autoencoders-for-data","repo_url":"https://github.com/dynamicslab/NormalFormAE","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[{"task_slug":"form","task_name":"Form"},{"task_slug":"model-discovery","task_name":"Model Discovery"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}