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This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD, we show how to perform gradient-based optimization over the learnable kernel parameters and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto\u2013Sivashinsky equation, showcasing the method\u2019s different strengths.<\/jats:p>","DOI":"10.1063\/5.0341144","type":"journal-article","created":{"date-parts":[[2026,10,14]],"date-time":"2026-10-14T12:46:06Z","timestamp":1791981966000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Dictionary learning for kernel EDMD"],"prefix":"10.1063","volume":"36","author":[{"ORCID":"https:\/\/orcid.org\/0009-0001-4735-9315","authenticated-orcid":false,"given":"Erik Lien","family":"Bolager","sequence":"first","affiliation":[{"name":"School of Computation, Information and Technology, Munich Data Science Institute, Munich Center for Machine Learning, Technical University of Munich 1 , Munich,","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9446-2614","authenticated-orcid":false,"given":"Boumediene","family":"Hamzi","sequence":"additional","affiliation":[{"name":"Department of Computing and Mathematical Sciences, Caltech 2 , Pasadena, California 91125,","place":["USA"]},{"name":"The Alan Turing Institute 3 , London,","place":["United Kingdom"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5677-1600","authenticated-orcid":false,"given":"Houman","family":"Owhadi","sequence":"additional","affiliation":[{"name":"Department of Computing and Mathematical Sciences, Caltech 2 , Pasadena, California 91125,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2220-3522","authenticated-orcid":false,"given":"Ioannis G.","family":"Kevrekidis","sequence":"additional","affiliation":[{"name":"Department of Chemical and Biomolecular Engineering and of Applied Mathematics and Statistics, Johns Hopkins University 4 , Baltimore, Maryland 21218,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2906-1769","authenticated-orcid":false,"given":"Felix","family":"Dietrich","sequence":"additional","affiliation":[{"name":"School of Computation, Information and Technology, Munich Data Science Institute, Munich Center for Machine Learning, Technical University of Munich 1 , Munich,","place":["Germany"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"317","published-online":{"date-parts":[[2026,10,14]]},"reference":[{"key":"2026101408460445600_c1","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/j.physd.2004.06.015","article-title":"Comparison of systems with complex behavior","volume":"197","year":"2004","journal-title":"Physica D"},{"key":"2026101408460445600_c2","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s11071-005-2824-x","article-title":"Spectral properties of dynamical systems, model reduction and decompositions","volume":"41","year":"2005","journal-title":"Nonlinear Dyn."},{"key":"2026101408460445600_c3","unstructured":"I.\n            Mezi\u0107\n          , \u201cOperator is the model,\u201d arXiv:2310.18516 (2023)."},{"key":"2026101408460445600_c4","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1073\/pnas.17.5.315","article-title":"Hamiltonian systems and transformation in Hilbert space","volume":"17","year":"1931","journal-title":"Proc. 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