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Learn.: Sci. Technol."],"published-print":{"date-parts":[[2022,3,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The <jats:italic>sparse identification of nonlinear dynamics<\/jats:italic> (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy <jats:italic>et al<\/jats:italic> (2019 <jats:italic>J. Computat. Phys.<\/jats:italic> \n                  <jats:bold>396<\/jats:bold> 483\u2013506) for simultaneously (1) denoising the data, (2) learning and parametrizing the noise probability distribution, and (3) identifying the underlying parsimonious dynamical system responsible for generating the time-series data. Thus within an integrated optimization framework, noise can be separated from signal, resulting in an architecture that is approximately twice as robust to noise as state-of-the-art methods, handling as much as 40% noise on a given time-series signal and explicitly parametrizing the noise probability distribution. We demonstrate this approach on several numerical examples, from Lotka-Volterra models to the spatio-temporal Lorenz 96 model. Further, we show the method can learn a diversity of probability distributions for the measurement noise, including Gaussian, uniform, Gamma, and Rayleigh distributions.<\/jats:p>","DOI":"10.1088\/2632-2153\/ac567a","type":"journal-article","created":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T22:26:57Z","timestamp":1645136817000},"page":"015031","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":52,"title":["Automatic differentiation to simultaneously identify nonlinear dynamics and extract noise probability distributions from data"],"prefix":"10.1088","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2279-2793","authenticated-orcid":true,"given":"Kadierdan","family":"Kaheman","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6565-5118","authenticated-orcid":false,"given":"Steven L","family":"Brunton","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6004-2275","authenticated-orcid":true,"given":"J","family":"Nathan Kutz","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2022,3,8]]},"reference":[{"key":"mlstac567abib1","author":"Nelles","year":"2013"},{"key":"mlstac567abib2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.arcontrol.2009.12.001","article-title":"Perspectives on system identification","volume":"34","author":"Ljung","year":"2010","journal-title":"Annu. 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