{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:16:14Z","timestamp":1787314574495,"version":"3.56.0"},"reference-count":28,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"9","license":[{"start":{"date-parts":[[2021,9,1]],"date-time":"2021-09-01T00:00:00Z","timestamp":1630454400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/legalcode"}],"funder":[{"DOI":"10.13039\/100000183","name":"Army Research Office under Cooperative Agreement","doi-asserted-by":"publisher","award":["W911NF-18-2-0048"],"award-info":[{"award-number":["W911NF-18-2-0048"]}],"id":[{"id":"10.13039\/100000183","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Department of Defense through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program"},{"name":"MIT\u2013SenseTime Alliance on Artificial Intelligence"},{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["HR00111890042"],"award-info":[{"award-number":["HR00111890042"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006602","name":"United States Air Force Research Laboratory","doi-asserted-by":"publisher","award":["FA8750-19-2-1000"],"award-info":[{"award-number":["FA8750-19-2-1000"]}],"id":[{"id":"10.13039\/100006602","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Trans. Neural Netw. Learning Syst."],"published-print":{"date-parts":[[2021,9]]},"DOI":"10.1109\/tnnls.2020.3017010","type":"journal-article","created":{"date-parts":[[2020,8,28]],"date-time":"2020-08-28T15:58:39Z","timestamp":1598630319000},"page":"4166-4177","source":"Crossref","is-referenced-by-count":174,"title":["Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery"],"prefix":"10.1109","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8248-2346","authenticated-orcid":false,"given":"Samuel","family":"Kim","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6183-5237","authenticated-orcid":false,"given":"Peter Y.","family":"Lu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Srijon","family":"Mukherjee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2503-1626","authenticated-orcid":false,"given":"Michael","family":"Gilbert","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8675-2390","authenticated-orcid":false,"given":"Li","family":"Jing","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vladimir","family":"Ceperic","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7184-5831","authenticated-orcid":false,"given":"Marin","family":"Soljacic","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"263","reference":[{"key":"ref10","first-page":"8035","article-title":"Neural arithmetic logic units","author":"trask","year":"2018","journal-title":"Advances in Neural IInformation Processing Systems"},{"key":"ref11","first-page":"497","article-title":"Unsupervised learning of latent physical properties using perception-prediction networks","volume":"1","author":"zheng","year":"2018","journal-title":"Proc 34th Conf Uncertainty Artif Intell (UAI)"},{"key":"ref12","article-title":"Extracting interpretable physical parameters from spatiotemporal systems using unsupervised learning","author":"lu","year":"2019","journal-title":"arXiv 1907 06011"},{"key":"ref13","article-title":"Visual physics: Discovering physical laws from videos","author":"chari","year":"2019","journal-title":"arXiv 1911 11893"},{"key":"ref14","article-title":"Learning sparse neural networks through $L_{0}$\n regularization","author":"louizos","year":"2017","journal-title":"arXiv 1712 01312"},{"key":"ref15","first-page":"3854","article-title":"Variational dropout sparsifies deep neural networks","volume":"5","author":"molchanov","year":"2017","journal-title":"Proc 34th Int Conf Mach Learn (ICML)"},{"key":"ref16","article-title":"To prune, or not to prune: Exploring the efficacy of pruning for model compression","author":"zhu","year":"2017","journal-title":"arXiv 1710 01878"},{"key":"ref17","article-title":"The state of sparsity in deep neural networks","author":"gale","year":"2019","journal-title":"arXiv 1902 09574"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539792240406"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1109\/CVPRW.2017.61"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1137\/18M1191944"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1126\/sciadv.1602614"},{"key":"ref27","article-title":"Gradient descent finds global minima of deep neural networks","author":"du","year":"2018","journal-title":"arXiv 1811 03804"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1517384113"},{"key":"ref6","article-title":"Extrapolation and learning equations","author":"martius","year":"2016","journal-title":"arXiv 1610 02995"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2016.0446"},{"key":"ref8","first-page":"3208","article-title":"PDE-net: Learning PDEs from data","author":"long","year":"2018","journal-title":"Mach Learn Res"},{"key":"ref7","article-title":"Learning equations for extrapolation and control","author":"sahoo","year":"2018","journal-title":"arXiv 1806 07259"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1165893"},{"key":"ref9","article-title":"PDE-net 2.0: Learning PDEs from data with a numeric-symbolic hybrid deep network","author":"long","year":"2018","journal-title":"arXiv 1812 04426"},{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BF00175355"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1007\/s11432-010-0090-0"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2013.10.023"},{"key":"ref21","first-page":"1225","article-title":"Representative of ${\\text{L}}_{1\/2}$\n Regularization among ${\\text{L}}_{q}$\n (0 < q $\\leq1$\n) regularizations: An experimental study based on phase diagram","volume":"38","author":"xu","year":"2012","journal-title":"ACTA Automatica Sinica"},{"key":"ref24","author":"abadi","year":"2015","journal-title":"TensorFlow Large-Scale Machine Learning on Heterogeneous Systems"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2013.11.006"},{"key":"ref26","first-page":"192","article-title":"The loss surfaces of multilayer networks","volume":"38","author":"choromanska","year":"2015","journal-title":"J Mach Learn Res"},{"key":"ref25","first-page":"26","article-title":"Lecture 6.5-RMSPROP: Divide the gradient by a running average of its recent magnitude","volume":"4","author":"tieleman","year":"2012","journal-title":"Neural Netw Mach Learning"}],"container-title":["IEEE Transactions on Neural Networks and Learning Systems"],"original-title":[],"link":[{"URL":"http:\/\/xplorestaging.ieee.org\/ielx7\/5962385\/9525619\/09180100.pdf?arnumber=9180100","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T11:45:49Z","timestamp":1641987949000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/9180100\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9]]},"references-count":28,"journal-issue":{"issue":"9"},"URL":"https:\/\/doi.org\/10.1109\/tnnls.2020.3017010","relation":{},"ISSN":["2162-237X","2162-2388"],"issn-type":[{"value":"2162-237X","type":"print"},{"value":"2162-2388","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,9]]}}}