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An optimum network is obtained by minimizing the normalized mean square error (NMSE) of training data, and L-BFGS is the optimized algorithm of second-order precision. Numerical experimental results show that 3D-PDE-Net can achieve the solution with good accuracy using few training samples, and it is of highly significant in solving linear and nonlinear unsteady PDEs.<\/jats:p>","DOI":"10.1162\/neco_a_01462","type":"journal-article","created":{"date-parts":[[2021,12,16]],"date-time":"2021-12-16T22:55:39Z","timestamp":1639695339000},"page":"518-540","update-policy":"https:\/\/doi.org\/10.1162\/mitpressjournals.corrections.policy","source":"Crossref","is-referenced-by-count":12,"title":["Convolution-Based Model-Solving Method for Three-Dimensional, Unsteady, Partial Differential Equations"],"prefix":"10.1162","volume":"34","author":[{"given":"Wenshu","family":"Zha","sequence":"first","affiliation":[{"name":"Hefei University of Technology, Hefei, Anhui, 230009, China wszha@hfut.edu.cn"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Wen","family":"Zhang","sequence":"additional","affiliation":[{"name":"Hefei University of Technology, Hefei, Anhui, 230009, China 2019111279@mail.hfut.edu.cn"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daolun","family":"Li","sequence":"additional","affiliation":[{"name":"Hefei University of Technology, Hefei, Anhui, 230009, China ldaol@ustc.edu.cn"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yan","family":"Xing","sequence":"additional","affiliation":[{"name":"Hefei University of Technology, Hefei, Anhui, 230009, China xy1128@126.com"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lei","family":"He","sequence":"additional","affiliation":[{"name":"Hefei University of Technology, Hefei, Anhui, 230009, China hlei80@163.com"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jieqing","family":"Tan","sequence":"additional","affiliation":[{"name":"Hefei University of Technology, Hefei, Anhui, 230009, China jqtan@mail.hf.ah.cn"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"281","published-online":{"date-parts":[[2022,1,14]]},"reference":[{"key":"2022040618224433900_B1","doi-asserted-by":"crossref","DOI":"10.1515\/9783112707173","volume-title":"Inverse problems in differential equations","author":"Anger","year":"1990"},{"key":"2022040618224433900_B2","doi-asserted-by":"publisher","first-page":"3932","DOI":"10.1073\/pnas.1517384113","article-title":"Discovering governing equations from data by sparse identification of nonlinear dynamical systems","volume":"113","author":"Brunton","year":"2016","journal-title":"Proceedings of the National Academy of Sciences"},{"key":"2022040618224433900_B3","doi-asserted-by":"publisher","first-page":"1033","DOI":"10.1090\/S0894-0347-2012-00740-1","article-title":"Image restoration: Total variation, wavelet frames, and beyond","volume":"25","author":"Cai","year":"2012","journal-title":"Journal of the American Mathematical Society"},{"key":"2022040618224433900_B4","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1109\/ICCIS.2011.35","article-title":"A high-order compact ADI scheme for the three-dimensional unsteady convection-diffusion equation","volume-title":"Proceedings of the 2011 International Conference on Computational and Information Sciences","author":"Cao","year":"2011"},{"key":"2022040618224433900_B5","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1016\/j.jcp.2019.05.008","article-title":"Identification of physical processes via combined data-driven and data-assimilation methods","volume":"393","author":"Chang","year":"2019","journal-title":"Journal of Computational Physics"},{"key":"2022040618224433900_B6","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1002\/(SICI)1098-2426(199803)14:2&lt;159::AID-NUM2&gt;3.0.CO;2-N","article-title":"A second-order ADI scheme for three-dimensional parabolic differential equations","volume":"14","author":"Dai","year":"1998","journal-title":"Numerical Methods for Partial Differential Equations"},{"key":"2022040618224433900_B7","doi-asserted-by":"publisher","first-page":"606","DOI":"10.1137\/15M1037457","article-title":"Image restoration: Wavelet frame shrinkage, nonlinear evolution PDEs, and beyond","volume":"15","author":"Dong","year":"2017","journal-title":"Multiscale Modeling and Simulation"},{"key":"2022040618224433900_B8","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1007\/BF01386093","article-title":"A general formulation of alternating direction methods, I. 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