{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T06:28:50Z","timestamp":1775197730517,"version":"3.50.1"},"reference-count":34,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2024,10,16]],"date-time":"2024-10-16T00:00:00Z","timestamp":1729036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Key Scientific Research Projects Plan of Henan Higher Education Institutions","award":["24A120009"],"award-info":[{"award-number":["24A120009"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Solving shallow water equations is crucial in science and engineering for understanding and predicting natural phenomena. To address the limitations of Physics-Informed Neural Network (PINN) in solving shallow water equations, we propose an improved PINN algorithm integrated with a deep learning framework. This algorithm introduces a regularization term as a penalty in the loss function, based on the PINN and Long Short-Term Memory (LSTM) models, and incorporates an attention mechanism to solve the original equation across the entire domain. Simulation experiments were conducted on one-dimensional and two-dimensional shallow water equations. The results indicate that, compared to the classical PINN algorithm, the improved algorithm shows significant advantages in handling discontinuities, such as sparse waves, in one-dimensional problems. It accurately captures sparse waves and avoids smoothing effects. In two-dimensional problems, the improved algorithm demonstrates good symmetry and effectively reduces non-physical oscillations. It also shows significant advantages in capturing details and handling complex phenomena, offering higher reliability and accuracy. The improved PINNs algorithm, which combines neural networks with physical mechanisms, can provide robust solutions and effectively avoid some of the shortcomings of classical PINNs methods. It also possesses high resolution and strong generalization capabilities, enabling accurate predictions at any given moment.<\/jats:p>","DOI":"10.3390\/sym16101376","type":"journal-article","created":{"date-parts":[[2024,10,18]],"date-time":"2024-10-18T09:14:48Z","timestamp":1729242888000},"page":"1376","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["An Improved PINN Algorithm for Shallow Water Equations Driven by Deep Learning"],"prefix":"10.3390","volume":"16","author":[{"given":"Yanling","family":"Li","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China"}]},{"given":"Qianxing","family":"Sun","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China"}]},{"given":"Junfang","family":"Wei","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China"}]},{"given":"Chunyan","family":"Huang","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,10,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1017\/S0962492918000028","article-title":"Finite-volume schemes for shallow-water equations","volume":"27","author":"Kurganov","year":"2018","journal-title":"Acta Numer."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"083508","DOI":"10.1063\/5.0031936","article-title":"Discrete shallow water equations preserving symmetries and conservation laws","volume":"62","author":"Dorodnitsyn","year":"2021","journal-title":"J. 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