{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,13]],"date-time":"2026-01-13T21:16:07Z","timestamp":1768338967138,"version":"3.49.0"},"reference-count":42,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,7,4]],"date-time":"2024-07-04T00:00:00Z","timestamp":1720051200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["52105504"],"award-info":[{"award-number":["52105504"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2022YFB3403800"],"award-info":[{"award-number":["2022YFB3403800"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["52105504"],"award-info":[{"award-number":["52105504"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2022YFB3403800"],"award-info":[{"award-number":["2022YFB3403800"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This paper proposed a two-dimensional steady-state field prediction approach that combines B-spline functions and a fully connected neural network. In this approach, field data, which are determined by corresponding control vectors, are fitted by a selected B-spline function set, yielding the corresponding best-fitting weight vectors, and then a fully connected neural network is trained using those weight vectors and control vectors. The trained neural network first predicts a weight vector using a given control vector, and then the corresponding field can be restored via the selected B-spline set. This method was applied to learn and predict two-dimensional steady advection\u2013diffusion physical fields with absorption and source terms, and its accuracy and performance were tested and verified by a series of numerical experiments with different B-spline sets, boundary conditions, field gradients, and field states. The proposed method was finally compared with a generative adversarial network (GAN) and a physics-informed neural network (PINN). The results indicated that the B-spline neural network could predict the tested physical fields well; the overall error can be reduced by expanding the selected B-spline set. Compared with GAN and PINN, the proposed method also presented the advantages of a high prediction accuracy, less demand for training data, and high training efficiency.<\/jats:p>","DOI":"10.3390\/e26070577","type":"journal-article","created":{"date-parts":[[2024,7,4]],"date-time":"2024-07-04T06:28:54Z","timestamp":1720074534000},"page":"577","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Best-Fitting B-Spline Neural Network Approach to the Prediction of Advection\u2013Diffusion Physical Fields with Absorption and Source Terms"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0009-0007-6622-1008","authenticated-orcid":false,"given":"Xuedong","family":"Zhu","sequence":"first","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4881-2193","authenticated-orcid":false,"given":"Jianhua","family":"Liu","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"},{"name":"Hebei Key Laboratory of Intelligent Assembly and Detection Technology, Tangshan Research Institute, Beijing Institute of Technology, Tangshan 063000, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-8609-3688","authenticated-orcid":false,"given":"Xiaohui","family":"Ao","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"},{"name":"Hebei Key Laboratory of Intelligent Assembly and Detection Technology, Tangshan Research Institute, Beijing Institute of Technology, Tangshan 063000, China"}]},{"given":"Sen","family":"He","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"}]},{"given":"Lei","family":"Tao","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"}]},{"given":"Feng","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China"}]}],"member":"1968","published-online":{"date-parts":[[2024,7,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"106553","DOI":"10.1016\/j.agwat.2020.106553","article-title":"Numerical model to predict water temperature distribution in a paddy rice field","volume":"245","author":"Nishida","year":"2021","journal-title":"Agric. 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