{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,8]],"date-time":"2026-03-08T17:15:58Z","timestamp":1772990158028,"version":"3.50.1"},"reference-count":43,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2025,11,6]],"date-time":"2025-11-06T00:00:00Z","timestamp":1762387200000},"content-version":"vor","delay-in-days":309,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"funder":[{"DOI":"10.13039\/501100015956","name":"Special Project for Research and Development in Key areas of Guangdong Province","doi-asserted-by":"publisher","award":["2024ZDZX2096"],"award-info":[{"award-number":["2024ZDZX2096"]}],"id":[{"id":"10.13039\/501100015956","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100015956","name":"Special Project for Research and Development in Key areas of Guangdong Province","doi-asserted-by":"publisher","award":["2024KCXTD054"],"award-info":[{"award-number":["2024KCXTD054"]}],"id":[{"id":"10.13039\/501100015956","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2025,1]]},"abstract":"<jats:p>\n                    This paper introduces a coupled physics\u2010informed neural network (C\u2013PINN) framework for simulating electron\u2013lattice thermal conduction. The framework uses an additive \u0394\u2010decomposition to couple the electron temperature\n                    <jats:italic>T<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>e<\/jats:italic>\n                    <\/jats:sub>\n                    and lattice temperature\n                    <jats:italic>T<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>l<\/jats:italic>\n                    <\/jats:sub>\n                    , with a learnable correction term \u0394(\n                    <jats:italic>t<\/jats:italic>\n                    ,\n                    <jats:italic>x<\/jats:italic>\n                    ) capturing spatial\u2013temporal deviations from thermal equilibrium. Additionally, a residual\u2010balanced and localized adaptive refinement (RLAR) strategy is employed to enhance model accuracy and training efficiency, especially in high\u2010error regions. Numerical experiments demonstrate the model\u2019s robustness across different Knudsen numbers (\n                    <jats:italic>K<\/jats:italic>\n                    <jats:sub>\n                      <jats:italic>n<\/jats:italic>\n                    <\/jats:sub>\n                    ) and boundary conditions, showing its ability to capture fine\u2010grained temperature deviations and optimize high\u2010error regions through dynamic adaptive sampling.\n                  <\/jats:p>","DOI":"10.1155\/ijmm\/5570716","type":"journal-article","created":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T06:42:16Z","timestamp":1762497736000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["C\u2013PINN: A Coupled Physics\u2010Informed Neural Network for Solving Parabolic Two\u2010Step Models in Electron\u2013Lattice Thermal Conduction"],"prefix":"10.1155","volume":"2025","author":[{"given":"Zhihong","family":"Che","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boyang","family":"Liu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qi","family":"Lin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zhecheng","family":"Yang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8635-0527","authenticated-orcid":false,"given":"Liangcai","family":"Mei","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2025,11,6]]},"reference":[{"key":"e_1_2_11_1_2","first-page":"173","article-title":"Relaxation Between Electrons and the Crystalline Lattice","volume":"4","author":"Kaganov M. 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