{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T12:51:02Z","timestamp":1762260662104,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T00:00:00Z","timestamp":1762214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"PhD Research Startup Foundation of Guangdong University of Science and Technology","award":["GKY-2022BSQD-36"],"award-info":[{"award-number":["GKY-2022BSQD-36"]}]},{"name":"Principal Investigator of the Guangdong Provincial Education Planning Project","award":["20180550996"],"award-info":[{"award-number":["20180550996"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["BDCC"],"abstract":"<jats:p>A Bose-Fermi mixture, consisting of both bosons and fermions, exhibits distinctive quantum coherence and phase transitions, offering valuable insights into many-body quantum systems. The ground state, as the system\u2019s lowest energy configuration, is essential for understanding its overall behavior. In this study, we introduce the Bose-Fermi Energy-based Deep Neural Network (BF-EnDNN), a novel deep learning approach designed to solve the ground-state problem of Bose-Fermi mixtures at zero temperature through energy minimization. This method incorporates three key innovations: point sampling pre-training, a Dynamic Symmetry Layer (DSL), and a Positivity Preserving Layer (PPL). These features significantly improve the network\u2019s accuracy and stability in quantum calculations. Our numerical results show that BF-EnDNN achieves accuracy comparable to traditional finite difference methods, with effective extension to two-dimensional systems. The method demonstrates high precision across various parameters, making it a promising tool for investigating complex quantum systems.<\/jats:p>","DOI":"10.3390\/bdcc9110279","type":"journal-article","created":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T11:11:16Z","timestamp":1762254676000},"page":"279","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Deep Learning-Based Method for a Ground-State Solution of Bose-Fermi Mixture at Zero Temperature"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-0131-2834","authenticated-orcid":false,"given":"Xianghong","family":"He","sequence":"first","affiliation":[{"name":"College of General Education, Guangdong University of Science and Technology, Dongguan 523668, China"}]},{"given":"Jidong","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 515200, China"}]},{"given":"Rentao","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 515200, China"}]},{"given":"Yuhan","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 515200, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1782-4922","authenticated-orcid":false,"given":"Rongpei","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Advanced Manufacturing, Guangdong University of Technology, Jieyang 515200, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1137\/120873820","article-title":"Analysis and Computation for Ground State Solutions of Bose\u2013Fermi Mixtures at Zero Temperature","volume":"73","author":"Cai","year":"2013","journal-title":"SIAM J. 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