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Here, to break this bottleneck, we have developed a deep equivariant neural network framework to represent the density functional theory Hamiltonian of magnetic materials for efficient electronic-structure calculation. A neural network architecture incorporating a priori knowledge of fundamental physical principles, especially the nearsightedness principle and the equivariance requirements of Euclidean and time-reversal symmetries (<jats:inline-formula><jats:alternatives><jats:tex-math>$$E(3)\\times \\{I,{{{\\mathcal{T}}}}\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>E<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mn>3<\/mml:mn>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u00d7<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>{<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>I<\/mml:mi>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mi>T<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mo>}<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), is designed, which is critical to capture the subtle magnetic effects. Systematic experiments on spin-spiral, nanotube and moir\u00e9 magnets were performed, making the challenging study of magnetic skyrmions feasible.<\/jats:p>","DOI":"10.1038\/s43588-023-00424-3","type":"journal-article","created":{"date-parts":[[2023,4,26]],"date-time":"2023-04-26T16:02:51Z","timestamp":1682524971000},"page":"321-327","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":49,"title":["Deep-learning electronic-structure calculation of magnetic superstructures"],"prefix":"10.1038","volume":"3","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5967-5251","authenticated-orcid":false,"given":"He","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zechen","family":"Tang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xiaoxun","family":"Gong","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8746-8970","authenticated-orcid":false,"given":"Nianlong","family":"Zou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9685-2547","authenticated-orcid":false,"given":"Wenhui","family":"Duan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4844-2460","authenticated-orcid":false,"given":"Yong","family":"Xu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,4,26]]},"reference":[{"key":"424_CR1","doi-asserted-by":"publisher","first-page":"899","DOI":"10.1038\/nnano.2013.243","volume":"8","author":"N Nagaosa","year":"2013","unstructured":"Nagaosa, N. & Tokura, Y. 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