{"id":88572,"date":"2026-08-11T08:44:00","date_gmt":"2026-08-11T05:44:00","guid":{"rendered":"https:\/\/twodots.gr\/?p=88572"},"modified":"2026-08-11T08:44:00","modified_gmt":"2026-08-11T05:44:00","slug":"eso-techniti-noimosyni-fysiki-akmes","status":"publish","type":"post","link":"https:\/\/twodots.gr\/bg\/eso-techniti-noimosyni-fysiki-akmes\/","title":{"rendered":"ESO: \u03cc\u03c4\u03b1\u03bd \u03b7 \u03c4\u03b5\u03c7\u03bd\u03b7\u03c4\u03ae \u03bd\u03bf\u03b7\u03bc\u03bf\u03c3\u03cd\u03bd\u03b7 \u03bc\u03b1\u03b8\u03b1\u03af\u03bd\u03b5\u03b9 \u03c4\u03b7 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ae \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03b1\u03ba\u03bc\u03ad\u03c2"},"content":{"rendered":"<div class=\"td-article-lede\">\n<p><strong>\u03a4\u03bf Edge-Conditioned Spectral Operator (ESO) \u03b5\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03b5\u03af \u03bd\u03b1 \u03bb\u03cd\u03c3\u03b5\u03b9 \u03ad\u03bd\u03b1 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03c9\u03bd neural operators: \u03b7 \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03bc\u03af\u03be\u03b7 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03c7\u03ac\u03bd\u03b5\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03c4\u03b9\u03c2 \u03b1\u03c0\u03cc\u03c4\u03bf\u03bc\u03b5\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ad\u03c2 \u03cc\u03c0\u03bf\u03c5 \u03b7 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ae \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03b9\u03bf \u03b1\u03c0\u03b1\u03b9\u03c4\u03b7\u03c4\u03b9\u03ba\u03ae.<\/strong><\/p>\n<p>\u0397 \u03c0\u03c1\u03ce\u03c4\u03b7 \u03ad\u03ba\u03b4\u03bf\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1\u03c2 \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03c4\u03bf \u03c7\u03b1\u03bc\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03bf \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03cc L2 error \u03ba\u03b1\u03b9 \u03c3\u03c4\u03b1 \u03b5\u03bd\u03bd\u03ad\u03b1 benchmarks \u03c0\u03bf\u03c5 \u03b5\u03be\u03b5\u03c4\u03ac\u03c3\u03c4\u03b7\u03ba\u03b1\u03bd. \u03a4\u03bf \u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03bf \u03b5\u03cd\u03c1\u03b7\u03bc\u03b1 \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf \u03ad\u03bd\u03b1\u03c2 \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf\u03c2 \u03bc\u03ad\u03c3\u03bf\u03c2 \u03cc\u03c1\u03bf\u03c2, \u03b1\u03bb\u03bb\u03ac \u03b7 \u03bc\u03b5\u03af\u03c9\u03c3\u03b7 \u03c3\u03c6\u03b1\u03bb\u03bc\u03ac\u03c4\u03c9\u03bd \u03ba\u03bf\u03bd\u03c4\u03ac \u03c3\u03b5 coefficient jumps, \u03b4\u03af\u03bd\u03b5\u03c2 \u03ba\u03b1\u03b9 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2 \u03c5\u03c8\u03b7\u03bb\u03ae\u03c2 \u03ba\u03bb\u03af\u03c3\u03b7\u03c2. \u03a0\u03b1\u03c1\u03b1\u03bc\u03ad\u03bd\u03b5\u03b9, \u03cc\u03bc\u03c9\u03c2, preprint \u03ba\u03b1\u03b9 \u03ba\u03ac\u03b8\u03b5 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03b9\u03ba\u03ae \u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b1\u03bd\u03b5\u03be\u03ac\u03c1\u03c4\u03b7\u03c4\u03b7 \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03ae.<\/p>\n<\/div>\n<div class=\"td-article-toc\">\n<div class=\"td-toc-title\">\u03a0\u03b5\u03c1\u03b9\u03b5\u03c7\u03cc\u03bc\u03b5\u03bd\u03b1<\/div>\n<ul>\n<li><a href=\"#pagkosmia-topiki-orasi\">\u0393\u03b9\u03b1\u03c4\u03af \u03bf\u03b9 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b9\u03ba\u03ad\u03c2 \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03b9\u03c2 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 \u03ba\u03b1\u03b9 \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ae \u03cc\u03c1\u03b1\u03c3\u03b7<\/a><\/li>\n<li><a href=\"#exomalynsi-krisimes-perioches\">\u03a4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03b7\u03c2 \u03b5\u03be\u03bf\u03bc\u03ac\u03bb\u03c5\u03bd\u03c3\u03b7\u03c2 \u03ba\u03bf\u03bd\u03c4\u03ac \u03c3\u03c4\u03b9\u03c2 \u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03b5\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2<\/a><\/li>\n<li><a href=\"#pvmm-geitonikes-diafores\">PVMM: \u03c0\u03ce\u03c2 \u03bf\u03b9 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03b3\u03b5\u03b9\u03c4\u03cc\u03bd\u03c9\u03bd \u03b5\u03c0\u03b7\u03c1\u03b5\u03ac\u03b6\u03bf\u03c5\u03bd \u03c4\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac modes<\/a><\/li>\n<li><a href=\"#par-physics-aware-reweighting\">PAR: \u03cc\u03c4\u03b1\u03bd \u03c4\u03bf loss \u03b4\u03af\u03bd\u03b5\u03b9 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf \u03b2\u03ac\u03c1\u03bf\u03c2 \u03b5\u03ba\u03b5\u03af \u03cc\u03c0\u03bf\u03c5 \u03bc\u03b5\u03c4\u03c1\u03ac \u03b7 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ae<\/a><\/li>\n<li><a href=\"#domimena-mi-domimena-plegmata\">\u0394\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 \u03ba\u03b1\u03b9 \u03bc\u03b7 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 \u03c0\u03bb\u03ad\u03b3\u03bc\u03b1\u03c4\u03b1 \u03bc\u03b5 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03ae \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ac<\/a><\/li>\n<li><a href=\"#ennea-pde-benchmarks\">\u0395\u03bd\u03bd\u03ad\u03b1 benchmarks, \u03b1\u03c0\u03cc Darcy \u03ba\u03b1\u03b9 Navier\u2013Stokes \u03ad\u03c9\u03c2 \u03c1\u03bf\u03ae \u03b1\u03af\u03bc\u03b1\u03c4\u03bf\u03c2<\/a><\/li>\n<li><a href=\"#posotika-optika-apotelesmata\">\u03a4\u03b9 \u03b4\u03b5\u03af\u03c7\u03bd\u03bf\u03c5\u03bd \u03c4\u03b1 \u03c0\u03bf\u03c3\u03bf\u03c4\u03b9\u03ba\u03ac \u03ba\u03b1\u03b9 \u03bf\u03c0\u03c4\u03b9\u03ba\u03ac \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03ad\u03c3\u03bc\u03b1\u03c4\u03b1<\/a><\/li>\n<li><a href=\"#ablations-d-q-par\">\u03a4\u03b1 ablations \u03b5\u03be\u03b7\u03b3\u03bf\u03cd\u03bd \u03b3\u03b9\u03b1\u03c4\u03af \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 D, Q \u03ba\u03b1\u03b9 PAR<\/a><\/li>\n<li><a href=\"#theoritiki-idiotita\">\u03a4\u03b9 \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03b7 \u00ab\u03cc\u03c7\u03b9 \u03c7\u03b5\u03b9\u03c1\u03cc\u03c4\u03b5\u03c1\u03b7\u00bb \u03b8\u03b5\u03c9\u03c1\u03b7\u03c4\u03b9\u03ba\u03ae \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1<\/a><\/li>\n<li><a href=\"#epicheirimatiki-anagnosi\">\u0397 \u03b5\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03b1\u03bd\u03ac\u03b3\u03bd\u03c9\u03c3\u03b7: \u03b1\u03ba\u03c1\u03af\u03b2\u03b5\u03b9\u03b1 \u03c3\u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03c0\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bd \u03c3\u03c5\u03bd\u03ad\u03c0\u03b5\u03b9\u03b5\u03c2<\/a><\/li>\n<li><a href=\"#oria-meletis-elegchos\">\u038c\u03c1\u03b9\u03b1 \u03c4\u03b7\u03c2 \u03bc\u03b5\u03bb\u03ad\u03c4\u03b7\u03c2 \u03ba\u03b1\u03b9 \u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ad\u03bb\u03b5\u03b3\u03c7\u03bf\u03c2 \u03c0\u03c1\u03b9\u03bd \u03b1\u03c0\u03cc \u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae<\/a><\/li>\n<li><a href=\"#symperasma\">\u0417\u0430\u043a\u043b\u044e\u0447\u0435\u043d\u0438\u0435<\/a><\/li>\n<\/ul>\n<\/div>\n<p>\u03a4\u03b1 \u03bd\u03b5\u03c5\u03c1\u03c9\u03bd\u03b9\u03ba\u03ac \u03bc\u03bf\u03bd\u03c4\u03ad\u03bb\u03b1 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03bd\u03b1 \u03c0\u03c1\u03bf\u03c3\u03b5\u03b3\u03b3\u03af\u03c3\u03bf\u03c5\u03bd \u03c4\u03b7 \u03bb\u03cd\u03c3\u03b7 \u03c3\u03cd\u03bd\u03b8\u03b5\u03c4\u03c9\u03bd \u03c6\u03c5\u03c3\u03b9\u03ba\u03ce\u03bd \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03c9\u03bd \u03c0\u03bf\u03bb\u03cd \u03c4\u03b1\u03c7\u03cd\u03c4\u03b5\u03c1\u03b1 \u03b1\u03c0\u03cc \u03bc\u03b9\u03b1 \u03bd\u03ad\u03b1 \u03b1\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03ae \u03b5\u03c0\u03af\u03bb\u03c5\u03c3\u03b7 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03c0\u03b5\u03c1\u03af\u03c0\u03c4\u03c9\u03c3\u03b7. \u03a4\u03bf \u03b4\u03cd\u03c3\u03ba\u03bf\u03bb\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf, \u03cc\u03bc\u03c9\u03c2, \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf \u03bd\u03b1 \u03ba\u03b1\u03c4\u03b1\u03bd\u03bf\u03ae\u03c3\u03bf\u03c5\u03bd \u03c4\u03b7 \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03ae \u03b5\u03b9\u03ba\u03cc\u03bd\u03b1 \u03b5\u03bd\u03cc\u03c2 \u03c0\u03b5\u03b4\u03af\u03bf\u03c5. \u03a0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b1\u03bd\u03c4\u03b9\u03bb\u03b7\u03c6\u03b8\u03bf\u03cd\u03bd \u03ba\u03b1\u03b9 \u03c4\u03b9 \u03c3\u03c5\u03bc\u03b2\u03b1\u03af\u03bd\u03b5\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03c3\u03c4\u03b1 \u03cc\u03c1\u03b9\u03b1: \u03b5\u03ba\u03b5\u03af \u03cc\u03c0\u03bf\u03c5 \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03b1\u03c0\u03cc\u03c4\u03bf\u03bc\u03b1 \u03ad\u03bd\u03b1 \u03c5\u03bb\u03b9\u03ba\u03cc, \u03c3\u03c7\u03b7\u03bc\u03b1\u03c4\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03bc\u03b9\u03b1 \u03b4\u03af\u03bd\u03b7 \u03ae \u03b5\u03bc\u03c6\u03b1\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b7 \u03b8\u03b5\u03c1\u03bc\u03bf\u03ba\u03c1\u03b1\u03c3\u03b9\u03b1\u03ba\u03ae \u03ba\u03bb\u03af\u03c3\u03b7. \u0397 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1 \u03c4\u03c9\u03bd Zhentao Tan, Ruijie Quan \u03ba\u03b1\u03b9 Yi Yang \u03c0\u03c1\u03bf\u03c4\u03b5\u03af\u03bd\u03b5\u03b9 \u03c4\u03bf <strong>Edge-Conditioned Spectral Operator (ESO)<\/strong>, \u03ad\u03bd\u03b1 \u03bc\u03bf\u03bd\u03c4\u03ad\u03bb\u03bf \u03c0\u03bf\u03c5 \u03b5\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03b5\u03af \u03bd\u03b1 \u03c3\u03c5\u03bd\u03b4\u03ad\u03c3\u03b5\u03b9 \u03b1\u03c5\u03c4\u03ad\u03c2 \u03c4\u03b9\u03c2 \u03b4\u03cd\u03bf \u03ba\u03bb\u03af\u03bc\u03b1\u03ba\u03b5\u03c2.<\/p>\n<p>\u0397 \u03b2\u03b1\u03c3\u03b9\u03ba\u03ae \u03b9\u03b4\u03ad\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b7: \u03b1\u03bd\u03c4\u03af \u03b7 \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae \u03c4\u03c9\u03bd \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u03c4\u03c1\u03cc\u03c0\u03c9\u03bd \u03bd\u03b1 \u03b5\u03be\u03b1\u03c1\u03c4\u03ac\u03c4\u03b1\u03b9 \u03ba\u03c5\u03c1\u03af\u03c9\u03c2 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03b5\u03bd\u03cc\u03c2 \u03bc\u03b5\u03bc\u03bf\u03bd\u03c9\u03bc\u03ad\u03bd\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u03c5, \u03c4\u03bf ESO \u03b5\u03be\u03b5\u03c4\u03ac\u03b6\u03b5\u03b9 \u03ba\u03b1\u03b9 \u03c4\u03b9\u03c2 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ad\u03c2 \u03c4\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u03c5 \u03b1\u03c0\u03cc \u03c4\u03bf\u03c5\u03c2 \u03b3\u03b5\u03af\u03c4\u03bf\u03bd\u03ad\u03c2 \u03c4\u03bf\u03c5. \u0388\u03c4\u03c3\u03b9, \u03b7 \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03bc\u03af\u03be\u03b7 \u03b4\u03b5\u03bd \u03bb\u03b5\u03b9\u03c4\u03bf\u03c5\u03c1\u03b3\u03b5\u03af \u03b1\u03c0\u03bf\u03ba\u03bf\u03bc\u03bc\u03ad\u03bd\u03b7 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ad\u03c2, \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03ad\u03c2 \u03b1\u03c3\u03c5\u03bd\u03ad\u03c7\u03b5\u03b9\u03b5\u03c2. \u0397 \u03bc\u03b5\u03bb\u03ad\u03c4\u03b7 \u03b1\u03be\u03b9\u03bf\u03bb\u03bf\u03b3\u03b5\u03af \u03c4\u03b7\u03bd \u03c0\u03c1\u03bf\u03c3\u03ad\u03b3\u03b3\u03b9\u03c3\u03b7 \u03c3\u03b5 \u03b5\u03bd\u03bd\u03ad\u03b1 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1 \u03bc\u03b5 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 \u03ba\u03b1\u03b9 \u03bc\u03b7 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 \u03c0\u03bb\u03ad\u03b3\u03bc\u03b1\u03c4\u03b1 \u03ba\u03b1\u03b9 \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03c4\u03b7\u03bd \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b7 \u03b5\u03c0\u03af\u03b4\u03bf\u03c3\u03b7 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03c4\u03c9\u03bd \u03c3\u03c5\u03b3\u03ba\u03c1\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03c9\u03bd \u03bc\u03bf\u03bd\u03c4\u03ad\u03bb\u03c9\u03bd \u03c3\u03b5 \u03cc\u03bb\u03b1 \u03c4\u03b1 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b1 tests.<\/p>\n<h2 id=\"pagkosmia-topiki-orasi\">\u0393\u03b9\u03b1\u03c4\u03af \u03bf\u03b9 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b9\u03ba\u03ad\u03c2 \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03b9\u03c2 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 \u03ba\u03b1\u03b9 \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ae \u03cc\u03c1\u03b1\u03c3\u03b7<\/h2>\n<p>\u039f\u03b9 \u03bc\u03b5\u03c1\u03b9\u03ba\u03ad\u03c2 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b9\u03ba\u03ad\u03c2 \u03b5\u03be\u03b9\u03c3\u03ce\u03c3\u03b5\u03b9\u03c2, \u03ae PDEs, \u03c0\u03b5\u03c1\u03b9\u03b3\u03c1\u03ac\u03c6\u03bf\u03c5\u03bd \u03c6\u03b1\u03b9\u03bd\u03cc\u03bc\u03b5\u03bd\u03b1 \u03cc\u03c0\u03c9\u03c2 \u03b7 \u03c1\u03bf\u03ae \u03c1\u03b5\u03c5\u03c3\u03c4\u03ce\u03bd, \u03b7 \u03bc\u03b5\u03c4\u03b1\u03c6\u03bf\u03c1\u03ac \u03b8\u03b5\u03c1\u03bc\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2 \u03ba\u03b1\u03b9 \u03b7 \u03bc\u03b7\u03c7\u03b1\u03bd\u03b9\u03ba\u03ae \u03c0\u03b1\u03c1\u03b1\u03bc\u03cc\u03c1\u03c6\u03c9\u03c3\u03b7. \u039f\u03b9 \u03ba\u03bb\u03b1\u03c3\u03b9\u03ba\u03ad\u03c2 \u03b1\u03c1\u03b9\u03b8\u03bc\u03b7\u03c4\u03b9\u03ba\u03ad\u03c2 \u03bc\u03ad\u03b8\u03bf\u03b4\u03bf\u03b9, \u03cc\u03c0\u03c9\u03c2 \u03c4\u03b1 \u03c0\u03b5\u03c0\u03b5\u03c1\u03b1\u03c3\u03bc\u03ad\u03bd\u03b1 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03b1, \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03bd\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b3\u03bf\u03c5\u03bd \u03b1\u03ba\u03c1\u03b9\u03b2\u03b5\u03af\u03c2 \u03bb\u03cd\u03c3\u03b5\u03b9\u03c2, \u03b1\u03bb\u03bb\u03ac \u03b7 \u03b5\u03c0\u03b1\u03bd\u03ac\u03bb\u03b7\u03c8\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03c0\u03af\u03bb\u03c5\u03c3\u03b7\u03c2 \u03b3\u03b9\u03b1 \u03c0\u03bf\u03bb\u03bb\u03ad\u03c2 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03ad\u03c2 \u03b5\u03b9\u03c3\u03cc\u03b4\u03bf\u03c5\u03c2 \u03ad\u03c7\u03b5\u03b9 \u03c5\u03c8\u03b7\u03bb\u03cc \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03c4\u03b9\u03ba\u03cc \u03ba\u03cc\u03c3\u03c4\u03bf\u03c2. \u039f\u03b9 neural operators \u03b5\u03ba\u03c0\u03b1\u03b9\u03b4\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03ce\u03c3\u03c4\u03b5 \u03bd\u03b1 \u03c0\u03c1\u03bf\u03c3\u03b5\u03b3\u03b3\u03af\u03b6\u03bf\u03c5\u03bd \u03b1\u03c0\u03b5\u03c5\u03b8\u03b5\u03af\u03b1\u03c2 \u03c4\u03b7 \u03c7\u03b1\u03c1\u03c4\u03bf\u03b3\u03c1\u03ac\u03c6\u03b7\u03c3\u03b7 \u03b1\u03c0\u03cc \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03b5\u03b9\u03c3\u03cc\u03b4\u03bf\u03c5 \u03c3\u03b5 \u03bc\u03b9\u03b1 \u03c3\u03c5\u03bd\u03ac\u03c1\u03c4\u03b7\u03c3\u03b7 \u03b5\u03be\u03cc\u03b4\u03bf\u03c5.<\/p>\n<p>\u039f\u03b9 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03af operators, \u03bc\u03b5 \u03b3\u03bd\u03c9\u03c3\u03c4\u03cc \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1 \u03c4\u03bf Fourier Neural Operator, \u03bc\u03b5\u03c4\u03b1\u03c6\u03ad\u03c1\u03bf\u03c5\u03bd \u03c4\u03bf \u03c0\u03b5\u03b4\u03af\u03bf \u03c3\u03b5 \u03ad\u03bd\u03b1\u03bd \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03c7\u03ce\u03c1\u03bf \u03ba\u03b1\u03b9 \u03b1\u03bd\u03b1\u03bc\u03b5\u03b9\u03b3\u03bd\u03cd\u03bf\u03c5\u03bd \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1 \u03c3\u03b5 \u03bc\u03b5\u03b3\u03ac\u03bb\u03b7 \u03b1\u03c0\u03cc\u03c3\u03c4\u03b1\u03c3\u03b7. \u0391\u03c5\u03c4\u03cc \u03c4\u03bf\u03c5\u03c2 \u03b4\u03af\u03bd\u03b5\u03b9 \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03b5\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03ba\u03ac\u03bb\u03c5\u03c8\u03b7. \u03a9\u03c3\u03c4\u03cc\u03c3\u03bf, \u03bc\u03b9\u03b1 \u03b4\u03b9\u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b1 \u03b4\u03b9\u03b1\u03c0\u03b5\u03c1\u03b1\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2 \u03c3\u03c4\u03b7 \u03c1\u03bf\u03ae Darcy \u03ae \u03bc\u03b9\u03b1 \u03ad\u03bd\u03c4\u03bf\u03bd\u03b7 \u03b4\u03af\u03bd\u03b7 \u03b4\u03b5\u03bd \u03bf\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03bc\u03cc\u03bd\u03bf \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c4\u03b9\u03bc\u03ae \u03c3\u03c4\u03bf \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03cc \u03c3\u03b7\u03bc\u03b5\u03af\u03bf. \u039f\u03c1\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7 \u03c3\u03c7\u03ad\u03c3\u03b7 \u03c4\u03bf\u03c5 \u03bc\u03b5 \u03c4\u03bf \u03ba\u03bf\u03bd\u03c4\u03b9\u03bd\u03cc \u03c0\u03b5\u03c1\u03b9\u03b2\u03ac\u03bb\u03bb\u03bf\u03bd. \u0397 \u03bc\u03b5\u03bb\u03ad\u03c4\u03b7 \u03b5\u03bd\u03c4\u03bf\u03c0\u03af\u03b6\u03b5\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03b1\u03c5\u03c4\u03cc \u03c4\u03bf \u03ba\u03b5\u03bd\u03cc \u03c3\u03c4\u03b7\u03bd point-centered \u03c0\u03c1\u03bf\u03c3\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae.<\/p>\n<div class=\"td-article-note\">\n<p><strong>\u0391\u03c0\u03ac\u03bd\u03c4\u03b7\u03c3\u03b7 \u03bc\u03b5 \u03bc\u03af\u03b1 \u03c6\u03c1\u03ac\u03c3\u03b7:<\/strong> \u03c4\u03bf ESO \u03b4\u03b5\u03bd \u03b1\u03bd\u03c4\u03b9\u03ba\u03b1\u03b8\u03b9\u03c3\u03c4\u03ac \u03c4\u03b7\u03bd \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03b5\u03b9\u03ba\u03cc\u03bd\u03b1\u00b7 \u03c0\u03c1\u03bf\u03c3\u03b8\u03ad\u03c4\u03b5\u03b9 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ae \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03b1\u03ba\u03bc\u03ad\u03c2 \u03ce\u03c3\u03c4\u03b5 \u03b7 \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae \u03c4\u03c9\u03bd modes \u03bd\u03b1 \u03b1\u03bd\u03c4\u03b9\u03b4\u03c1\u03ac \u03c3\u03b5 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03ad\u03c2 \u03b1\u03bb\u03bb\u03b1\u03b3\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ba\u03ce\u03bd \u03c3\u03b7\u03bc\u03b5\u03af\u03c9\u03bd.<\/p>\n<\/div>\n<h2 id=\"exomalynsi-krisimes-perioches\">\u03a4\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1 \u03c4\u03b7\u03c2 \u03b5\u03be\u03bf\u03bc\u03ac\u03bb\u03c5\u03bd\u03c3\u03b7\u03c2 \u03ba\u03bf\u03bd\u03c4\u03ac \u03c3\u03c4\u03b9\u03c2 \u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03b5\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2<\/h2>\n<p>\u038c\u03c4\u03b1\u03bd \u03b4\u03cd\u03bf \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ba\u03ac \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03ad\u03c7\u03bf\u03c5\u03bd \u03c0\u03b1\u03c1\u03cc\u03bc\u03bf\u03b9\u03b1 \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03ae \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03b1\u03bb\u03bb\u03ac \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03ae \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ba\u03ae \u03b4\u03bf\u03bc\u03ae, \u03ad\u03bd\u03b1\u03c2 \u03bc\u03b7\u03c7\u03b1\u03bd\u03b9\u03c3\u03bc\u03cc\u03c2 \u03c0\u03bf\u03c5 \u03ba\u03bf\u03b9\u03c4\u03ac \u03bc\u03cc\u03bd\u03bf \u03c4\u03bf \u03ba\u03ad\u03bd\u03c4\u03c1\u03bf \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03c4\u03bf\u03c5\u03c2 \u03b4\u03ce\u03c3\u03b5\u03b9 \u03c0\u03b1\u03c1\u03cc\u03bc\u03bf\u03b9\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac \u03b2\u03ac\u03c1\u03b7. \u0391\u03c5\u03c4\u03cc \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03c1\u03bf\u03b2\u03bb\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03cc\u03c4\u03b1\u03bd \u03c4\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03c6\u03c5\u03c3\u03b9\u03ba\u03cc \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03b4\u03b9\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03bb\u03cc\u03b3\u03c9 \u03bc\u03b9\u03b1\u03c2 \u03b1\u03c0\u03cc\u03c4\u03bf\u03bc\u03b7\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ae\u03c2 \u03b1\u03bb\u03bb\u03b1\u03b3\u03ae\u03c2. \u0397 \u03c3\u03c5\u03bd\u03ad\u03c0\u03b5\u03b9\u03b1 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c5\u03c0\u03b5\u03c1\u03b2\u03bf\u03bb\u03b9\u03ba\u03ae \u03b5\u03be\u03bf\u03bc\u03ac\u03bb\u03c5\u03bd\u03c3\u03b7 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03b5\u03ba\u03b5\u03af \u03cc\u03c0\u03bf\u03c5 \u03b7 \u03bb\u03cd\u03c3\u03b7 \u03bf\u03c6\u03b5\u03af\u03bb\u03b5\u03b9 \u03bd\u03b1 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03bd\u03b5\u03b9 \u03b1\u03b9\u03c7\u03bc\u03b7\u03c1\u03ae.<\/p>\n<p>\u039f\u03b9 \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03b5\u03af\u03c2 \u03b4\u03af\u03bd\u03bf\u03c5\u03bd \u03ba\u03b1\u03b9 \u03b8\u03b5\u03c9\u03c1\u03b7\u03c4\u03b9\u03ba\u03ae \u03b4\u03b9\u03b1\u03c4\u03cd\u03c0\u03c9\u03c3\u03b7 \u03b1\u03c5\u03c4\u03bf\u03cd \u03c4\u03bf\u03c5 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd. \u0391\u03bd \u03b4\u03cd\u03bf \u03b5\u03af\u03c3\u03bf\u03b4\u03bf\u03b9 \u03ad\u03c7\u03bf\u03c5\u03bd \u03af\u03b4\u03b9\u03b1 \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03ae \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c3\u03b5 \u03ad\u03bd\u03b1\u03bd \u03ba\u03cc\u03bc\u03b2\u03bf \u03b1\u03bb\u03bb\u03ac \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03cc edge descriptor, \u03ad\u03bd\u03b1 \u03bc\u03bf\u03bd\u03c4\u03ad\u03bb\u03bf \u03c0\u03bf\u03c5 \u03ba\u03b1\u03b8\u03bf\u03c1\u03af\u03b6\u03b5\u03b9 \u03c4\u03b7\u03bd \u03c0\u03c1\u03cc\u03b2\u03bb\u03b5\u03c8\u03b7 \u03bc\u03cc\u03bd\u03bf \u03b1\u03c0\u03cc \u03c4\u03bf \u03ba\u03ad\u03bd\u03c4\u03c1\u03bf \u03b4\u03b5\u03bd \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03c4\u03b9\u03c2 \u03b4\u03b9\u03b1\u03c7\u03c9\u03c1\u03af\u03c3\u03b5\u03b9 \u03c3\u03c4\u03bf \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03bf \u03c3\u03b7\u03bc\u03b5\u03af\u03bf. \u03a4\u03bf ESO \u03b4\u03b5\u03bd \u03b4\u03b5\u03c3\u03bc\u03b5\u03cd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03b1\u03c5\u03c4\u03ae \u03c4\u03b7\u03bd \u03b1\u03b4\u03c5\u03bd\u03b1\u03bc\u03af\u03b1, \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae \u03c4\u03c9\u03bd modes \u03b5\u03be\u03b1\u03c1\u03c4\u03ac\u03c4\u03b1\u03b9 \u03c4\u03cc\u03c3\u03bf \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03ae \u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03cc\u03c3\u03bf \u03ba\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03c0\u03b5\u03c1\u03b9\u03b3\u03c1\u03b1\u03c6\u03ae \u03c4\u03c9\u03bd \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ce\u03bd.<\/p>\n<h2 id=\"pvmm-geitonikes-diafores\">PVMM: \u03c0\u03ce\u03c2 \u03bf\u03b9 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03b3\u03b5\u03b9\u03c4\u03cc\u03bd\u03c9\u03bd \u03b5\u03c0\u03b7\u03c1\u03b5\u03ac\u03b6\u03bf\u03c5\u03bd \u03c4\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac modes<\/h2>\n<p>\u039f \u03c0\u03c5\u03c1\u03ae\u03bd\u03b1\u03c2 \u03c4\u03bf\u03c5 ESO \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf <strong>Pairwise-Variation Modal Mixer (PVMM)<\/strong>. \u0393\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03af\u03b6\u03b5\u03b9 \u03b4\u03cd\u03bf \u03c3\u03c5\u03bc\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac \u03c3\u03c4\u03b1\u03c4\u03b9\u03c3\u03c4\u03b9\u03ba\u03ac. \u03a4\u03bf \u03c0\u03c1\u03ce\u03c4\u03bf, \u03bf \u03c4\u03bf\u03c0\u03b9\u03ba\u03cc\u03c2 \u03bc\u03b5\u03c4\u03b1\u03c4\u03bf\u03c0\u03b9\u03c3\u03c4\u03b9\u03ba\u03cc\u03c2 \u03cc\u03c1\u03bf\u03c2 D, \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03bc\u03ad\u03c3\u03b7 \u03c0\u03c1\u03bf\u03c3\u03b7\u03bc\u03b1\u03c3\u03bc\u03ad\u03bd\u03b7 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ac \u03b1\u03bd\u03ac\u03bc\u03b5\u03c3\u03b1 \u03c3\u03c4\u03bf \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03cc \u03c3\u03b7\u03bc\u03b5\u03af\u03bf \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c5\u03c2 \u03b3\u03b5\u03af\u03c4\u03bf\u03bd\u03ad\u03c2 \u03c4\u03bf\u03c5. \u0394\u03b9\u03b1\u03c4\u03b7\u03c1\u03b5\u03af \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1 \u03ba\u03b1\u03c4\u03b5\u03cd\u03b8\u03c5\u03bd\u03c3\u03b7\u03c2 \u03ba\u03b1\u03b9 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b1\u03c0\u03bf\u03c4\u03c5\u03c0\u03ce\u03c3\u03b5\u03b9 \u03b1\u03c3\u03cd\u03bc\u03bc\u03b5\u03c4\u03c1\u03b5\u03c2 \u03b1\u03bb\u03bb\u03b1\u03b3\u03ad\u03c2 \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03bc\u03b9\u03b1 \u03b4\u03b9\u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b1.<\/p>\n<p>\u039f\u03b9 \u03c0\u03c1\u03bf\u03c3\u03b7\u03bc\u03b1\u03c3\u03bc\u03ad\u03bd\u03b5\u03c2 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ad\u03c2 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03cc\u03bc\u03c9\u03c2 \u03bd\u03b1 \u03b1\u03bb\u03bb\u03b7\u03bb\u03bf\u03b1\u03bd\u03b1\u03b9\u03c1\u03b5\u03b8\u03bf\u03cd\u03bd. \u0393\u03b9\u03b1 \u03b1\u03c5\u03c4\u03cc \u03c4\u03bf PVMM \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03ba\u03b1\u03b9 \u03c4\u03bf\u03bd \u03cc\u03c1\u03bf Q, \u03c4\u03b7 \u03bc\u03ad\u03c3\u03b7 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03c3\u03bc\u03ad\u03bd\u03b7 \u03ad\u03bd\u03c4\u03b1\u03c3\u03b7 \u03c4\u03c9\u03bd \u03c4\u03bf\u03c0\u03b9\u03ba\u03ce\u03bd \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ce\u03bd. \u039f Q \u03b5\u03c0\u03b9\u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03bc\u03b9\u03b1 \u03b9\u03c3\u03c7\u03c5\u03c1\u03ae \u03b1\u03bd\u03c4\u03af\u03b8\u03b5\u03c3\u03b7 \u03b1\u03bd\u03b5\u03be\u03ac\u03c1\u03c4\u03b7\u03c4\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03b5\u03cd\u03b8\u03c5\u03bd\u03c3\u03ae \u03c4\u03b7\u03c2. \u03a4\u03bf \u03bc\u03bf\u03bd\u03c4\u03ad\u03bb\u03bf \u03c3\u03c5\u03bd\u03b5\u03bd\u03ce\u03bd\u03b5\u03b9 \u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c4\u03bf\u03c5 \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03bf\u03cd \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u03c5, \u03c4\u03bf D \u03ba\u03b1\u03b9 \u03c4\u03bf Q \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03bc\u03b5\u03c4\u03b1\u03c4\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bc\u03ad\u03c3\u03c9 \u03b5\u03bd\u03cc\u03c2 gating network \u03c3\u03b5 logits \u03b3\u03b9\u03b1 \u03c4\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ac modes.<\/p>\n<p>\u0397 \u03c4\u03b5\u03bb\u03b9\u03ba\u03ae \u03b2\u03b1\u03b8\u03bc\u03bf\u03bb\u03bf\u03b3\u03af\u03b1 \u03c3\u03c5\u03bd\u03b4\u03c5\u03ac\u03b6\u03b5\u03b9 \u03c4\u03bf point-conditioned \u03ba\u03b1\u03b9 \u03c4\u03bf edge-conditioned \u03c3\u03ba\u03ad\u03bb\u03bf\u03c2. \u0394\u03b5\u03bd \u03b5\u03b3\u03ba\u03b1\u03c4\u03b1\u03bb\u03b5\u03af\u03c0\u03b5\u03b9 \u03b5\u03c0\u03bf\u03bc\u03ad\u03bd\u03c9\u03c2 \u03c4\u03b7 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03b2\u03ac\u03c3\u03b7 \u03bf\u03cd\u03c4\u03b5 \u03c4\u03b7\u03bd \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03bc\u03af\u03be\u03b7. \u03a4\u03b7\u03bd \u03c0\u03c1\u03bf\u03c3\u03b1\u03c1\u03bc\u03cc\u03b6\u03b5\u03b9 \u03bc\u03b5 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03b1\u03ba\u03bc\u03ad\u03c2 \u03c4\u03bf\u03c5 \u03c4\u03bf\u03c0\u03b9\u03ba\u03bf\u03cd \u03b3\u03c1\u03ac\u03c6\u03bf\u03c5. \u039f\u03b9 \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03b5\u03af\u03c2 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03bf\u03cd\u03bd \u03b9\u03b4\u03b9\u03bf\u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 Laplace\u2013Beltrami, \u03ce\u03c3\u03c4\u03b5 \u03b7 \u03af\u03b4\u03b9\u03b1 \u03bb\u03bf\u03b3\u03b9\u03ba\u03ae \u03bd\u03b1 \u03b5\u03c6\u03b1\u03c1\u03bc\u03cc\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c4\u03cc\u03c3\u03bf \u03c3\u03b5 \u03ba\u03b1\u03bd\u03bf\u03bd\u03b9\u03ba\u03ac grids \u03cc\u03c3\u03bf \u03ba\u03b1\u03b9 \u03c3\u03b5 \u03b1\u03ba\u03b1\u03bd\u03cc\u03bd\u03b9\u03c3\u03c4\u03b1 meshes.<\/p>\n<h2 id=\"par-physics-aware-reweighting\">PAR: \u03cc\u03c4\u03b1\u03bd \u03c4\u03bf loss \u03b4\u03af\u03bd\u03b5\u03b9 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf \u03b2\u03ac\u03c1\u03bf\u03c2 \u03b5\u03ba\u03b5\u03af \u03cc\u03c0\u03bf\u03c5 \u03bc\u03b5\u03c4\u03c1\u03ac \u03b7 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ae<\/h2>\n<p>\u03a4\u03bf \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03bf \u03c3\u03c5\u03c3\u03c4\u03b1\u03c4\u03b9\u03ba\u03cc \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf <strong>Physics-Aware Reweighting (PAR)<\/strong>. \u0395\u03bd\u03ce \u03c4\u03bf PVMM \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03b5\u03c3\u03c9\u03c4\u03b5\u03c1\u03b9\u03ba\u03ac \u03c4\u03bf\u03bd \u03c4\u03c1\u03cc\u03c0\u03bf \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae\u03c2 \u03c4\u03c9\u03bd modes, \u03c4\u03bf PAR \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c4\u03bf \u03b5\u03ba\u03c0\u03b1\u03b9\u03b4\u03b5\u03c5\u03c4\u03b9\u03ba\u03cc \u03c3\u03ae\u03bc\u03b1. \u039a\u03b1\u03c4\u03b1\u03c3\u03ba\u03b5\u03c5\u03ac\u03b6\u03b5\u03b9 \u03ad\u03bd\u03b1\u03bd \u03c7\u03ac\u03c1\u03c4\u03b7 \u03b5\u03c5\u03b1\u03b9\u03c3\u03b8\u03b7\u03c3\u03af\u03b1\u03c2 \u03b1\u03c0\u03cc \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03c0\u03b5\u03b4\u03af\u03b1 \u03c0\u03bf\u03c5 \u03c5\u03c0\u03ac\u03c1\u03c7\u03bf\u03c5\u03bd \u03ae\u03b4\u03b7 \u03c3\u03c4\u03bf \u03b5\u03ba\u03ac\u03c3\u03c4\u03bf\u03c4\u03b5 benchmark \u03ba\u03b1\u03b9 \u03b4\u03af\u03bd\u03b5\u03b9 \u03c5\u03c8\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03bf \u03b2\u03ac\u03c1\u03bf\u03c2 \u03c3\u03c4\u03b1 \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u03c4\u03c9\u03bd \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03ce\u03bd \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ce\u03bd.<\/p>\n<p>\u0397 \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae \u03c4\u03bf\u03c5 \u03c0\u03b5\u03b4\u03af\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 task-specific. \u03a3\u03c4\u03b7 \u03c1\u03bf\u03ae Darcy \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03b5\u03af\u03c4\u03b1\u03b9 \u03bf \u03c3\u03c5\u03bd\u03c4\u03b5\u03bb\u03b5\u03c3\u03c4\u03ae\u03c2 \u03b4\u03b9\u03b1\u03c0\u03b5\u03c1\u03b1\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2, \u03c3\u03c4\u03b7 Navier\u2013Stokes \u03b7 \u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03b7 \u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c3\u03c4\u03c1\u03bf\u03b2\u03b9\u03bb\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2 \u03ba\u03b1\u03c4\u03ac \u03c4\u03bf rollout, \u03c3\u03c4\u03b7 \u03bc\u03b5\u03c4\u03b1\u03c6\u03bf\u03c1\u03ac \u03b8\u03b5\u03c1\u03bc\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2 \u03b7 \u03b8\u03b5\u03c1\u03bc\u03bf\u03ba\u03c1\u03b1\u03c3\u03b9\u03b1\u03ba\u03ae \u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03ba\u03b1\u03b9 \u03c3\u03c4\u03b7 \u03c1\u03bf\u03ae \u03b1\u03af\u03bc\u03b1\u03c4\u03bf\u03c2 \u03b7 \u03c7\u03c1\u03bf\u03bd\u03b9\u03ba\u03ae \u03c4\u03c1\u03bf\u03c7\u03b9\u03ac \u03c4\u03b7\u03c2 \u03c4\u03b1\u03c7\u03cd\u03c4\u03b7\u03c4\u03b1\u03c2. \u03a4\u03b1 \u03b2\u03ac\u03c1\u03b7 \u03ba\u03b1\u03bd\u03bf\u03bd\u03b9\u03ba\u03bf\u03c0\u03bf\u03b9\u03bf\u03cd\u03bd\u03c4\u03b1\u03b9 \u03b1\u03bd\u03ac \u03b4\u03b5\u03af\u03b3\u03bc\u03b1, \u03ce\u03c3\u03c4\u03b5 \u03bd\u03b1 \u03b5\u03ba\u03c6\u03c1\u03ac\u03b6\u03bf\u03c5\u03bd \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03ae \u03b5\u03c5\u03b1\u03b9\u03c3\u03b8\u03b7\u03c3\u03af\u03b1 \u03ba\u03b1\u03b9 \u03bd\u03b1 \u03bc\u03b7\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03bf\u03c5\u03bd \u03b1\u03bd\u03b5\u03be\u03ad\u03bb\u03b5\u03b3\u03ba\u03c4\u03b1 \u03c4\u03b7 \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03ae \u03ba\u03bb\u03af\u03bc\u03b1\u03ba\u03b1 \u03c4\u03bf\u03c5 loss.<\/p>\n<p>PVMM \u03ba\u03b1\u03b9 PAR \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03b4\u03cd\u03bf \u03bf\u03bd\u03bf\u03bc\u03b1\u03c3\u03af\u03b5\u03c2 \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd \u03af\u03b4\u03b9\u03b1 \u03c0\u03b1\u03c1\u03ad\u03bc\u03b2\u03b1\u03c3\u03b7. \u03a4\u03bf \u03c0\u03c1\u03ce\u03c4\u03bf \u03bc\u03b1\u03b8\u03b1\u03af\u03bd\u03b5\u03b9 \u03bd\u03b1 \u03b4\u03b9\u03b1\u03bc\u03bf\u03c1\u03c6\u03ce\u03bd\u03b5\u03b9 \u03c4\u03b7 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03b5\u03c0\u03b5\u03be\u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ad\u03c2. \u03a4\u03bf \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03bf \u03b4\u03b7\u03bb\u03ce\u03bd\u03b5\u03b9 \u03c1\u03b7\u03c4\u03ac \u03ba\u03b1\u03c4\u03ac \u03c4\u03b7\u03bd \u03b5\u03ba\u03c0\u03b1\u03af\u03b4\u03b5\u03c5\u03c3\u03b7 \u03cc\u03c4\u03b9 \u03c4\u03b1 \u03bb\u03ac\u03b8\u03b7 \u03c3\u03b5 \u03bf\u03c1\u03b9\u03c3\u03bc\u03ad\u03bd\u03b5\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2 \u03ba\u03bf\u03c3\u03c4\u03af\u03b6\u03bf\u03c5\u03bd \u03c0\u03b5\u03c1\u03b9\u03c3\u03c3\u03cc\u03c4\u03b5\u03c1\u03bf. \u03a4\u03b1 ablations \u03c4\u03b7\u03c2 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1\u03c2 \u03b4\u03b5\u03af\u03c7\u03bd\u03bf\u03c5\u03bd \u03cc\u03c4\u03b9 \u03ba\u03ac\u03b8\u03b5 \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03bf \u03c3\u03c5\u03bc\u03b2\u03ac\u03bb\u03bb\u03b5\u03b9 \u03ba\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03bf \u03c3\u03c5\u03bd\u03b4\u03c5\u03b1\u03c3\u03bc\u03cc\u03c2 \u03c4\u03bf\u03c5\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf \u03b9\u03c3\u03c7\u03c5\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03c2.<\/p>\n<h2 id=\"domimena-mi-domimena-plegmata\">\u0394\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 \u03ba\u03b1\u03b9 \u03bc\u03b7 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 \u03c0\u03bb\u03ad\u03b3\u03bc\u03b1\u03c4\u03b1 \u03bc\u03b5 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03ae \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ac<\/h2>\n<p>\u03a3\u03b5 \u03ad\u03bd\u03b1 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03bf \u03c0\u03bb\u03ad\u03b3\u03bc\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03ba\u03cc\u03bc\u03b2\u03bf\u03c2 \u03ad\u03c7\u03b5\u03b9 \u03bb\u03bf\u03b3\u03b9\u03ba\u03ad\u03c2 \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03b5\u03c2. \u0397 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1 \u03bf\u03c1\u03af\u03b6\u03b5\u03b9 \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ac 3 \u03b5\u03c0\u03af 3 \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03ba\u03ac\u03b8\u03b5 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf, \u03c7\u03c9\u03c1\u03af\u03c2 \u03c4\u03bf \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03cc \u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03af\u03bf, \u03ba\u03b1\u03b9 \u03c5\u03bb\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03c4\u03b7 \u03c3\u03c5\u03b3\u03ba\u03ad\u03bd\u03c4\u03c1\u03c9\u03c3\u03b7 \u03bc\u03b5 grouped convolution. \u0391\u03c5\u03c4\u03cc \u03c4\u03b1\u03b9\u03c1\u03b9\u03ac\u03b6\u03b5\u03b9 \u03c3\u03b5 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1 \u03cc\u03c0\u03c9\u03c2 Darcy Flow, Navier\u2013Stokes, Airfoil \u03ba\u03b1\u03b9 Plasticity, \u03b1\u03ba\u03cc\u03bc\u03b7 \u03ba\u03b1\u03b9 \u03cc\u03c4\u03b1\u03bd \u03bf\u03b9 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ad\u03c2 \u03b1\u03c0\u03bf\u03c3\u03c4\u03ac\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 grid \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf\u03bc\u03bf\u03b9\u03cc\u03bc\u03bf\u03c1\u03c6\u03b5\u03c2.<\/p>\n<p>\u03a3\u03c4\u03b1 \u03bc\u03b7 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 meshes \u03b4\u03b5\u03bd \u03c5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 \u03b1\u03bd\u03c4\u03af\u03c3\u03c4\u03bf\u03b9\u03c7\u03bf \u03ba\u03b1\u03bd\u03bf\u03bd\u03b9\u03ba\u03cc \u03c0\u03bb\u03ad\u03b3\u03bc\u03b1. \u0395\u03ba\u03b5\u03af \u03bf\u03b9 \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03b5\u03af\u03c2 \u03b4\u03b7\u03bc\u03b9\u03bf\u03c5\u03c1\u03b3\u03bf\u03cd\u03bd \u03ad\u03bd\u03b1 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03cc k-nearest-neighbor graph \u03c3\u03b5 \u03c7\u03ce\u03c1\u03bf \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ce\u03bd \u03c3\u03c5\u03bd\u03c4\u03b5\u03c4\u03b1\u03b3\u03bc\u03ad\u03bd\u03c9\u03bd \u03c0\u03bf\u03c5 \u03c0\u03c1\u03bf\u03ba\u03cd\u03c0\u03c4\u03b5\u03b9 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03c0\u03c1\u03ce\u03c4\u03b5\u03c2 \u03b9\u03b4\u03b9\u03bf\u03c3\u03c5\u03bd\u03b1\u03c1\u03c4\u03ae\u03c3\u03b5\u03b9\u03c2 Laplace\u2013Beltrami. \u03a3\u03c4\u03b1 \u03c0\u03b5\u03b9\u03c1\u03ac\u03bc\u03b1\u03c4\u03b1 \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03bf\u03cd\u03bd \u03bf\u03ba\u03c4\u03ce \u03b3\u03b5\u03af\u03c4\u03bf\u03bd\u03b5\u03c2, \u03b5\u03ba\u03c4\u03cc\u03c2 \u03b1\u03bd \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03ac. \u039c\u03b5 \u03b1\u03c5\u03c4\u03cc \u03c4\u03bf\u03bd \u03c4\u03c1\u03cc\u03c0\u03bf \u03c4\u03bf \u00abedge\u00bb \u03b4\u03b5\u03bd \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03b1\u03c0\u03bb\u03ce\u03c2 \u03bf\u03c1\u03b9\u03b6\u03cc\u03bd\u03c4\u03b9\u03b1 \u03ae \u03ba\u03ac\u03b8\u03b5\u03c4\u03b7 \u03b3\u03b5\u03b9\u03c4\u03bd\u03af\u03b1\u03c3\u03b7, \u03b1\u03bb\u03bb\u03ac \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03b9\u03ba\u03ae \u03c3\u03c7\u03ad\u03c3\u03b7 \u03c0\u03ac\u03bd\u03c9 \u03c3\u03c4\u03bf mesh.<\/p>\n<h2 id=\"ennea-pde-benchmarks\">\u0395\u03bd\u03bd\u03ad\u03b1 benchmarks, \u03b1\u03c0\u03cc Darcy \u03ba\u03b1\u03b9 Navier\u2013Stokes \u03ad\u03c9\u03c2 \u03c1\u03bf\u03ae \u03b1\u03af\u03bc\u03b1\u03c4\u03bf\u03c2<\/h2>\n<p>\u0397 \u03b1\u03be\u03b9\u03bf\u03bb\u03cc\u03b3\u03b7\u03c3\u03b7 \u03ba\u03b1\u03bb\u03cd\u03c0\u03c4\u03b5\u03b9 \u03c4\u03ad\u03c3\u03c3\u03b5\u03c1\u03b1 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03bf\u03c5 mesh \u03ba\u03b1\u03b9 \u03c0\u03ad\u03bd\u03c4\u03b5 \u03bc\u03b7 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03bf\u03c5. \u03a3\u03c4\u03b7\u03bd \u03c0\u03c1\u03ce\u03c4\u03b7 \u03bf\u03bc\u03ac\u03b4\u03b1 \u03b2\u03c1\u03af\u03c3\u03ba\u03bf\u03bd\u03c4\u03b1\u03b9 Darcy Flow, Navier\u2013Stokes, Airfoil \u03ba\u03b1\u03b9 Plasticity. \u03a3\u03c4\u03b7 \u03b4\u03b5\u03cd\u03c4\u03b5\u03c1\u03b7 \u03b2\u03c1\u03af\u03c3\u03ba\u03bf\u03bd\u03c4\u03b1\u03b9 Irregular Darcy, Pipe Turbulence, Heat Transfer, Composite \u03ba\u03b1\u03b9 Blood Flow. \u039f\u03b9 \u03b5\u03af\u03c3\u03bf\u03b4\u03bf\u03b9 \u03ba\u03b1\u03b9 \u03bf\u03b9 \u03ad\u03be\u03bf\u03b4\u03bf\u03b9 \u03b4\u03b9\u03b1\u03c6\u03ad\u03c1\u03bf\u03c5\u03bd \u03bf\u03c5\u03c3\u03b9\u03b1\u03c3\u03c4\u03b9\u03ba\u03ac: \u03b1\u03c0\u03cc \u03b4\u03b9\u03b1\u03c0\u03b5\u03c1\u03b1\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c0\u03c1\u03bf\u03c2 \u03c0\u03af\u03b5\u03c3\u03b7, \u03b1\u03c0\u03cc \u03b9\u03c3\u03c4\u03bf\u03c1\u03b9\u03ba\u03ae \u03c0\u03c1\u03bf\u03c2 \u03bc\u03b5\u03bb\u03bb\u03bf\u03bd\u03c4\u03b9\u03ba\u03ae \u03c3\u03c4\u03c1\u03bf\u03b2\u03b9\u03bb\u03cc\u03c4\u03b7\u03c4\u03b1, \u03b1\u03c0\u03cc \u03b8\u03b5\u03c1\u03bc\u03b9\u03ba\u03cc \u03c6\u03bf\u03c1\u03c4\u03af\u03bf \u03c0\u03c1\u03bf\u03c2 \u03c0\u03b1\u03c1\u03b1\u03bc\u03cc\u03c1\u03c6\u03c9\u03c3\u03b7 \u03ba\u03b1\u03b9 \u03b1\u03c0\u03cc \u03bf\u03c1\u03b9\u03b1\u03ba\u03ad\u03c2 \u03c3\u03c5\u03bd\u03b8\u03ae\u03ba\u03b5\u03c2 \u03c0\u03c1\u03bf\u03c2 \u03c4\u03c1\u03b9\u03c3\u03b4\u03b9\u03ac\u03c3\u03c4\u03b1\u03c4\u03b7 \u03b8\u03b5\u03c1\u03bc\u03bf\u03ba\u03c1\u03b1\u03c3\u03af\u03b1.<\/p>\n<p>\u03a9\u03c2 \u03b2\u03b1\u03c3\u03b9\u03ba\u03cc metric \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03b5\u03af\u03c4\u03b1\u03b9 \u03c4\u03bf \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03cc \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 L2 \u03c3\u03c4\u03bf test set. \u039f\u03b9 \u03c3\u03c5\u03b3\u03ba\u03c1\u03af\u03c3\u03b5\u03b9\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03bf\u03c5\u03bd \u03c3\u03c5\u03bc\u03b2\u03b1\u03c4\u03b9\u03ba\u03bf\u03cd\u03c2 spectral operators, adaptive spectral \u03bc\u03bf\u03bd\u03c4\u03ad\u03bb\u03b1, graph-based operators \u03ba\u03b1\u03b9 attention-based operators. \u039f\u03b9 \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03b5\u03af\u03c2 \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03bf\u03c5\u03bd \u03c3\u03c5\u03b3\u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03bf\u03c5\u03c2 \u03c0\u03c1\u03bf\u03cb\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u03bf\u03cd\u03c2 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03c4\u03c1\u03c9\u03bd \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b5\u03c0\u03b5\u03af\u03c2 \u03c1\u03c5\u03b8\u03bc\u03af\u03c3\u03b5\u03b9\u03c2 \u03b5\u03ba\u03c0\u03b1\u03af\u03b4\u03b5\u03c5\u03c3\u03b7\u03c2, \u03bc\u03b5 \u03c4\u03b9\u03c2 \u03b1\u03bd\u03b1\u03bb\u03c5\u03c4\u03b9\u03ba\u03ad\u03c2 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03c4\u03c1\u03bf\u03c5\u03c2 \u03bd\u03b1 \u03c0\u03b1\u03c1\u03ad\u03c7\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf \u03c0\u03b1\u03c1\u03ac\u03c1\u03c4\u03b7\u03bc\u03b1.<\/p>\n<h2 id=\"posotika-optika-apotelesmata\">\u03a4\u03b9 \u03b4\u03b5\u03af\u03c7\u03bd\u03bf\u03c5\u03bd \u03c4\u03b1 \u03c0\u03bf\u03c3\u03bf\u03c4\u03b9\u03ba\u03ac \u03ba\u03b1\u03b9 \u03bf\u03c0\u03c4\u03b9\u03ba\u03ac \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03ad\u03c3\u03bc\u03b1\u03c4\u03b1<\/h2>\n<p>\u03a3\u03c4\u03bf\u03c5\u03c2 \u03c0\u03af\u03bd\u03b1\u03ba\u03b5\u03c2 \u03c4\u03b7\u03c2 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1\u03c2, \u03c4\u03bf ESO \u03b5\u03bc\u03c6\u03b1\u03bd\u03af\u03b6\u03b5\u03b9 \u03c4\u03bf \u03c7\u03b1\u03bc\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03bf \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03cc L2 error \u03ba\u03b1\u03b9 \u03c3\u03c4\u03b1 \u03b5\u03bd\u03bd\u03ad\u03b1 benchmarks. \u03a3\u03c4\u03b1 \u03bc\u03b7 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 meshes, \u03b3\u03b9\u03b1 \u03c0\u03b1\u03c1\u03ac\u03b4\u03b5\u03b9\u03b3\u03bc\u03b1, \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03bf\u03bd\u03c4\u03b1\u03b9 \u03c4\u03b9\u03bc\u03ad\u03c2 6,71e-3 \u03c3\u03c4\u03bf Irregular Darcy, 4,32e-3 \u03c3\u03c4\u03bf Pipe Turbulence, 9,75e-5 \u03c3\u03c4\u03bf Heat Transfer, 7,93e-3 \u03c3\u03c4\u03bf Composite \u03ba\u03b1\u03b9 2,76e-2 \u03c3\u03c4\u03bf Blood Flow. \u0391\u03c5\u03c4\u03ad\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03b5\u03c4\u03c1\u03ae\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03bf\u03c5 \u03c0\u03b5\u03b9\u03c1\u03b1\u03bc\u03b1\u03c4\u03b9\u03ba\u03bf\u03cd \u03c0\u03c1\u03c9\u03c4\u03bf\u03ba\u03cc\u03bb\u03bb\u03bf\u03c5 \u03ba\u03b1\u03b9 \u03b4\u03b5\u03bd \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b3\u03b5\u03bd\u03b9\u03ba\u03b5\u03cd\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b1\u03c5\u03c4\u03cc\u03bc\u03b1\u03c4\u03b1 \u03c3\u03b5 \u03ba\u03ac\u03b8\u03b5 PDE \u03ae dataset.<\/p>\n<p>\u0397 \u03bf\u03c0\u03c4\u03b9\u03ba\u03ae \u03b1\u03bd\u03ac\u03bb\u03c5\u03c3\u03b7 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03be\u03af\u03c3\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03ae \u03bc\u03b5 \u03c4\u03bf\u03bd \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03cc \u03bc\u03ad\u03c3\u03bf \u03cc\u03c1\u03bf. \u03a3\u03c4\u03b7 Darcy Flow \u03c4\u03bf HPM \u03b5\u03bc\u03c6\u03b1\u03bd\u03af\u03b6\u03b5\u03b9 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b1 \u03bb\u03ac\u03b8\u03b7 \u03ba\u03bf\u03bd\u03c4\u03ac \u03c3\u03b5 \u03b1\u03c0\u03cc\u03c4\u03bf\u03bc\u03b5\u03c2 \u03b4\u03b9\u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b5\u03c2 \u03b4\u03b9\u03b1\u03c0\u03b5\u03c1\u03b1\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2, \u03b5\u03bd\u03ce \u03c4\u03bf ESO \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03b9 \u03c4\u03bf \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03b1\u03c5\u03c4\u03ac \u03c4\u03b1 coefficient jumps. \u03a3\u03c4\u03b7 Navier\u2013Stokes \u03b7 \u03b2\u03b5\u03bb\u03c4\u03af\u03c9\u03c3\u03b7 \u03c3\u03c5\u03b3\u03ba\u03b5\u03bd\u03c4\u03c1\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9 \u03c3\u03b5 \u03b4\u03af\u03bd\u03b5\u03c2 \u03ba\u03b1\u03b9 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2 \u03c5\u03c8\u03b7\u03bb\u03ae\u03c2 \u03ba\u03bb\u03af\u03c3\u03b7\u03c2. \u0391\u03c5\u03c4\u03cc \u03c5\u03c0\u03bf\u03c3\u03c4\u03b7\u03c1\u03af\u03b6\u03b5\u03b9 \u03c4\u03bf\u03bd \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03cc \u03b9\u03c3\u03c7\u03c5\u03c1\u03b9\u03c3\u03bc\u03cc \u03cc\u03c4\u03b9 \u03b7 edge conditioning \u03b2\u03bf\u03b7\u03b8\u03ac \u03b5\u03ba\u03b5\u03af \u03cc\u03c0\u03bf\u03c5 \u03b7 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ae \u03b4\u03bf\u03bc\u03ae \u03ad\u03c7\u03b5\u03b9 \u03c6\u03c5\u03c3\u03b9\u03ba\u03cc \u03bd\u03cc\u03b7\u03bc\u03b1.<\/p>\n<div class=\"td-chart td-chart--metrics\">\n<p class=\"td-chart-title\">\u039c\u03b5\u03af\u03c9\u03c3\u03b7 \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03bf\u03cd L2 error \u03c3\u03c4\u03b1 \u03c4\u03ad\u03c3\u03c3\u03b5\u03c1\u03b1 structured benchmarks<\/p>\n<p class=\"td-chart-intro\">\u039f\u03b9 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03bc\u03ad\u03bd\u03b5\u03c2 \u03b1\u03c0\u03cc \u03c4\u03bf\u03bd \u03a0\u03af\u03bd\u03b1\u03ba\u03b1 2 \u03c4\u03b7\u03c2 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1\u03c2 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 \u03c4\u03bf\u03c5 \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf\u03c5 baseline \u03ba\u03ac\u03b8\u03b5 benchmark. \u0394\u03b5\u03af\u03c7\u03bd\u03bf\u03c5\u03bd \u03c4\u03bf \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03bf \u03c0\u03b5\u03b9\u03c1\u03b1\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03c0\u03c1\u03c9\u03c4\u03cc\u03ba\u03bf\u03bb\u03bb\u03bf, \u03cc\u03c7\u03b9 \u03b3\u03b5\u03bd\u03b9\u03ba\u03ae \u03c5\u03c0\u03b5\u03c1\u03bf\u03c7\u03ae \u03c3\u03b5 \u03ba\u03ac\u03b8\u03b5 PDE.<\/p>\n<div class=\"td-metric-grid\">\n<div class=\"td-metric-card\">\n<div class=\"td-metric-value\">18,1%<\/div>\n<div class=\"td-metric-label\">Darcy Flow<\/div>\n<p>\u03a4\u03bf ESO \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 3,52e-3 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 4,30e-3 \u03c4\u03bf\u03c5 \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf\u03c5 baseline.<\/p>\n<\/div>\n<div class=\"td-metric-card\">\n<div class=\"td-metric-value\">6,4%<\/div>\n<div class=\"td-metric-label\">Airfoil<\/div>\n<p>\u03a4\u03bf \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03cc L2 error \u03b5\u03af\u03bd\u03b1\u03b9 3,93e-3 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 4,20e-3.<\/p>\n<\/div>\n<div class=\"td-metric-card\">\n<div class=\"td-metric-value\">1,4%<\/div>\n<div class=\"td-metric-label\">Navier\u2013Stokes<\/div>\n<p>\u0397 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ac \u03b5\u03af\u03bd\u03b1\u03b9 \u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03b7: 4,77e-2 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 4,84e-2.<\/p>\n<\/div>\n<div class=\"td-metric-card\">\n<div class=\"td-metric-value\">13,6%<\/div>\n<div class=\"td-metric-label\">Plasticity<\/div>\n<p>\u03a4\u03bf ESO \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 4,90e-4 \u03ad\u03bd\u03b1\u03bd\u03c4\u03b9 5,67e-4.<\/p>\n<\/div>\n<\/div>\n<p class=\"td-chart-note\">\u03a3\u03c4\u03b1 \u03c0\u03ad\u03bd\u03c4\u03b5 unstructured benchmarks \u03c4\u03bf ESO \u03ad\u03c7\u03b5\u03b9 \u03b5\u03c0\u03af\u03c3\u03b7\u03c2 \u03c4\u03bf \u03c7\u03b1\u03bc\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03bf \u03b1\u03bd\u03b1\u03c6\u03b5\u03c1\u03cc\u03bc\u03b5\u03bd\u03bf error, \u03bc\u03b5 \u03c4\u03b7 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b7 \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03ae \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ac \u03c4\u03bf\u03c5 \u03c0\u03af\u03bd\u03b1\u03ba\u03b1 \u03bd\u03b1 \u03b5\u03bc\u03c6\u03b1\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c3\u03c4\u03bf Heat Transfer.<\/p>\n<\/div>\n<h2 id=\"ablations-d-q-par\">\u03a4\u03b1 ablations \u03b5\u03be\u03b7\u03b3\u03bf\u03cd\u03bd \u03b3\u03b9\u03b1\u03c4\u03af \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 D, Q \u03ba\u03b1\u03b9 PAR<\/h2>\n<p>\u0397 \u03b1\u03c6\u03b1\u03af\u03c1\u03b5\u03c3\u03b7 \u03c4\u03c9\u03bd edge statistics \u03b5\u03c0\u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03bf\u03c5\u03c3\u03b9\u03b1\u03c3\u03c4\u03b9\u03ba\u03ac \u03c4\u03b7\u03bd point-only \u03b5\u03ba\u03b4\u03bf\u03c7\u03ae. \u03a3\u03c4\u03bf Darcy Flow \u03c4\u03bf \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03cc L2 error \u03c0\u03bf\u03c5 \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03c4\u03b1\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 4,59e-3 \u03b3\u03b9\u03b1 points only, 4,16e-3 \u03bc\u03b5 \u03bc\u03cc\u03bd\u03bf D, 3,92e-3 \u03bc\u03b5 \u03bc\u03cc\u03bd\u03bf Q \u03ba\u03b1\u03b9 3,78e-3 \u03cc\u03c4\u03b1\u03bd \u03c3\u03c5\u03bd\u03b4\u03c5\u03ac\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 D \u03ba\u03b1\u03b9 Q. \u03a3\u03c4\u03b7 Navier\u2013Stokes \u03bf\u03b9 \u03b1\u03bd\u03c4\u03af\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03c2 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 7,34e-2, 5,43e-2, 5,25e-2 \u03ba\u03b1\u03b9 5,06e-2.<\/p>\n<p>\u03a4\u03bf \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03b4\u03b5\u03af\u03c7\u03bd\u03b5\u03b9 \u03c3\u03c5\u03bc\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c4\u03b7\u03c4\u03b1: \u03b7 \u03ba\u03b1\u03c4\u03b5\u03cd\u03b8\u03c5\u03bd\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03bc\u03ad\u03c3\u03b7\u03c2 \u03b1\u03bb\u03bb\u03b1\u03b3\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03bf \u03bc\u03ad\u03b3\u03b5\u03b8\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ae\u03c2 \u03b1\u03bd\u03c4\u03af\u03b8\u03b5\u03c3\u03b7\u03c2 \u03b4\u03b5\u03bd \u03bc\u03b5\u03c4\u03b1\u03c6\u03ad\u03c1\u03bf\u03c5\u03bd \u03c4\u03b7\u03bd \u03af\u03b4\u03b9\u03b1 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1. \u03a0\u03c1\u03cc\u03c3\u03b8\u03b5\u03c4\u03b5\u03c2 \u03b4\u03bf\u03ba\u03b9\u03bc\u03ad\u03c2 \u03b5\u03be\u03b5\u03c4\u03ac\u03b6\u03bf\u03c5\u03bd \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03b5\u03c4\u03b9\u03ba\u03ad\u03c2 \u03ba\u03bb\u03af\u03bc\u03b1\u03ba\u03b5\u03c2 \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ac\u03c2 \u03ba\u03b1\u03b9 \u03b3\u03b5\u03bd\u03af\u03ba\u03b5\u03c5\u03c3\u03b7 \u03b1\u03bd\u03ac\u03bb\u03c5\u03c3\u03b7\u03c2. \u03a3\u03cd\u03bc\u03c6\u03c9\u03bd\u03b1 \u03bc\u03b5 \u03c4\u03b7 \u03bc\u03b5\u03bb\u03ad\u03c4\u03b7, \u03b7 \u03c0\u03bb\u03ae\u03c1\u03b7\u03c2 \u03b4\u03b9\u03b1\u03bc\u03cc\u03c1\u03c6\u03c9\u03c3\u03b7 PVMM \u03bc\u03b1\u03b6\u03af \u03bc\u03b5 PAR \u03b4\u03af\u03bd\u03b5\u03b9 \u03c4\u03b7\u03bd \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b7 \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03ae \u03b9\u03c3\u03bf\u03c1\u03c1\u03bf\u03c0\u03af\u03b1.<\/p>\n<div class=\"td-comparison\">\n<p class=\"td-comparison-title\">\u03a4\u03b9 \u03c0\u03c1\u03bf\u03c3\u03b8\u03ad\u03c4\u03b5\u03b9 \u03ba\u03ac\u03b8\u03b5 \u03c3\u03c5\u03c3\u03c4\u03b1\u03c4\u03b9\u03ba\u03cc \u03c3\u03c4\u03bf baseline<\/p>\n<div class=\"td-comparison-cards td-comparison-cards--horizontal\">\n<div class=\"td-comparison-grid td-comparison-grid--three\">\n<div class=\"td-platform-card\">\n<h3>PAR \u03bc\u03cc\u03bd\u03bf<\/h3>\n<p>\u0391\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c4\u03bf \u03b5\u03ba\u03c0\u03b1\u03b9\u03b4\u03b5\u03c5\u03c4\u03b9\u03ba\u03cc \u03b2\u03ac\u03c1\u03bf\u03c2 \u03c3\u03c4\u03b9\u03c2 \u03b5\u03c5\u03b1\u03af\u03c3\u03b8\u03b7\u03c4\u03b5\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2. \u03a3\u03c4\u03b1 ablations \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03b9 \u03c4\u03bf error \u03c4\u03bf\u03c5 Darcy \u03b1\u03c0\u03cc 4,59e-3 \u03c3\u03b5 4,32e-3 \u03ba\u03b1\u03b9 \u03c4\u03bf\u03c5 Navier\u2013Stokes \u03b1\u03c0\u03cc 7,34e-2 \u03c3\u03b5 6,91e-2.<\/p>\n<div class=\"td-badge-row\"><span class=\"td-badge\">\u0395\u03c0\u03ad\u03bc\u03b2\u03b1\u03c3\u03b7 \u03c3\u03c4\u03bf loss<\/span><\/div>\n<\/div>\n<div class=\"td-platform-card\">\n<h3>PVMM \u03bc\u03cc\u03bd\u03bf<\/h3>\n<p>\u0391\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c4\u03b7 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03bc\u03af\u03be\u03b7 \u03bc\u03b5 D \u03ba\u03b1\u03b9 Q. \u039f\u03b9 \u03b1\u03bd\u03c4\u03af\u03c3\u03c4\u03bf\u03b9\u03c7\u03b5\u03c2 \u03c4\u03b9\u03bc\u03ad\u03c2 \u03b3\u03af\u03bd\u03bf\u03bd\u03c4\u03b1\u03b9 3,78e-3 \u03ba\u03b1\u03b9 5,06e-2, \u03ac\u03c1\u03b1 \u03c4\u03bf PVMM \u03b5\u03af\u03bd\u03b1\u03b9 \u03bf \u03b9\u03c3\u03c7\u03c5\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03c2 \u03bc\u03b5\u03bc\u03bf\u03bd\u03c9\u03bc\u03ad\u03bd\u03bf\u03c2 \u03c3\u03c5\u03bd\u03c4\u03b5\u03bb\u03b5\u03c3\u03c4\u03ae\u03c2.<\/p>\n<div class=\"td-badge-row\"><span class=\"td-badge\">\u0395\u03c0\u03ad\u03bc\u03b2\u03b1\u03c3\u03b7 \u03c3\u03c4\u03b1 modes<\/span><\/div>\n<\/div>\n<div class=\"td-platform-card td-platform-card--navy\">\n<h3>PVMM + PAR<\/h3>\n<p>\u0397 \u03c0\u03bb\u03ae\u03c1\u03b7\u03c2 ESO \u03b4\u03b9\u03b1\u03bc\u03cc\u03c1\u03c6\u03c9\u03c3\u03b7 \u03c6\u03c4\u03ac\u03bd\u03b5\u03b9 3,52e-3 \u03c3\u03c4\u03bf Darcy \u03ba\u03b1\u03b9 4,77e-2 \u03c3\u03c4\u03bf Navier\u2013Stokes, \u03c4\u03bf \u03c7\u03b1\u03bc\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03bf error \u03c4\u03bf\u03c5 ablation.<\/p>\n<div class=\"td-badge-row\"><span class=\"td-badge\">\u03a3\u03c5\u03bc\u03c0\u03bb\u03b7\u03c1\u03c9\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc\u03c2 \u03c3\u03c5\u03bd\u03b4\u03c5\u03b1\u03c3\u03bc\u03cc\u03c2<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2 id=\"theoritiki-idiotita\">\u03a4\u03b9 \u03c3\u03b7\u03bc\u03b1\u03af\u03bd\u03b5\u03b9 \u03b7 \u00ab\u03cc\u03c7\u03b9 \u03c7\u03b5\u03b9\u03c1\u03cc\u03c4\u03b5\u03c1\u03b7\u00bb \u03b8\u03b5\u03c9\u03c1\u03b7\u03c4\u03b9\u03ba\u03ae \u03b9\u03b4\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1<\/h2>\n<p>\u0397 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03b9\u03ba\u03bd\u03cd\u03b5\u03b9 \u03cc\u03c4\u03b9 \u03b7 \u03ba\u03bb\u03ac\u03c3\u03b7 \u03c4\u03c9\u03bd edge-conditioned \u03bc\u03bf\u03bd\u03c4\u03ad\u03bb\u03c9\u03bd \u03c0\u03b5\u03c1\u03b9\u03bb\u03b1\u03bc\u03b2\u03ac\u03bd\u03b5\u03b9 \u03c4\u03b7 \u03c3\u03c5\u03bc\u03b2\u03b1\u03c4\u03b9\u03ba\u03ae \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03ba\u03bb\u03ac\u03c3\u03b7, \u03b5\u03c0\u03b5\u03b9\u03b4\u03ae \u03b7 edge \u03b4\u03b9\u03cc\u03c1\u03b8\u03c9\u03c3\u03b7 \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03bc\u03b7\u03b4\u03b5\u03bd\u03b9\u03c3\u03c4\u03b5\u03af. \u03a5\u03c0\u03cc \u03b1\u03c5\u03c4\u03ae \u03c4\u03b7\u03bd \u03c5\u03c0\u03cc\u03b8\u03b5\u03c3\u03b7 \u03b5\u03b3\u03ba\u03bb\u03b5\u03b9\u03c3\u03bc\u03bf\u03cd, \u03c4\u03bf \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf \u03b8\u03b5\u03c9\u03c1\u03b7\u03c4\u03b9\u03ba\u03ac \u03b5\u03c6\u03b9\u03ba\u03c4\u03cc \u03c1\u03af\u03c3\u03ba\u03bf \u03c4\u03b7\u03c2 \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b7\u03c2 \u03ba\u03bb\u03ac\u03c3\u03b7\u03c2 \u03b4\u03b5\u03bd \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c7\u03b5\u03b9\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf \u03b1\u03c0\u03cc \u03b5\u03ba\u03b5\u03af\u03bd\u03bf \u03c4\u03b7\u03c2 \u03b2\u03b1\u03c3\u03b9\u03ba\u03ae\u03c2. \u0391\u03c5\u03c4\u03cc \u03b5\u03af\u03bd\u03b1\u03b9 \u03b4\u03ae\u03bb\u03c9\u03c3\u03b7 \u03b3\u03b9\u03b1 \u03c4\u03b7 \u03b4\u03c5\u03bd\u03b1\u03c4\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c0\u03c1\u03bf\u03c3\u03ad\u03b3\u03b3\u03b9\u03c3\u03b7\u03c2, \u03cc\u03c7\u03b9 \u03b5\u03b3\u03b3\u03cd\u03b7\u03c3\u03b7 \u03cc\u03c4\u03b9 \u03ba\u03ac\u03b8\u03b5 \u03b5\u03ba\u03c0\u03b1\u03af\u03b4\u03b5\u03c5\u03c3\u03b7 \u03b8\u03b1 \u03c0\u03b5\u03c4\u03cd\u03c7\u03b5\u03b9 \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1.<\/p>\n<p>\u0397 \u03b4\u03b9\u03ac\u03ba\u03c1\u03b9\u03c3\u03b7 \u03ad\u03c7\u03b5\u03b9 \u03c3\u03b7\u03bc\u03b1\u03c3\u03af\u03b1. \u0397 \u03b1\u03c5\u03be\u03b7\u03bc\u03ad\u03bd\u03b7 \u03b5\u03ba\u03c6\u03c1\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b4\u03b5\u03bd \u03b5\u03be\u03b1\u03bb\u03b5\u03af\u03c6\u03b5\u03b9 \u03b6\u03b7\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 optimization, \u03c0\u03bf\u03b9\u03cc\u03c4\u03b7\u03c4\u03b1\u03c2 \u03b4\u03b5\u03b4\u03bf\u03bc\u03ad\u03bd\u03c9\u03bd \u03ae \u03bc\u03b5\u03c4\u03b1\u03c6\u03bf\u03c1\u03ac\u03c2 \u03c3\u03b5 \u03ac\u03b3\u03bd\u03c9\u03c3\u03c4\u03b1 \u03c0\u03c1\u03bf\u03b2\u03bb\u03ae\u03bc\u03b1\u03c4\u03b1. \u0397 \u03b5\u03bc\u03c0\u03b5\u03b9\u03c1\u03b9\u03ba\u03ae \u03c4\u03b5\u03ba\u03bc\u03b7\u03c1\u03af\u03c9\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1\u03c2 \u03b1\u03c6\u03bf\u03c1\u03ac \u03c4\u03b1 \u03b5\u03bd\u03bd\u03ad\u03b1 benchmarks \u03ba\u03b1\u03b9 \u03c4\u03b9\u03c2 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b5\u03c2 \u03c1\u03c5\u03b8\u03bc\u03af\u03c3\u03b5\u03b9\u03c2 \u03c4\u03bf\u03c5\u03c2. \u0393\u03b9\u03b1 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03b9\u03ba\u03ae \u03c7\u03c1\u03ae\u03c3\u03b7 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 <a href=\"https:\/\/twodots.gr\/klinika-oria-ai-provlepseis-katanohites\/\">\u03b1\u03bd\u03b5\u03be\u03ac\u03c1\u03c4\u03b7\u03c4\u03b7 \u03b1\u03be\u03b9\u03bf\u03bb\u03cc\u03b3\u03b7\u03c3\u03b7 \u03b1\u03bd\u03ac \u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03b7 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ae \u03ba\u03b1\u03b9 \u03c4\u03cd\u03c0\u03bf \u03bb\u03ac\u03b8\u03bf\u03c5\u03c2<\/a> \u03c3\u03c4\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc mesh, \u03c3\u03c4\u03b9\u03c2 \u03bf\u03c1\u03b9\u03b1\u03ba\u03ad\u03c2 \u03c3\u03c5\u03bd\u03b8\u03ae\u03ba\u03b5\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03c4\u03b1 \u03b1\u03c0\u03bf\u03b4\u03b5\u03ba\u03c4\u03ac \u03cc\u03c1\u03b9\u03b1 \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1\u03c4\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03bf\u03c1\u03b3\u03b1\u03bd\u03b9\u03c3\u03bc\u03bf\u03cd.<\/p>\n<h2 id=\"epicheirimatiki-anagnosi\">\u0397 \u03b5\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03b1\u03bd\u03ac\u03b3\u03bd\u03c9\u03c3\u03b7: \u03b1\u03ba\u03c1\u03af\u03b2\u03b5\u03b9\u03b1 \u03c3\u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03c0\u03bf\u03c5 \u03ad\u03c7\u03bf\u03c5\u03bd \u03c3\u03c5\u03bd\u03ad\u03c0\u03b5\u03b9\u03b5\u03c2<\/h2>\n<p>\u0393\u03b9\u03b1 \u03ad\u03bd\u03b1\u03bd business owner \u03c4\u03bf ESO \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03b5\u03c1\u03b3\u03b1\u03bb\u03b5\u03af\u03bf marketing \u03bf\u03cd\u03c4\u03b5 \u03ad\u03c4\u03bf\u03b9\u03bc\u03b7 \u03b5\u03bc\u03c0\u03bf\u03c1\u03b9\u03ba\u03ae \u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae. \u0397 \u03c3\u03cd\u03bd\u03b4\u03b5\u03c3\u03b7 \u03bc\u03b5 \u03c4\u03bf\u03c5\u03c2 <a href=\"https:\/\/twodots.gr\/aftomatismoi-epicheiriseon-ai\/\">\u0391\u03c5\u03c4\u03bf\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u03bf\u03cd\u03c2 \u0395\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03ae\u03c3\u03b5\u03c9\u03bd &amp; AI<\/a> \u03b1\u03c6\u03bf\u03c1\u03ac \u03c4\u03b7 \u03bc\u03b5\u03b8\u03bf\u03b4\u03bf\u03bb\u03bf\u03b3\u03af\u03b1 \u03b1\u03be\u03b9\u03bf\u03bb\u03cc\u03b3\u03b7\u03c3\u03b7\u03c2 \u03ba\u03b1\u03b9 \u03cc\u03c7\u03b9 \u03c5\u03c0\u03cc\u03c3\u03c7\u03b5\u03c3\u03b7 \u03cc\u03c4\u03b9 \u03b7 TWO DOTS \u03c0\u03b1\u03c1\u03ad\u03c7\u03b5\u03b9 \u03ad\u03c4\u03bf\u03b9\u03bc\u03bf solver ESO. \u0395\u03af\u03bd\u03b1\u03b9 \u03b5\u03c1\u03b5\u03c5\u03bd\u03b7\u03c4\u03b9\u03ba\u03ae \u03b1\u03c1\u03c7\u03b9\u03c4\u03b5\u03ba\u03c4\u03bf\u03bd\u03b9\u03ba\u03ae \u03b3\u03b9\u03b1 surrogate modeling \u03c6\u03c5\u03c3\u03b9\u03ba\u03ce\u03bd \u03c3\u03c5\u03c3\u03c4\u03b7\u03bc\u03ac\u03c4\u03c9\u03bd. \u0397 \u03c7\u03c1\u03ae\u03c3\u03b9\u03bc\u03b7 \u03b1\u03c1\u03c7\u03ae, \u03c9\u03c2 \u03c0\u03c1\u03bf\u03c3\u03b5\u03ba\u03c4\u03b9\u03ba\u03ae \u03b5\u03c1\u03bc\u03b7\u03bd\u03b5\u03af\u03b1, \u03b5\u03af\u03bd\u03b1\u03b9 \u03cc\u03c4\u03b9 \u03ad\u03bd\u03b1 \u03bc\u03ad\u03c3\u03bf metric \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03ba\u03c1\u03cd\u03b2\u03b5\u03b9 \u03bb\u03ac\u03b8\u03b7 \u03c3\u03b5 \u03bc\u03b9\u03ba\u03c1\u03ad\u03c2 \u03b1\u03bb\u03bb\u03ac \u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03b5\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2. \u03a3\u03b5 \u03ad\u03c1\u03b3\u03b1 engineering AI, \u03b7 \u03b1\u03be\u03b9\u03bf\u03bb\u03cc\u03b3\u03b7\u03c3\u03b7 \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b5\u03be\u03b5\u03c4\u03ac\u03b6\u03b5\u03b9 \u03cc\u03c7\u03b9 \u03bc\u03cc\u03bd\u03bf \u03c4\u03bf \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03cc error \u03b1\u03bb\u03bb\u03ac \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c0\u03bf\u03cd \u03b5\u03bc\u03c6\u03b1\u03bd\u03af\u03b6\u03b5\u03c4\u03b1\u03b9.<\/p>\n<p>\u039f\u03bc\u03ac\u03b4\u03b5\u03c2 \u03c0\u03bf\u03c5 \u03b4\u03b9\u03b5\u03c1\u03b5\u03c5\u03bd\u03bf\u03cd\u03bd real-time simulation, digital twins \u03ae \u03b5\u03c0\u03b1\u03bd\u03b1\u03bb\u03b1\u03bc\u03b2\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03b5\u03c2 \u03bc\u03b5\u03bb\u03ad\u03c4\u03b5\u03c2 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03c4\u03c1\u03c9\u03bd \u03bc\u03ad\u03c3\u03b1 \u03b1\u03c0\u03cc <a href=\"https:\/\/twodots.gr\/ta-kalytera-ergaleia-technitis-noimosynis-gia-michanikoys\/\">AI \u03b5\u03c1\u03b3\u03b1\u03bb\u03b5\u03af\u03b1 \u03b3\u03b9\u03b1 \u03bc\u03b7\u03c7\u03b1\u03bd\u03b9\u03ba\u03bf\u03cd\u03c2<\/a> \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03bd\u03b1 \u03b5\u03be\u03b5\u03c4\u03ac\u03c3\u03bf\u03c5\u03bd \u03b1\u03bd \u03bf\u03b9 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03b5\u03c5\u03b1\u03af\u03c3\u03b8\u03b7\u03c4\u03b5\u03c2 \u03b6\u03ce\u03bd\u03b5\u03c2 \u03c4\u03bf\u03c5\u03c2 \u03bf\u03c1\u03af\u03b6\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ba\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ad\u03c2 \u03ba\u03b1\u03b9 \u03b1\u03bd \u03c5\u03c0\u03ac\u03c1\u03c7\u03bf\u03c5\u03bd \u03ba\u03b1\u03c4\u03ac\u03bb\u03bb\u03b7\u03bb\u03b1 \u03c0\u03b5\u03b4\u03af\u03b1 \u03b3\u03b9\u03b1 loss reweighting.<\/p>\n<div class=\"td-decision-band\">\n<p class=\"td-decision-label\">\u03a0\u03cc\u03c4\u03b5 \u03b1\u03be\u03af\u03b6\u03b5\u03b9 \u03ad\u03bd\u03b1 \u03c0\u03b9\u03bb\u03bf\u03c4\u03b9\u03ba\u03cc ESO;<\/p>\n<p><strong>\u038c\u03c4\u03b1\u03bd \u03c5\u03c0\u03ac\u03c1\u03c7\u03bf\u03c5\u03bd \u03c0\u03bf\u03bb\u03bb\u03ad\u03c2 \u03b5\u03c0\u03b1\u03bd\u03b1\u03bb\u03b1\u03bc\u03b2\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03b5\u03c2 \u03bb\u03cd\u03c3\u03b5\u03b9\u03c2 \u03c4\u03b7\u03c2 \u03af\u03b4\u03b9\u03b1\u03c2 \u03bf\u03b9\u03ba\u03bf\u03b3\u03ad\u03bd\u03b5\u03b9\u03b1\u03c2 PDE, \u03bf \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 inference \u03ad\u03c7\u03b5\u03b9 \u03bf\u03c5\u03c3\u03b9\u03b1\u03c3\u03c4\u03b9\u03ba\u03ae \u03b1\u03be\u03af\u03b1 \u03ba\u03b1\u03b9 \u03bf\u03b9 \u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03b5\u03c2 \u03b6\u03ce\u03bd\u03b5\u03c2 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03bd\u03b1 \u03bf\u03c1\u03b9\u03c3\u03c4\u03bf\u03cd\u03bd \u03c0\u03c1\u03b9\u03bd \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd \u03b5\u03ba\u03c0\u03b1\u03af\u03b4\u03b5\u03c5\u03c3\u03b7.<\/strong><\/p>\n<p>\u03a7\u03c9\u03c1\u03af\u03c2 \u03b1\u03be\u03b9\u03cc\u03c0\u03b9\u03c3\u03c4\u03bf solver baseline, region-wise metrics, \u03b4\u03b5\u03b4\u03bf\u03bc\u03ad\u03bd\u03b1 \u03b5\u03ba\u03c4\u03cc\u03c2 \u03ba\u03b1\u03c4\u03b1\u03bd\u03bf\u03bc\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03b1\u03c6\u03ae \u03cc\u03c1\u03b9\u03b1 \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1\u03c4\u03bf\u03c2, \u03c4\u03bf \u03c7\u03b1\u03bc\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03bf test average \u03b4\u03b5\u03bd \u03b1\u03c1\u03ba\u03b5\u03af \u03b3\u03b9\u03b1 go-live.<\/p>\n<\/div>\n<p>\u0391\u03c5\u03c4\u03cc \u03b4\u03b5\u03bd \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03b5\u03af \u03b1\u03c0\u03cc\u03b4\u03b5\u03b9\u03be\u03b7 \u03cc\u03c4\u03b9 \u03c4\u03bf ESO \u03c4\u03b1\u03b9\u03c1\u03b9\u03ac\u03b6\u03b5\u03b9 \u03c3\u03b5 \u03ba\u03ac\u03b8\u03b5 \u03c0\u03b5\u03c1\u03af\u03c0\u03c4\u03c9\u03c3\u03b7. \u03a0\u03c1\u03bf\u03c4\u03b5\u03af\u03bd\u03b5\u03b9 \u03cc\u03bc\u03c9\u03c2 \u03bc\u03b9\u03b1 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b7 \u03bb\u03af\u03c3\u03c4\u03b1 \u03b5\u03bb\u03ad\u03b3\u03c7\u03bf\u03c5: geometry, \u03bf\u03c1\u03b9\u03c3\u03bc\u03cc\u03c2 \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ac\u03c2, \u03b4\u03b9\u03b1\u03b8\u03ad\u03c3\u03b9\u03bc\u03b1 physical fields, region-wise metrics \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bc\u03c0\u03b5\u03c1\u03b9\u03c6\u03bf\u03c1\u03ac \u03c3\u03b5 \u03bd\u03ad\u03b1 resolutions.<\/p>\n<h2 id=\"oria-meletis-elegchos\">\u038c\u03c1\u03b9\u03b1 \u03c4\u03b7\u03c2 \u03bc\u03b5\u03bb\u03ad\u03c4\u03b7\u03c2 \u03ba\u03b1\u03b9 \u03b5\u03c0\u03cc\u03bc\u03b5\u03bd\u03bf\u03c2 \u03ad\u03bb\u03b5\u03b3\u03c7\u03bf\u03c2 \u03c0\u03c1\u03b9\u03bd \u03b1\u03c0\u03cc \u03b5\u03c6\u03b1\u03c1\u03bc\u03bf\u03b3\u03ae<\/h2>\n<p>\u03a4\u03bf paper \u03b5\u03af\u03bd\u03b1\u03b9 preprint \u03c0\u03c1\u03ce\u03c4\u03b7\u03c2 \u03ad\u03ba\u03b4\u03bf\u03c3\u03b7\u03c2 \u03c3\u03c4\u03bf arXiv \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03ad\u03c3\u03bc\u03b1\u03c4\u03b1 \u03c0\u03c1\u03bf\u03ad\u03c1\u03c7\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b1\u03c0\u03cc \u03c4\u03bf\u03c5\u03c2 \u03af\u03b4\u03b9\u03bf\u03c5\u03c2 \u03c4\u03bf\u03c5\u03c2 \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03b5\u03af\u03c2. \u03a0\u03b1\u03c1\u03cc\u03c4\u03b9 \u03c0\u03b1\u03c1\u03ad\u03c7\u03b5\u03c4\u03b1\u03b9 \u03ba\u03ce\u03b4\u03b9\u03ba\u03b1\u03c2, \u03b1\u03c0\u03b1\u03b9\u03c4\u03b5\u03af\u03c4\u03b1\u03b9 \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03ae \u03c0\u03c1\u03b9\u03bd \u03b1\u03c0\u03cc \u03b9\u03c3\u03c7\u03c5\u03c1\u03bf\u03cd\u03c2 \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03b9\u03ba\u03bf\u03cd\u03c2 \u03b9\u03c3\u03c7\u03c5\u03c1\u03b9\u03c3\u03bc\u03bf\u03cd\u03c2. \u03a4\u03b1 benchmarks \u03b5\u03af\u03bd\u03b1\u03b9 \u03c0\u03bf\u03b9\u03ba\u03af\u03bb\u03b1, \u03b1\u03bb\u03bb\u03ac \u03b4\u03b5\u03bd \u03ba\u03b1\u03bb\u03cd\u03c0\u03c4\u03bf\u03c5\u03bd \u03ba\u03ac\u03b8\u03b5 PDE, \u03ba\u03ac\u03b8\u03b5 \u03ba\u03b1\u03c4\u03b1\u03bd\u03bf\u03bc\u03ae \u03b4\u03b5\u03b4\u03bf\u03bc\u03ad\u03bd\u03c9\u03bd \u03ae \u03ba\u03ac\u03b8\u03b5 \u03bc\u03bf\u03c1\u03c6\u03ae mesh. \u0395\u03c0\u03af\u03c3\u03b7\u03c2, \u03c4\u03bf PAR \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1 \u03c0\u03c1\u03bf\u03c3\u03b1\u03c1\u03bc\u03bf\u03c3\u03bc\u03ad\u03bd\u03b7 \u03c3\u03c4\u03bf task, \u03ac\u03c1\u03b1 \u03b7 \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae \u03b5\u03c5\u03b1\u03b9\u03c3\u03b8\u03b7\u03c3\u03af\u03b1\u03c2 \u03c7\u03c1\u03b5\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ae \u03b3\u03bd\u03ce\u03c3\u03b7.<\/p>\n<p>\u0397 \u03c3\u03c9\u03c3\u03c4\u03ae \u03c3\u03c5\u03bd\u03ad\u03c7\u03b5\u03b9\u03b1 \u03b4\u03b5\u03bd \u03b5\u03af\u03bd\u03b1\u03b9 \u03bd\u03b1 \u03b1\u03bd\u03c4\u03b9\u03ba\u03b1\u03c4\u03b1\u03c3\u03c4\u03b1\u03b8\u03b5\u03af \u03b1\u03c5\u03c4\u03cc\u03bc\u03b1\u03c4\u03b1 \u03ad\u03bd\u03b1\u03c2 \u03c5\u03c0\u03ac\u03c1\u03c7\u03c9\u03bd solver \u03bf\u03cd\u03c4\u03b5 \u03bd\u03b1 \u03bc\u03b5\u03c4\u03b1\u03c4\u03c1\u03b1\u03c0\u03b5\u03af \u03ad\u03bd\u03b1 \u03b5\u03c1\u03b5\u03c5\u03bd\u03b7\u03c4\u03b9\u03ba\u03cc \u03b1\u03c0\u03bf\u03c4\u03ad\u03bb\u03b5\u03c3\u03bc\u03b1 \u03c3\u03b5 \u03b1\u03bd\u03b5\u03be\u03ad\u03bb\u03b5\u03b3\u03ba\u03c4\u03bf automation. \u0397 <a href=\"https:\/\/twodots.gr\/ai-agents-poios-ftaiei-otan-aftomatopoiisi-apotygchanei\/\">\u03b5\u03c5\u03b8\u03cd\u03bd\u03b7 \u03cc\u03c4\u03b1\u03bd \u03ad\u03bd\u03b1 AI workflow \u03b1\u03c0\u03bf\u03c4\u03c5\u03b3\u03c7\u03ac\u03bd\u03b5\u03b9<\/a> \u03c0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03bd\u03b5\u03b9 \u03c3\u03b1\u03c6\u03ae\u03c2. \u0395\u03af\u03bd\u03b1\u03b9 \u03bd\u03b1 \u03c3\u03c5\u03b3\u03ba\u03c1\u03b9\u03b8\u03b5\u03af \u03c4\u03bf ESO \u03bc\u03b5 \u03c4\u03bf \u03c4\u03c1\u03ad\u03c7\u03bf\u03bd baseline \u03c3\u03c4\u03bf \u03af\u03b4\u03b9\u03bf dataset, \u03bc\u03b5 \u03ba\u03bf\u03b9\u03bd\u03cc parameter budget, \u03c0\u03bf\u03bb\u03bb\u03b1\u03c0\u03bb\u03ac seeds \u03ba\u03b1\u03b9 metrics \u03c4\u03cc\u03c3\u03bf \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03ac \u03cc\u03c3\u03bf \u03ba\u03b1\u03b9 \u03b1\u03bd\u03ac \u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03b7 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ae. \u03a0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03b5\u03c0\u03af\u03c3\u03b7\u03c2 \u03bd\u03b1 \u03bc\u03b5\u03c4\u03c1\u03b7\u03b8\u03bf\u03cd\u03bd \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 \u03b5\u03ba\u03c0\u03b1\u03af\u03b4\u03b5\u03c5\u03c3\u03b7\u03c2, inference, \u03bc\u03bd\u03ae\u03bc\u03b7, \u03c3\u03c5\u03bc\u03c0\u03b5\u03c1\u03b9\u03c6\u03bf\u03c1\u03ac \u03b5\u03ba\u03c4\u03cc\u03c2 \u03ba\u03b1\u03c4\u03b1\u03bd\u03bf\u03bc\u03ae\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03cc\u03c4\u03b7\u03c4\u03b1 \u03c3\u03b5 \u03b1\u03bb\u03bb\u03b1\u03b3\u03ad\u03c2 \u03b1\u03bd\u03ac\u03bb\u03c5\u03c3\u03b7\u03c2. \u0397 \u03b4\u03b7\u03bc\u03bf\u03c3\u03b9\u03b5\u03c5\u03bc\u03ad\u03bd\u03b7 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1 \u03b4\u03af\u03bd\u03b5\u03b9 \u03bc\u03b9\u03b1 \u03b9\u03c3\u03c7\u03c5\u03c1\u03ae \u03c5\u03c0\u03cc\u03b8\u03b5\u03c3\u03b7 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1\u03c2\u00b7 \u03b7 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ae \u03b5\u03c0\u03b9\u03ba\u03cd\u03c1\u03c9\u03c3\u03b7 \u03ba\u03b1\u03b8\u03bf\u03c1\u03af\u03b6\u03b5\u03b9 \u03b1\u03bd \u03b1\u03c5\u03c4\u03ae \u03bc\u03b5\u03c4\u03b1\u03c4\u03c1\u03ad\u03c0\u03b5\u03c4\u03b1\u03b9 \u03c3\u03b5 \u03b5\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03b7\u03c3\u03b9\u03b1\u03ba\u03ae \u03b1\u03be\u03af\u03b1.<\/p>\n<div class=\"td-step-list\">\n<p class=\"td-step-list-title\">\u03a0\u03ad\u03bd\u03c4\u03b5 \u03ad\u03bb\u03b5\u03b3\u03c7\u03bf\u03b9 \u03c0\u03c1\u03b9\u03bd \u03b1\u03c0\u03cc \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03b9\u03ba\u03ae \u03c7\u03c1\u03ae\u03c3\u03b7<\/p>\n<ol>\n<li><span class=\"td-step-kicker\">\u0392\u03ae\u03bc\u03b1 1<\/span><strong>\u0391\u03bd\u03b1\u03c0\u03b1\u03c1\u03ac\u03b3\u03b5\u03c4\u03b5 \u03c4\u03bf \u03b5\u03c0\u03af\u03c3\u03b7\u03bc\u03bf baseline<\/strong>\n<p>\u03a4\u03c1\u03ad\u03be\u03c4\u03b5 \u03c4\u03bf\u03bd \u03ba\u03ce\u03b4\u03b9\u03ba\u03b1 \u03c4\u03c9\u03bd \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03ad\u03c9\u03bd \u03c3\u03c4\u03bf \u03b4\u03b9\u03ba\u03cc \u03c3\u03b1\u03c2 dataset \u03ba\u03b1\u03b9 \u03ba\u03b1\u03c4\u03b1\u03b3\u03c1\u03ac\u03c8\u03c4\u03b5 seeds, preprocessing, parameter budget \u03ba\u03b1\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ae \u03ad\u03ba\u03b4\u03bf\u03c3\u03b7 \u03c4\u03bf\u03c5 solver \u03b1\u03bd\u03b1\u03c6\u03bf\u03c1\u03ac\u03c2.<\/p>\n<\/li>\n<li><span class=\"td-step-kicker\">\u0392\u03ae\u03bc\u03b1 2<\/span><strong>\u039f\u03c1\u03af\u03c3\u03c4\u03b5 \u03c4\u03b9\u03c2 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03b5\u03c5\u03b1\u03af\u03c3\u03b8\u03b7\u03c4\u03b5\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2<\/strong>\n<p>\u0394\u03b7\u03bb\u03ce\u03c3\u03c4\u03b5 \u03c0\u03c1\u03b9\u03bd \u03b1\u03c0\u03cc \u03c4\u03b7 \u03c3\u03cd\u03b3\u03ba\u03c1\u03b9\u03c3\u03b7 \u03c0\u03bf\u03cd \u03b2\u03c1\u03af\u03c3\u03ba\u03bf\u03bd\u03c4\u03b1\u03b9 \u03b4\u03b9\u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b5\u03c2, \u03b4\u03af\u03bd\u03b5\u03c2, \u03bc\u03b5\u03b3\u03ac\u03bb\u03b5\u03c2 \u03ba\u03bb\u03af\u03c3\u03b5\u03b9\u03c2 \u03ae \u03b6\u03ce\u03bd\u03b5\u03c2 \u03c5\u03c8\u03b7\u03bb\u03bf\u03cd \u03ba\u03cc\u03c3\u03c4\u03bf\u03c5\u03c2 \u03ba\u03b1\u03b9 \u03c0\u03bf\u03b9\u03bf physical field \u03c4\u03c1\u03bf\u03c6\u03bf\u03b4\u03bf\u03c4\u03b5\u03af \u03c4\u03bf PAR.<\/p>\n<\/li>\n<li><span class=\"td-step-kicker\">\u0392\u03ae\u03bc\u03b1 3<\/span><strong>\u039c\u03b5\u03c4\u03c1\u03ae\u03c3\u03c4\u03b5 \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03cc \u03ba\u03b1\u03b9 region-wise error<\/strong>\n<p>\u03a3\u03c5\u03b3\u03ba\u03c1\u03af\u03bd\u03b5\u03c4\u03b5 relative L2, worst-case error, \u03c3\u03c4\u03b1\u03b8\u03b5\u03c1\u03cc\u03c4\u03b7\u03c4\u03b1 \u03b1\u03bd\u03ac \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ae \u03ba\u03b1\u03b9 \u03c4\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03ba\u03cc\u03c3\u03c4\u03bf\u03c2 \u03c4\u03b7\u03c2 \u03b1\u03c0\u03cc\u03ba\u03bb\u03b9\u03c3\u03b7\u03c2 \u03b1\u03bd\u03c4\u03af \u03bd\u03b1 \u03b2\u03b1\u03c3\u03b9\u03c3\u03c4\u03b5\u03af\u03c4\u03b5 \u03c3\u03b5 \u03ad\u03bd\u03b1\u03bd \u03bc\u03ad\u03c3\u03bf \u03cc\u03c1\u03bf.<\/p>\n<\/li>\n<li><span class=\"td-step-kicker\">\u0392\u03ae\u03bc\u03b1 4<\/span><strong>\u0394\u03bf\u03ba\u03b9\u03bc\u03ac\u03c3\u03c4\u03b5 \u03bd\u03ad\u03b5\u03c2 \u03b1\u03bd\u03b1\u03bb\u03cd\u03c3\u03b5\u03b9\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03c5\u03bd\u03b8\u03ae\u03ba\u03b5\u03c2<\/strong>\n<p>\u0395\u03bb\u03ad\u03b3\u03be\u03c4\u03b5 mesh resolution, \u03b3\u03b5\u03c9\u03bc\u03b5\u03c4\u03c1\u03af\u03b5\u03c2 \u03ba\u03b1\u03b9 boundary conditions \u03b5\u03ba\u03c4\u03cc\u03c2 \u03b5\u03ba\u03c0\u03b1\u03af\u03b4\u03b5\u03c5\u03c3\u03b7\u03c2, \u03ba\u03b1\u03b8\u03ce\u03c2 \u03ba\u03b1\u03b9 \u03c7\u03c1\u03cc\u03bd\u03bf inference, \u03bc\u03bd\u03ae\u03bc\u03b7 \u03ba\u03b1\u03b9 \u03c5\u03c0\u03bf\u03bb\u03bf\u03b3\u03b9\u03c3\u03c4\u03b9\u03ba\u03cc \u03ba\u03cc\u03c3\u03c4\u03bf\u03c2.<\/p>\n<\/li>\n<li><span class=\"td-step-kicker\">\u0392\u03ae\u03bc\u03b1 5<\/span><strong>\u0398\u03ad\u03c3\u03c4\u03b5 \u03cc\u03c1\u03b9\u03b1, fallback \u03ba\u03b1\u03b9 monitoring<\/strong>\n<p>\u039f\u03c1\u03af\u03c3\u03c4\u03b5 \u03c0\u03cc\u03c4\u03b5 \u03c4\u03bf surrogate \u03b1\u03c0\u03bf\u03c1\u03c1\u03af\u03c0\u03c4\u03b5\u03b9 \u03c4\u03b7\u03bd \u03c0\u03c1\u03cc\u03b2\u03bb\u03b5\u03c8\u03b7, \u03c0\u03cc\u03c4\u03b5 \u03b5\u03c0\u03b9\u03c3\u03c4\u03c1\u03ad\u03c6\u03b5\u03b9 \u03c3\u03c4\u03bf\u03bd \u03ba\u03bb\u03b1\u03c3\u03b9\u03ba\u03cc solver \u03ba\u03b1\u03b9 \u03c0\u03bf\u03b9\u03b1 \u03b1\u03c0\u03cc\u03ba\u03bb\u03b9\u03c3\u03b7 \u03b5\u03bd\u03b5\u03c1\u03b3\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03b1\u03bd\u03b8\u03c1\u03ce\u03c0\u03b9\u03bd\u03b7 \u03b5\u03c0\u03b1\u03bd\u03b5\u03be\u03ad\u03c4\u03b1\u03c3\u03b7.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<h2 id=\"symperasma\">\u0417\u0430\u043a\u043b\u044e\u0447\u0435\u043d\u0438\u0435<\/h2>\n<p>\u03a4\u03bf Edge-Conditioned Spectral Operator \u03bc\u03b5\u03c4\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03c4\u03b7\u03bd \u03c0\u03c1\u03bf\u03c3\u03bf\u03c7\u03ae \u03b1\u03c0\u03cc \u03c4\u03b1 \u03bc\u03b5\u03bc\u03bf\u03bd\u03c9\u03bc\u03ad\u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03c3\u03c4\u03b9\u03c2 \u03c3\u03c7\u03ad\u03c3\u03b5\u03b9\u03c2 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03c4\u03bf\u03c5\u03c2, \u03c7\u03c9\u03c1\u03af\u03c2 \u03bd\u03b1 \u03c7\u03ac\u03bd\u03b5\u03b9 \u03c4\u03b7\u03bd \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03b4\u03cd\u03bd\u03b1\u03bc\u03b7 \u03c4\u03b7\u03c2 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae\u03c2 \u03bc\u03af\u03be\u03b7\u03c2. \u03a4\u03bf PVMM \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03c0\u03c1\u03bf\u03c3\u03b7\u03bc\u03b1\u03c3\u03bc\u03ad\u03bd\u03b5\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b5\u03c4\u03c1\u03b1\u03b3\u03c9\u03bd\u03b9\u03c3\u03bc\u03ad\u03bd\u03b5\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ad\u03c2 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ad\u03c2 \u03b3\u03b9\u03b1 \u03bd\u03b1 \u03b5\u03c0\u03b7\u03c1\u03b5\u03ac\u03c3\u03b5\u03b9 \u03c4\u03b7\u03bd \u03b5\u03c0\u03b9\u03bb\u03bf\u03b3\u03ae modes, \u03b5\u03bd\u03ce \u03c4\u03bf PAR \u03b5\u03bd\u03b9\u03c3\u03c7\u03cd\u03b5\u03b9 \u03c4\u03bf \u03b5\u03ba\u03c0\u03b1\u03b9\u03b4\u03b5\u03c5\u03c4\u03b9\u03ba\u03cc \u03c3\u03ae\u03bc\u03b1 \u03c3\u03c4\u03b9\u03c2 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03c3\u03b7\u03bc\u03b1\u03bd\u03c4\u03b9\u03ba\u03ad\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2.<\/p>\n<p>\u03a3\u03c4\u03b1 \u03b5\u03bd\u03bd\u03ad\u03b1 benchmarks \u03c4\u03b7\u03c2 \u03bc\u03b5\u03bb\u03ad\u03c4\u03b7\u03c2, \u03b1\u03c5\u03c4\u03ae \u03b7 \u03c3\u03cd\u03bd\u03b8\u03b5\u03c3\u03b7 \u03bf\u03b4\u03b7\u03b3\u03b5\u03af \u03c3\u03c4\u03b9\u03c2 \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b5\u03c2 \u03b1\u03bd\u03b1\u03c6\u03b5\u03c1\u03cc\u03bc\u03b5\u03bd\u03b5\u03c2 \u03b5\u03c0\u03b9\u03b4\u03cc\u03c3\u03b5\u03b9\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03b5 \u03bc\u03b9\u03ba\u03c1\u03cc\u03c4\u03b5\u03c1\u03b1 \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u03ba\u03bf\u03bd\u03c4\u03ac \u03c3\u03b5 \u03b4\u03b9\u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b5\u03c2, \u03b4\u03af\u03bd\u03b5\u03c2 \u03ba\u03b1\u03b9 \u03ac\u03bb\u03bb\u03b5\u03c2 \u03b5\u03c5\u03b1\u03af\u03c3\u03b8\u03b7\u03c4\u03b5\u03c2 \u03b4\u03bf\u03bc\u03ad\u03c2. \u03a4\u03bf \u03bf\u03c5\u03c3\u03b9\u03b1\u03c3\u03c4\u03b9\u03ba\u03cc \u03bc\u03ae\u03bd\u03c5\u03bc\u03b1 \u03b3\u03b9\u03b1 \u03ba\u03ac\u03b8\u03b5 \u03bf\u03bc\u03ac\u03b4\u03b1 \u03c0\u03bf\u03c5 \u03b1\u03be\u03b9\u03bf\u03bb\u03bf\u03b3\u03b5\u03af AI \u03c3\u03b5 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03c3\u03c5\u03c3\u03c4\u03ae\u03bc\u03b1\u03c4\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03b1\u03c6\u03ad\u03c2: \u03b4\u03b5\u03bd \u03b1\u03c1\u03ba\u03b5\u03af \u03bd\u03b1 \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03b5\u03b9 \u03c0\u03cc\u03c3\u03bf \u03bb\u03ac\u03b8\u03bf\u03c2 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c3\u03c5\u03bd\u03bf\u03bb\u03b9\u03ba\u03ac \u03c4\u03bf \u03bc\u03bf\u03bd\u03c4\u03ad\u03bb\u03bf. \u03a0\u03c1\u03ad\u03c0\u03b5\u03b9 \u03bd\u03b1 \u03b3\u03bd\u03c9\u03c1\u03af\u03b6\u03b5\u03b9 \u03b1\u03bd \u03c4\u03bf \u03bb\u03ac\u03b8\u03bf\u03c2 \u03c3\u03c5\u03b3\u03ba\u03b5\u03bd\u03c4\u03c1\u03ce\u03bd\u03b5\u03c4\u03b1\u03b9 \u03b1\u03ba\u03c1\u03b9\u03b2\u03ce\u03c2 \u03c3\u03c4\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03cc\u03c0\u03bf\u03c5 \u03b7 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ae \u03b4\u03b5\u03bd \u03b5\u03c0\u03b9\u03c4\u03c1\u03ad\u03c0\u03b5\u03b9 \u03b1\u03c3\u03c4\u03bf\u03c7\u03af\u03b1.<\/p>\n<section class=\"td-service-cta\">\n<div class=\"td-service-cta-content\">\n<p class=\"td-service-cta-eyebrow\">\u0391\u03c5\u03c4\u03bf\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u03bf\u03af \u0395\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03ae\u03c3\u03b5\u03c9\u03bd &amp; AI \u03b1\u03c0\u03cc \u03c4\u03b7\u03bd TWO DOTS<\/p>\n<p class=\"td-service-cta-title\">\u039c\u03b5\u03c4\u03b1\u03c4\u03c1\u03ad\u03c8\u03c4\u03b5 \u03ad\u03bd\u03b1 AI pilot \u03c3\u03b5 \u03bc\u03b5\u03c4\u03c1\u03ae\u03c3\u03b9\u03bc\u03b7 \u03b5\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03b1\u03c0\u03cc\u03c6\u03b1\u03c3\u03b7.<\/p>\n<p>\u0397 TWO DOTS \u03c7\u03b1\u03c1\u03c4\u03bf\u03b3\u03c1\u03b1\u03c6\u03b5\u03af \u03c4\u03bf workflow, \u03c4\u03b1 \u03b4\u03b5\u03b4\u03bf\u03bc\u03ad\u03bd\u03b1, \u03c4\u03b1 \u03cc\u03c1\u03b9\u03b1 \u03b1\u03c0\u03bf\u03c4\u03c5\u03c7\u03af\u03b1\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03b1\u03bd\u03b8\u03c1\u03ce\u03c0\u03b9\u03bd\u03b1 \u03c3\u03b7\u03bc\u03b5\u03af\u03b1 \u03b5\u03bb\u03ad\u03b3\u03c7\u03bf\u03c5. \u0394\u03b5\u03bd \u03c0\u03b1\u03c1\u03ad\u03c7\u03b5\u03b9 \u03ad\u03c4\u03bf\u03b9\u03bc\u03bf solver ESO, \u03b1\u03bb\u03bb\u03ac \u03bc\u03c0\u03bf\u03c1\u03b5\u03af \u03bd\u03b1 \u03b2\u03bf\u03b7\u03b8\u03ae\u03c3\u03b5\u03b9 \u03bd\u03b1 \u03c3\u03c4\u03b7\u03b8\u03b5\u03af \u03ad\u03bd\u03b1 \u03b1\u03c3\u03c6\u03b1\u03bb\u03ad\u03c2 \u03c0\u03bb\u03b1\u03af\u03c3\u03b9\u03bf \u03b1\u03be\u03b9\u03bf\u03bb\u03cc\u03b3\u03b7\u03c3\u03b7\u03c2 \u03ba\u03b1\u03b9 \u03b1\u03c5\u03c4\u03bf\u03bc\u03b1\u03c4\u03bf\u03c0\u03bf\u03af\u03b7\u03c3\u03b7\u03c2 \u03b3\u03cd\u03c1\u03c9 \u03b1\u03c0\u03cc \u03c4\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc use case.<\/p>\n<div class=\"td-service-cta-actions\"><a class=\"td-service-cta-button\" href=\"https:\/\/twodots.gr\/aftomatismoi-epicheiriseon-ai\/\">\u0394\u03b5\u03af\u03c4\u03b5 \u03c4\u03bf\u03c5\u03c2 \u03b1\u03c5\u03c4\u03bf\u03bc\u03b1\u03c4\u03b9\u03c3\u03bc\u03bf\u03cd\u03c2 \u03b5\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03ae\u03c3\u03b5\u03c9\u03bd<\/a><\/div>\n<\/div>\n<\/section>\n<section id=\"sychnes-erotiseis\" class=\"td-faq-section\">\n<div class=\"td-faq\">\n<p class=\"td-faq-heading\">\u0427\u0435\u0441\u0442\u043e \u0437\u0430\u0434\u0430\u0432\u0430\u043d\u0438 \u0432\u044a\u043f\u0440\u043e\u0441\u0438<\/p>\n<details class=\"td-faq-item\">\n<summary class=\"td-faq-title\">\u03a4\u03b9 \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf Edge-Conditioned Spectral Operator;<\/summary>\n<div class=\"td-faq-content\">\n<p>\u03a4\u03bf Edge-Conditioned Spectral Operator \u03ae ESO \u03b5\u03af\u03bd\u03b1\u03b9 neural operator \u03c0\u03bf\u03c5 \u03c3\u03c5\u03bd\u03b4\u03c5\u03ac\u03b6\u03b5\u03b9 \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03bc\u03af\u03be\u03b7 \u03bc\u03b5 \u03c0\u03bb\u03b7\u03c1\u03bf\u03c6\u03bf\u03c1\u03af\u03b1 \u03b1\u03c0\u03cc \u03c4\u03b9\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ad\u03c2 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03b3\u03b5\u03b9\u03c4\u03bf\u03bd\u03b9\u03ba\u03ce\u03bd \u03c3\u03b7\u03bc\u03b5\u03af\u03c9\u03bd. \u03a3\u03c4\u03cc\u03c7\u03bf\u03c2 \u03c4\u03bf\u03c5 \u03b5\u03af\u03bd\u03b1\u03b9 \u03bd\u03b1 \u03b4\u03b9\u03b1\u03c4\u03b7\u03c1\u03b5\u03af \u03bc\u03b5\u03b3\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03b7 \u03b1\u03ba\u03c1\u03af\u03b2\u03b5\u03b9\u03b1 \u03ba\u03bf\u03bd\u03c4\u03ac \u03c3\u03b5 \u03b4\u03b9\u03b5\u03c0\u03b9\u03c6\u03ac\u03bd\u03b5\u03b9\u03b5\u03c2, \u03b4\u03af\u03bd\u03b5\u03c2 \u03ba\u03b1\u03b9 \u03ac\u03bb\u03bb\u03b5\u03c2 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03b5\u03c5\u03b1\u03af\u03c3\u03b8\u03b7\u03c4\u03b5\u03c2 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2.<\/p>\n<\/div>\n<\/details>\n<details class=\"td-faq-item\">\n<summary class=\"td-faq-title\">\u03a0\u03bf\u03b9\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03b7 \u03b4\u03b9\u03b1\u03c6\u03bf\u03c1\u03ac \u03c4\u03bf\u03c5 PVMM \u03b1\u03c0\u03cc \u03ad\u03bd\u03b1 point-centered gate;<\/summary>\n<div class=\"td-faq-content\">\n<p>\u03a4\u03bf PVMM \u03c7\u03c1\u03b7\u03c3\u03b9\u03bc\u03bf\u03c0\u03bf\u03b9\u03b5\u03af \u03c4\u03b7\u03bd \u03ba\u03b5\u03bd\u03c4\u03c1\u03b9\u03ba\u03ae \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03bc\u03b1\u03b6\u03af \u03bc\u03b5 \u03b4\u03cd\u03bf edge statistics, \u03c4\u03bf D \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd \u03c0\u03c1\u03bf\u03c3\u03b7\u03bc\u03b1\u03c3\u03bc\u03ad\u03bd\u03b7 \u03bc\u03ad\u03c3\u03b7 \u03b1\u03bb\u03bb\u03b1\u03b3\u03ae \u03ba\u03b1\u03b9 \u03c4\u03bf Q \u03b3\u03b9\u03b1 \u03c4\u03b7\u03bd \u03ad\u03bd\u03c4\u03b1\u03c3\u03b7 \u03c4\u03b7\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ae\u03c2 \u03b1\u03bd\u03c4\u03af\u03b8\u03b5\u03c3\u03b7\u03c2. \u0388\u03bd\u03b1 point-centered gate \u03b2\u03b1\u03c3\u03af\u03b6\u03b5\u03c4\u03b1\u03b9 \u03ba\u03c5\u03c1\u03af\u03c9\u03c2 \u03c3\u03c4\u03b7\u03bd \u03ba\u03b1\u03c4\u03ac\u03c3\u03c4\u03b1\u03c3\u03b7 \u03c4\u03bf\u03c5 \u03af\u03b4\u03b9\u03bf\u03c5 \u03c4\u03bf\u03c5 \u03c3\u03b7\u03bc\u03b5\u03af\u03bf\u03c5.<\/p>\n<\/div>\n<\/details>\n<details class=\"td-faq-item\">\n<summary class=\"td-faq-title\">\u03a4\u03b9 \u03ba\u03ac\u03bd\u03b5\u03b9 \u03c4\u03bf Physics-Aware Reweighting;<\/summary>\n<div class=\"td-faq-content\">\n<p>\u03a4\u03bf Physics-Aware Reweighting \u03ae PAR \u03b1\u03c5\u03be\u03ac\u03bd\u03b5\u03b9 \u03c4\u03bf \u03b5\u03ba\u03c0\u03b1\u03b9\u03b4\u03b5\u03c5\u03c4\u03b9\u03ba\u03cc \u03b2\u03ac\u03c1\u03bf\u03c2 \u03c4\u03c9\u03bd \u03c3\u03c6\u03b1\u03bb\u03bc\u03ac\u03c4\u03c9\u03bd \u03c3\u03b5 \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ad\u03c2 \u03c0\u03bf\u03c5 \u03b8\u03b5\u03c9\u03c1\u03bf\u03cd\u03bd\u03c4\u03b1\u03b9 \u03c6\u03c5\u03c3\u03b9\u03ba\u03ac \u03b5\u03c5\u03b1\u03af\u03c3\u03b8\u03b7\u03c4\u03b5\u03c2 \u03b1\u03c0\u03cc \u03c0\u03b5\u03b4\u03af\u03b1 \u03b4\u03b9\u03b1\u03b8\u03ad\u03c3\u03b9\u03bc\u03b1 \u03c3\u03c4\u03bf \u03b5\u03ba\u03ac\u03c3\u03c4\u03bf\u03c4\u03b5 benchmark. \u0394\u03b5\u03bd \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c4\u03b7 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03bc\u03af\u03be\u03b7\u00b7 \u03b1\u03bb\u03bb\u03ac\u03b6\u03b5\u03b9 \u03c4\u03bf loss.<\/p>\n<\/div>\n<\/details>\n<details class=\"td-faq-item\">\n<summary class=\"td-faq-title\">\u03a3\u03b5 \u03c0\u03cc\u03c3\u03b1 benchmarks \u03b1\u03be\u03b9\u03bf\u03bb\u03bf\u03b3\u03ae\u03b8\u03b7\u03ba\u03b5 \u03c4\u03bf ESO;<\/summary>\n<div class=\"td-faq-content\">\n<p>\u0397 \u03b5\u03c1\u03b3\u03b1\u03c3\u03af\u03b1 \u03b1\u03bd\u03b1\u03c6\u03ad\u03c1\u03b5\u03b9 \u03b5\u03bd\u03bd\u03ad\u03b1 PDE benchmarks: \u03c4\u03ad\u03c3\u03c3\u03b5\u03c1\u03b1 \u03bc\u03b5 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 \u03ba\u03b1\u03b9 \u03c0\u03ad\u03bd\u03c4\u03b5 \u03bc\u03b5 \u03bc\u03b7 \u03b4\u03bf\u03bc\u03b7\u03bc\u03ad\u03bd\u03b1 meshes. \u03a4\u03bf ESO \u03b5\u03af\u03c7\u03b5 \u03c4\u03bf \u03c7\u03b1\u03bc\u03b7\u03bb\u03cc\u03c4\u03b5\u03c1\u03bf \u03c3\u03c7\u03b5\u03c4\u03b9\u03ba\u03cc L2 error \u03c3\u03c4\u03bf\u03c5\u03c2 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03bf\u03c5\u03c2 \u03c0\u03af\u03bd\u03b1\u03ba\u03b5\u03c2 \u03c3\u03cd\u03b3\u03ba\u03c1\u03b9\u03c3\u03b7\u03c2.<\/p>\n<\/div>\n<\/details>\n<details class=\"td-faq-item\">\n<summary class=\"td-faq-title\">\u03a4\u03bf ESO \u03b1\u03bd\u03c4\u03b9\u03ba\u03b1\u03b8\u03b9\u03c3\u03c4\u03ac \u03ad\u03bd\u03b1\u03bd \u03ba\u03bb\u03b1\u03c3\u03b9\u03ba\u03cc numerical solver;<\/summary>\n<div class=\"td-faq-content\">\n<p>\u038c\u03c7\u03b9 \u03c9\u03c2 \u03b3\u03b5\u03bd\u03b9\u03ba\u03cc\u03c2 \u03ba\u03b1\u03bd\u03cc\u03bd\u03b1\u03c2. \u0397 \u03bc\u03b5\u03bb\u03ad\u03c4\u03b7 \u03c4\u03bf \u03b1\u03be\u03b9\u03bf\u03bb\u03bf\u03b3\u03b5\u03af \u03c9\u03c2 learned surrogate \u03b3\u03b9\u03b1 PDE solution operators. \u03a0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03b9\u03ba\u03ae \u03c7\u03c1\u03ae\u03c3\u03b7 \u03b1\u03c0\u03b1\u03b9\u03c4\u03b5\u03af \u03c3\u03cd\u03b3\u03ba\u03c1\u03b9\u03c3\u03b7 \u03bc\u03b5 \u03c4\u03bf\u03bd \u03c5\u03c0\u03ac\u03c1\u03c7\u03bf\u03bd\u03c4\u03b1 solver \u03c3\u03c4\u03bf \u03af\u03b4\u03b9\u03bf \u03c0\u03c1\u03cc\u03b2\u03bb\u03b7\u03bc\u03b1, \u03c3\u03c4\u03b1 \u03af\u03b4\u03b9\u03b1 \u03cc\u03c1\u03b9\u03b1 \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1\u03c4\u03bf\u03c2 \u03ba\u03b1\u03b9 \u03c3\u03b5 \u03b4\u03b5\u03b4\u03bf\u03bc\u03ad\u03bd\u03b1 \u03b5\u03ba\u03c4\u03cc\u03c2 \u03b5\u03ba\u03c0\u03b1\u03af\u03b4\u03b5\u03c5\u03c3\u03b7\u03c2.<\/p>\n<\/div>\n<\/details>\n<details class=\"td-faq-item\">\n<summary class=\"td-faq-title\">\u03a5\u03c0\u03ac\u03c1\u03c7\u03b5\u03b9 \u03b4\u03b9\u03b1\u03b8\u03ad\u03c3\u03b9\u03bc\u03bf\u03c2 \u03ba\u03ce\u03b4\u03b9\u03ba\u03b1\u03c2 \u03b3\u03b9\u03b1 \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03ae;<\/summary>\n<div class=\"td-faq-content\">\n<p>\u039d\u03b1\u03b9. \u039f\u03b9 \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03b5\u03af\u03c2 \u03c0\u03b1\u03c1\u03b1\u03c0\u03ad\u03bc\u03c0\u03bf\u03c5\u03bd \u03c3\u03c4\u03bf \u03b4\u03b7\u03bc\u03cc\u03c3\u03b9\u03bf repository Tanpig-X\/ESO \u03c3\u03c4\u03bf GitHub. \u0397 \u03cd\u03c0\u03b1\u03c1\u03be\u03b7 \u03ba\u03ce\u03b4\u03b9\u03ba\u03b1 \u03b4\u03b9\u03b5\u03c5\u03ba\u03bf\u03bb\u03cd\u03bd\u03b5\u03b9 \u03c4\u03b7\u03bd \u03b1\u03bd\u03b1\u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03ae, \u03b1\u03bb\u03bb\u03ac \u03b4\u03b5\u03bd \u03c5\u03c0\u03bf\u03ba\u03b1\u03b8\u03b9\u03c3\u03c4\u03ac \u03c4\u03b7\u03bd \u03b1\u03bd\u03b5\u03be\u03ac\u03c1\u03c4\u03b7\u03c4\u03b7 \u03b5\u03c0\u03b9\u03ba\u03cd\u03c1\u03c9\u03c3\u03b7.<\/p>\n<\/div>\n<\/details>\n<details class=\"td-faq-item\">\n<summary class=\"td-faq-title\">\u03a0\u03bf\u03b9\u03bf \u03b5\u03af\u03bd\u03b1\u03b9 \u03c4\u03bf \u03b2\u03b1\u03c3\u03b9\u03ba\u03cc \u03cc\u03c1\u03b9\u03bf \u03c4\u03c9\u03bd \u03b1\u03c0\u03bf\u03c4\u03b5\u03bb\u03b5\u03c3\u03bc\u03ac\u03c4\u03c9\u03bd;<\/summary>\n<div class=\"td-faq-content\">\n<p>\u03a0\u03c1\u03cc\u03ba\u03b5\u03b9\u03c4\u03b1\u03b9 \u03b3\u03b9\u03b1 arXiv preprint \u03c0\u03c1\u03ce\u03c4\u03b7\u03c2 \u03ad\u03ba\u03b4\u03bf\u03c3\u03b7\u03c2 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03c3\u03c5\u03bc\u03c0\u03b5\u03c1\u03ac\u03c3\u03bc\u03b1\u03c4\u03b1 \u03b1\u03c6\u03bf\u03c1\u03bf\u03cd\u03bd \u03c4\u03b1 \u03c3\u03c5\u03b3\u03ba\u03b5\u03ba\u03c1\u03b9\u03bc\u03ad\u03bd\u03b1 datasets, baselines \u03ba\u03b1\u03b9 \u03c0\u03c1\u03c9\u03c4\u03cc\u03ba\u03bf\u03bb\u03bb\u03b1. \u0394\u03b5\u03bd \u03b1\u03c0\u03bf\u03b4\u03b5\u03b9\u03ba\u03bd\u03cd\u03bf\u03c5\u03bd \u03cc\u03c4\u03b9 \u03c4\u03bf ESO \u03b8\u03b1 \u03b5\u03af\u03bd\u03b1\u03b9 \u03ba\u03b1\u03bb\u03cd\u03c4\u03b5\u03c1\u03bf \u03c3\u03b5 \u03ba\u03ac\u03b8\u03b5 PDE, mesh \u03ae \u03c0\u03b1\u03c1\u03b1\u03b3\u03c9\u03b3\u03b9\u03ba\u03cc \u03c0\u03b5\u03c1\u03b9\u03b2\u03ac\u03bb\u03bb\u03bf\u03bd.<\/p>\n<\/div>\n<\/details>\n<details class=\"td-faq-item\">\n<summary class=\"td-faq-title\">\u03a0\u03cc\u03c4\u03b5 \u03ad\u03c7\u03b5\u03b9 \u03b5\u03c0\u03b9\u03c7\u03b5\u03b9\u03c1\u03b7\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc \u03bd\u03cc\u03b7\u03bc\u03b1 \u03bd\u03b1 \u03b5\u03be\u03b5\u03c4\u03b1\u03c3\u03c4\u03b5\u03af;<\/summary>\n<div class=\"td-faq-content\">\n<p>\u0388\u03c7\u03b5\u03b9 \u03bd\u03cc\u03b7\u03bc\u03b1 \u03c3\u03b5 \u03b5\u03c0\u03b1\u03bd\u03b1\u03bb\u03b1\u03bc\u03b2\u03b1\u03bd\u03cc\u03bc\u03b5\u03bd\u03b5\u03c2 \u03c0\u03c1\u03bf\u03c3\u03bf\u03bc\u03bf\u03b9\u03ce\u03c3\u03b5\u03b9\u03c2, digital twins \u03ae \u03bc\u03b5\u03bb\u03ad\u03c4\u03b5\u03c2 \u03c0\u03b1\u03c1\u03b1\u03bc\u03ad\u03c4\u03c1\u03c9\u03bd \u03cc\u03c0\u03bf\u03c5 \u03bf \u03c7\u03c1\u03cc\u03bd\u03bf\u03c2 inference \u03ad\u03c7\u03b5\u03b9 \u03b1\u03be\u03af\u03b1 \u03ba\u03b1\u03b9 \u03c4\u03b1 \u03ba\u03c1\u03af\u03c3\u03b9\u03bc\u03b1 \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1\u03c4\u03b1 \u03bc\u03c0\u03bf\u03c1\u03bf\u03cd\u03bd \u03bd\u03b1 \u03bf\u03c1\u03b9\u03c3\u03c4\u03bf\u03cd\u03bd \u03b1\u03bd\u03ac \u03c0\u03b5\u03c1\u03b9\u03bf\u03c7\u03ae. \u03a7\u03c1\u03b5\u03b9\u03ac\u03b6\u03b5\u03c4\u03b1\u03b9 \u03c0\u03b9\u03bb\u03bf\u03c4\u03b9\u03ba\u03ae \u03c3\u03cd\u03b3\u03ba\u03c1\u03b9\u03c3\u03b7 \u03bc\u03b5 \u03c4\u03bf \u03c0\u03c1\u03b1\u03b3\u03bc\u03b1\u03c4\u03b9\u03ba\u03cc baseline \u03ba\u03b1\u03b9 \u03c3\u03b1\u03c6\u03ae \u03ba\u03c1\u03b9\u03c4\u03ae\u03c1\u03b9\u03b1 go\/no-go.<\/p>\n<\/div>\n<\/details>\n<\/div>\n<\/section>\n<div class=\"td-source-list\">\n<p id=\"piges\" class=\"td-source-list-title\">\u0418\u0437\u0442\u043e\u0447\u043d\u0438\u0446\u0438<\/p>\n<ul>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2608.06894\" target=\"_blank\" rel=\"noopener\">Tan, Quan &amp; Yang \u2014 From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning<\/a><\/li>\n<li><a href=\"https:\/\/github.com\/Tanpig-X\/ESO\" target=\"_blank\" rel=\"noopener\">Tanpig-X\/ESO \u2014 \u03b5\u03c0\u03af\u03c3\u03b7\u03bc\u03bf repository \u03c4\u03c9\u03bd \u03c3\u03c5\u03b3\u03b3\u03c1\u03b1\u03c6\u03ad\u03c9\u03bd<\/a><\/li>\n<li><a href=\"https:\/\/arxiv.org\/abs\/2010.08895\" target=\"_blank\" rel=\"noopener\">Li et al. \u2014 Fourier Neural Operator for Parametric Partial Differential Equations<\/a><\/li>\n<li><a href=\"https:\/\/airc.nist.gov\/airmf-resources\/airmf\/5-sec-core\/\" target=\"_blank\" rel=\"noopener\">NIST AI RMF Core \u2014 \u03c0\u03bb\u03b1\u03af\u03c3\u03b9\u03bf \u03b3\u03b9\u03b1 context-aware \u03b1\u03be\u03b9\u03bf\u03bb\u03cc\u03b3\u03b7\u03c3\u03b7, \u03bc\u03ad\u03c4\u03c1\u03b7\u03c3\u03b7 \u03ba\u03b1\u03b9 \u03b4\u03b9\u03b1\u03c7\u03b5\u03af\u03c1\u03b9\u03c3\u03b7 AI \u03ba\u03b9\u03bd\u03b4\u03cd\u03bd\u03c9\u03bd<\/a><\/li>\n<\/ul>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>\u03a4\u03bf ESO \u03c3\u03c5\u03bd\u03b4\u03ad\u03b5\u03b9 \u03c0\u03b1\u03b3\u03ba\u03cc\u03c3\u03bc\u03b9\u03b1 \u03c6\u03b1\u03c3\u03bc\u03b1\u03c4\u03b9\u03ba\u03ae \u03bc\u03af\u03be\u03b7 \u03bc\u03b5 \u03c4\u03b9\u03c2 \u03c4\u03bf\u03c0\u03b9\u03ba\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03b2\u03bf\u03bb\u03ad\u03c2 \u03bc\u03b5\u03c4\u03b1\u03be\u03cd \u03b3\u03b5\u03b9\u03c4\u03cc\u03bd\u03c9\u03bd \u03ba\u03b1\u03b9 \u03bc\u03b5\u03b9\u03ce\u03bd\u03b5\u03b9 \u03c4\u03bf \u03c3\u03c6\u03ac\u03bb\u03bc\u03b1 \u03c3\u03b5 \u03b5\u03bd\u03bd\u03ad\u03b1 PDE benchmarks.<\/p>","protected":false},"author":1,"featured_media":88605,"comment_status":"open","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"_gspb_post_css":"","content-type":"","footnotes":""},"categories":[17366],"tags":[7477,19198,19200,3597,19199],"class_list":["post-88572","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-digital-marketing","tag-machine-learning","tag-neural-operators","tag-pde","tag-techniti-noimosyni","tag-fysiki-prosomoiosi"],"blocksy_meta":{"styles_descriptor":{"styles":{"desktop":"","tablet":"","mobile":""},"google_fonts":[],"version":7}},"_links":{"self":[{"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/posts\/88572","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/comments?post=88572"}],"version-history":[{"count":1,"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/posts\/88572\/revisions"}],"predecessor-version":[{"id":88606,"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/posts\/88572\/revisions\/88606"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/media\/88605"}],"wp:attachment":[{"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/media?parent=88572"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/categories?post=88572"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/twodots.gr\/bg\/wp-json\/wp\/v2\/tags?post=88572"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}