{"id":46030,"date":"2023-10-25T16:43:32","date_gmt":"2023-10-25T08:43:32","guid":{"rendered":"https:\/\/swarma.org\/?p=46030"},"modified":"2023-10-25T16:43:32","modified_gmt":"2023-10-25T08:43:32","slug":"%e6%95%b0%e5%ad%a6%e6%8e%a2%e7%b4%a2%e7%9a%84%e6%9c%aa%e6%9d%a5%ef%bc%9a%e4%bb%8eai%e5%bc%95%e5%af%bc%e4%ba%ba%e7%b1%bb%e7%9b%b4%e8%a7%89%e5%88%b0%e6%95%b0%e5%ad%a6%e5%a4%a7%e8%af%ad%e8%a8%80%e6%a8%a1","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=46030","title":{"rendered":"\u6570\u5b66\u63a2\u7d22\u7684\u672a\u6765\uff1a\u4eceAI\u5f15\u5bfc\u4eba\u7c7b\u76f4\u89c9\u5230\u6570\u5b66\u5927\u8bed\u8a00\u6a21\u578b"},"content":{"rendered":"
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\u8d85\u5f26\u7406\u8bba\u6700\u5927\u7684\u4e00\u4e2a\u6311\u6218\u53eb\u505a\u771f\u7a7a\u7b80\u5e76<\/strong><\/span>\uff08vacuum degeneracy\uff09<\/span>\u95ee\u9898<\/strong>\uff0c\u5373\u600e\u6837\u5728\u4e0d\u540c\u7684\u51e0\u4f55\u4e2d\u627e\u5230\u4e00\u4e2a\u80fd\u4ea7\u751f\u6211\u4eec\u8fd9\u6837\u7684\u5b87\u5b99\u7684\u51e0\u4f55\u3002Douglas \u57282003\u5e74\u7814\u7a76\u8fc7\u8d85\u5f26\u5728\u6240\u6709\u771f\u7a7a\u91cc\u7684\u7edf\u8ba1\u884c\u4e3a\u3002\u540c\u65f6\uff0cKKLT<\/span>\uff08Kachru-Kallosh-Linde-Trivedi (2003)\uff09<\/span>\u53d1\u73b0\u5982\u679c\u53bb\u627e\u8fd9\u4e9b\u51e0\u4f55\u7684\u8bdd\uff0c\u9700\u8981\u572810500<\/sup>\u7684\u6570\u636e\u91cc\u53bb\u627e\u3002\u4e3a\u4ec0\u4e48\u8981\u8fd9\u4e48\u5927\u7684\u6570\u636e\u5462\uff1f\u57fa\u672c\u539f\u56e0\u662f\u8d85\u5f26\u7406\u8bba\u9700\u8981\u4ee3\u6570\u51e0\u4f55\u6765\u505a\uff0c\u4ee3\u6570\u51e0\u4f55\u9700\u8981\u7ec4\u5408\u51e0\u4f55\u6765\u505a\uff0c\u800c\u4e00\u8fdb\u5165\u7ec4\u5408\u6570\u5b66\u7684\u8303\u7574\uff0c\u5c31\u4f1a\u6709\u8d85\u6307\u6570\u7684\u7ef4\u5ea6\u589e\u957f\u3002<\/span><\/section>\n

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\u4f8b\u5982\uff0c\u57fa\u4e8eBorisov-Batyrev\u7684\u4e00\u4e2a\u5b9a\u7406\uff0c\u6700\u7b80\u5355\u7684\u6784\u9020\u5361\u62c9\u6bd4\u4e18\u6d41\u5f62\u7684\u65b9\u6cd5\u5c31\u662f\u7528\u81ea\u53cd\u591a\u9762\u4f53<\/span>\uff08reflexive polyhedra\uff09<\/span>\u3002\u81ea\u53cd\u591a\u9762\u4f53\u662f\u5185\u90e8\u53ea\u6709\u4e00\u4e2a\u70b9\uff0c\u4e14\u8ddd\u79bb\u8fd9\u4e2a\u70b9\u7684\u6240\u6709\u8fb9\u7684\u8ddd\u79bb\u662f1\u7684\u591a\u9762\u4f53\u3002\u6709\u4e00\u4e2a\u5047\u8bf4\u662f\u8bf4\u5728\u4efb\u4f55\u7ef4\u5ea6\u91cc\uff0c\u81ea\u53cd\u591a\u9762\u4f53\u7684\u6570\u91cf\u662f\u6709\u9650\u7684<\/span>\uff08up to SL(N;Z)\uff09<\/span>\u3002\u8fd9\u4e2a\u5047\u8bf4\u57281890\u5e74\u5728\u4e8c\u7ef4\u7a7a\u95f4\u91cc\u88ab\u8bc1\u660e\u662f\u6b63\u786e\u7684\uff0c\u8fd9\u6837\u7684\u591a\u9762\u4f53\u53ea\u670916\u4e2a\uff1b100\u591a\u5e74\u540e\uff0c\u8fd9\u4e2a\u5047\u8bf4\u624d\u5728\u4e09\u7ef4\u7a7a\u95f4\u91cc\u88ab\u8bc1\u660e\u662f\u6b63\u786e\uff0c\u4e14\u662f\u7528\u5f53\u65f6\u7b97\u529b\u6700\u5927\u7684\u8ba1\u7b97\u673a\u82b1\u4e86\u4e00\u5e74\u65f6\u95f4\u624d\u7b97\u51fa\u6765\u8fd9\u4e2a\u6570\u5b57<\/span>\uff084130\uff09<\/span>\u3002\u800c\u5230\u56db\u7ef4\u7a7a\u95f4\uff0c\u8fd9\u4e2a\u6570\u5b57\u5c31\u589e\u957f\u5230\u4e864.7\u4ebf\u591a\u3002\u5bf9\u4e8e\u56db\u7ef4\u4ee5\u4e0a\u7684\u7a7a\u95f4\uff0c\u8fd9\u4e2a\u5047\u8bf4\u8fd8\u662f\u4e00\u4e2a\u5f00\u653e\u95ee\u9898\u3002\u4ece\u8fd9\u4e2a\u4f8b\u5b50\u6211\u4eec\u53ef\u4ee5\u770b\u5230\uff0c\u4e3a\u4ec0\u4e48\u4e00\u4e2a\u6570\u5b66\u7269\u7406\u95ee\u9898\u4f7f\u6211\u4eec\u8003\u8651\u5230\u4e00\u4e2a\u4ee3\u6570\u51e0\u4f55\u95ee\u9898\uff0c\u7136\u540e\u8fd9\u4e2a\u4ee3\u6570\u51e0\u4f55\u95ee\u9898\u53c8\u4f7f\u6211\u4eec\u8003\u8651\u5230\u4e00\u4e2a\u7ec4\u5408\u51e0\u4f55\u95ee\u9898\u3002\u7136\u800c\u4e00\u65e6\u8fdb\u5165\u7ec4\u5408\u6570\u5b66\u7684\u4e16\u754c\uff0c\u4f60\u5c31\u4f1a\u53d1\u73b0\u8fd9\u4e9b\u6570\u5b57\u589e\u957f\u5f97\u975e\u5e38\u5feb\u3002<\/span><\/section>\n

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2017\u5e74\uff0c\u4f55\u6768\u8f89\u8001\u5e08\u5f00\u59cb\u601d\u8003\u662f\u5426\u53ef\u4ee5\u7528\u673a\u5668\u5b66\u4e60\u53bb\u7406\u89e3\u51e0\u4f55\uff0c\u4e8e\u662f\u542f\u52a8\u4e86\u4e00\u7cfb\u5217\u7684\u8d85\u5f26\u4f1a\u8bae String Data<\/span><\/p>\n

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He, Yang-Hui. “Machine-learning the string landscape.” Physics Letters B<\/em> 774 (2017): 564-568.<\/span><\/section>\n<\/blockquote>\n

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2. \u4ee3\u6570\u51e0\u4f55\u7684\u8ba1\u7b97\u95ee\u9898<\/strong><\/strong><\/strong><\/p>\n<\/section>\n

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\u6700\u5e38\u89c1\u7684\u8ba1\u7b97\u95ee\u9898\uff1a\u8ba1\u7b97\u4e00\u4e2a\u6d41\u5f62\u6216\u8005\u4ee3\u6570\u7c07\u7684\u62d3\u6251\u4e0d\u53d8\u91cf\u3002Bourbaki \u544a\u8bc9\u6211\u4eec\u7528\u540c\u8c03\u53bb\u62fc\u547d\u8ba1\u7b97\u3002<\/span><\/p>\n


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\u603b\u7684\u6765\u8bf4\uff0c\u6570\u5b66\u4e0a\u6709\u4e00\u5957\u6210\u719f\u7684\u65b9\u6cd5\u6765\u8ba1\u7b97\u8fd9\u4e9b\u62d3\u6251\u4e0d\u53d8\u91cf\u3002\u7136\u800c\uff0c\u8fd9\u5957\u65b9\u6cd5\u9700\u8981\u5148\u5c06\u4ee3\u6570\u7c07\u53d8\u6210 Gr\u00f6bner basis\uff0c\u800c\u53d8\u6362\u7b97\u6cd5\u7684\u590d\u6742\u5ea6\u662f\u53cc\u6307\u6570<\/span>\uff08double exponential\uff09<\/span>\u7684\u3002\u5982\u679c\u6240\u6709\u7684\u65b9\u7a0b\u90fd\u662f\u7ebf\u6027\u7684\u8bdd\uff0c\u90a3 Gr\u00f6bner basis \u53d8\u6362\u5c31\u4f1a\u53d8\u6210\u4e00\u4e2a\u666e\u901a\u7684\u9ad8\u65af reduction\uff0c\u9ad8\u65af reduction \u7684\u590d\u6742\u5ea6\u662f\u591a\u9879\u5f0f\u7684\u3002\u4f46\u4e00\u65e6\u8fdb\u5165\u975e\u7ebf\u6027\u4e16\u754c\uff0c\u590d\u6742\u5ea6\u5c31\u4f1a\u662f\u53cc\u6307\u6570\u65f6\u95f4\u3002\u8fd9\u5957\u7b97\u6cd5\u5df2\u7ecf\u88ab\u7f16\u7a0b\u5230 Sage Math\uff0cM2 \u8fd9\u4e9b\u8f6f\u4ef6\u91cc\u4e86\u3002\u5bf9\u4e8e\u7b80\u5355\u7684\u4ee3\u6570\u7c07\uff0c\u53ef\u4ee5\u7528\u8fd9\u5957\u7b97\u6cd5\u3002\u4f46\u662f\u53ea\u8981\u4ee3\u6570\u7c07\u7a0d\u53d8\u590d\u6742\uff0c\u6bd4\u5982\u5d4c\u5165\u5230\u4e24\u4e2a\u6216\u8005\u4e09\u4e2a\u5c04\u5f71\u7a7a\u95f4\u7684\u4e58\u79ef\uff0c\u90a3\u76ee\u524d\u4efb\u4f55\u8ba1\u7b97\u673a\u90fd\u65e0\u6cd5\u7b97\u51fa\u6765\u3002\u6bd4\u5982\u5982\u679c\u4f60\u60f3\u505a\u4e00\u4e2a\u4e94\u6b21\u65b9\u7a0b\u5728\u56db\u7ef4\u7684\u590d\u5c04\u5f71\u7a7a\u95f4<\/span>\uff08cp4\uff09<\/span>\u91cc\u9762\uff0c\u4e5f\u5c31\u662f\u5f88\u6709\u540d\u7684\u5361\u62c9\u6bd4\u4e18\u6d41\u5f62\uff0c\u5728\u8fd9\u91cc\u9762\u53bb\u7b97 Betti \u6570\uff1f\u7528Sage Math\u53ef\u4ee5\u7b97\u51fa\u6765\u3002\u4f46\u662f\u5982\u679c\u4f60\u60f3\u505a\u4e24\u4e2a\u65b9\u7a0b\u5728\u4e94\u4e2a\u6216\u8005\u4e03\u4e2a\u590d\u5c04\u5f71\u7a7a\u95f4\u91cc\u9762\u7684\u8bdd\uff0c\u7528Sage Math\u5c31\u6ca1\u529e\u6cd5\u7b97\u51fa\u6765\u4e86\u3002<\/span><\/section>\n

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\u4f55\u6559\u6388\u5bf9\u6b64\u7c7b\u95ee\u9898\u6709\u8fd9\u6837\u7684\u6d1e\u89c1\uff1a\u662f\u4e0d\u662f\u53ef\u4ee5\u628a\u5341\u4e07\u4e2a\u5982\u56fe\u4e2dX\u8fd9\u6837\u7684\u4ee3\u6570\u7c07<\/span>\uff08\u8f93\u5165\uff09<\/span>\u548c\u5b83\u4eec\u7684 Betti \u6570<\/span>\uff08\u8f93\u51fa\uff09<\/span>\u4f5c\u4e3a\u4e00\u4e2a\u7b80\u5355\u7684\u524d\u9988\u7f51\u7edc<\/span>\uff08feedforward network\uff09<\/span>\u7684\u8bad\u7ec3\u6570\u636e\uff0c\u7136\u540e\u770b\u795e\u7ecf\u7f51\u7edc\u7684\u8868\u73b0\u5982\u4f55\uff1f<\/span><\/section>\n

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\u4ee4\u4eba\u60ca\u559c\u7684\u662f\uff0c\u795e\u7ecf\u7f51\u7edc\u7684\u7cbe\u786e\u5ea6\u5f88\u5feb\u5c31\u80fd\u8fbe\u5230\u5f88\u9ad8\u3002\u7136\u540e\uff0c\u6700\u8fd1\u6709\u4e00\u6279\u4eba\u7528\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u548c Google \u7684 inception network \u628a\u7cbe\u786e\u5ea6\u63d0\u9ad8\u5230\u4e8699.9%\u3002\u8fd9\u4e2a\u5b9e\u9a8c\u544a\u8bc9\u6211\u4eec\uff0c\u4ee3\u6570\u51e0\u4f55\u6709\u8db3\u591f\u597d\u7684\u7ed3\u6784\uff0c\u4f7f\u5f97\u6211\u4eec\u80fd\u7528\u7b80\u5355\u7684\u795e\u7ecf\u7f51\u7edc\u6765\u627e\u5230\u5b83\u7684\u5185\u5728\u89c4\u5f8b<\/strong>\u3002\u53d7\u5230\u795e\u7ecf\u7f51\u7edc\u5728\u4ee3\u6570\u51e0\u4f55\u9886\u57df\u7684\u542f\u53d1\uff0c2020-2021\u5e74\u671f\u95f4\uff0c\u4f55\u6559\u6388\u548c\u5408\u4f5c\u8005\u5199\u4e86\u4e00\u7bc7\u7efc\u8ff0\uff0c\u6765\u7ec6\u6570\u5404\u4e2a\u6570\u5b66\u5206\u652f\u91cc\u53ef\u80fd\u7528\u4eba\u5de5\u667a\u80fd\u53bb\u627e\u7c7b\u4f3c\u89c4\u5f8b\u7684\u95ee\u9898\u3002<\/span><\/section>\n

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\u4e3e\u4e00\u4e2a\u7b80\u5355\u7684\u4f8b\u5b50\uff1a\u7ed9\u51fa\u4e00\u4e2a\u6570\u5217\uff0c\u63a8\u6d4b\u4e0b\u4e00\u4e2a\u6570\u5b57\u662f\u4ec0\u4e48\u3002\u4e0b\u9762\u4e09\u4e2a\u6570\u5217\uff0c\u4eba\u773c\u6216\u8bb8\u5c31\u80fd\u53d1\u73b0\u7b2c\u4e00\u4e2a\u548c\u7b2c\u4e8c\u4e2a\u6570\u5217\u7684\u89c4\u5f8b\uff0c\u7b2c\u4e09\u4e2a\u6570\u5217<\/span>\uff08Liouville Lambda\u51fd\u6570\uff0c\u4e00\u4e2a\u6b63\u6574\u6570\u7684\u8d28\u56e0\u6570\u4e2a\u6570\u662f\u5947\u6570\u8fd8\u662f\u5076\u6570\uff09<\/span>\u7528\u4eba\u773c\u53ef\u80fd\u5f88\u96be\u770b\u51fa\u6765\u3002\u795e\u7ecf\u7f51\u7edc\u80fd\u5f88\u597d\u5730\u53d1\u73b0\u8fd9\u4e9b\u6570\u5217\u7684\u89c4\u5f8b\uff0c\u5bf9\u4e0b\u4e00\u4e2a\u6570\u5b57\u505a\u7cbe\u786e\u7684\u9884\u6d4b\u5417\uff1f<\/span><\/section>\n

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\u5bf9\u6b64\uff0c\u4f55\u6559\u6388\u505a\u4e86\u4e00\u4e2a\u5b9e\u9a8c\uff0c\u521b\u5efa\u4e86\u5bf9\u4e00\u4e2a\u4e8c\u5143\u5411\u91cf\u505a\u4e8c\u5143\u5206\u7c7b\u7684\u4efb\u52a1\u3002\u6bd4\u5982\uff0c\u5982\u679c\u75280\u548c1\u6765\u8868\u793a\u4e00\u4e2a\u6b63\u6574\u6570\u662f\u4e0d\u662f\u7d20\u6570\uff0c\u8fd9\u6837\u5c31\u80fd\u83b7\u5f97\u4e00\u4e2a\u65e0\u7a77\u7684\u4e8c\u5143\u6570\u5217\uff0c\u7136\u540e\u5c1d\u8bd5\u7528\u76d1\u7763\u5b66\u4e60\u9884\u6d4b\u4e0b\u4e00\u4e2a\u6b63\u6574\u6570\u662f\u5426\u662f\u7d20\u6570\u3002\u8bad\u7ec3\u96c6\u662f\u8fd9\u6837\u7ec4\u6210\u7684\uff0c\u7528\u957f\u5ea6\u4e3a100\u7684 sliding window \u4ece\u5934\u5f00\u59cb\u6ed1\u52a8\uff0c\u6bcf\u4e00\u4e2a window \u91cc\u9762\u7684100\u4e2a\u6570\u4f5c\u4e3a\u8f93\u5165\uff0c\u4e0b\u4e00\u4e2a\u6570\u4f5c\u4e3a\u8f93\u51fa\u3002\u7136\u540e\uff0c\u53d65\u4e07\u4e2a\u6837\u672c\uff0c\u628a\u8fd9\u4e9b\u6570\u636e\u505a20-80\u7684\u4ea4\u53c9\u9a8c\u8bc1\uff0c\u7528\u652f\u6301\u5411\u91cf\u673a\u3001\u6734\u7d20\u8d1d\u53f6\u65af\u3001\u6700\u8fd1\u90bb\u3001\u524d\u9988\u7f51\u7edc\u7b49\u6a21\u578b\u8fdb\u884c\u8bad\u7ec3\uff0c\u7528\u51c6\u786e\u7387\u548c f-score \u4f5c\u4e3a\u8bc4\u4f30\u6307\u6807\u3002<\/span><\/section>\n

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\u7ed3\u679c\u662f\u8fd9\u6837\u7684\uff0c\u5bf9\u4e8e\u7b2c\u4e00\u4e2a\u6570\u5217\uff0c\u4efb\u4f55\u4e00\u4e2a\u5206\u7c7b\u5668\u90fd\u80fd\u5728\u51e0\u79d2\u949f\u4e4b\u5185\u8fbe\u5230100%\u7684\u51c6\u786e\u7387\u548c f-score\u3002\u7b2c\u4e8c\u4e2a\u6570\u5217\u6d89\u53ca\u5230\u7d20\u6570\u7684\u5206\u5e03\u4e0d\u5747\u5300\u95ee\u9898\uff0c\u56e0\u4e3a\u7d20\u6570\u662f\u6309\u7167 x\/log(x) \u8fd9\u6837\u5206\u5e03\u7684\uff0c\u6240\u4ee5\u5230\u540e\u9762\u4f1a\u6709\u8d8a\u6765\u8d8a\u591a\u76840\uff0c\u9700\u8981\u628a\u7a97\u6237\u6309\u6bd4\u4f8b\u62c9\u957f\u3002\u8fd9\u6837\u5904\u7406\u4e4b\u540e\uff0c\u795e\u7ecf\u7f51\u7edc\u548c\u652f\u6301\u5411\u91cf\u673a\u80fd\u8fbe\u523080%\u7684\u51c6\u786e\u7387\u548c0.6\u7684 f-score\u3002\u4f46\u7b2c\u4e09\u4e2a\u6570\u5217\uff0c\u4e5f\u5c31\u662f Liouville Lambda \u51fd\u6570<\/span>\uff08\u8fd9\u662f\u4e2a\u975e\u5e38\u6df1\u5965\u7684\u6570\u8bba\u95ee\u9898\uff0c\u8fd9\u4e2a\u51fd\u6570\u7684\u8868\u8fbe\u8ddf\u9ece\u66fc\u731c\u60f3\u662f\u7b49\u6548\u7684\uff0c\u77e5\u9053\u8fd9\u4e2a\u5c31\u80fd\u89e3\u51b3\u9ece\u66fc\u731c\u60f3\u4e86\uff09<\/span>\uff0c\u4efb\u4f55\u6a21\u578b\u80fd\u8fbe\u5230\u7684\u6700\u9ad8\u9884\u6d4b\u51c6\u786e\u7387\u4e5f\u53ea\u670950%\uff0c\u8ddf\u6295\u786c\u5e01\u7684\u6982\u7387\u4e00\u6837\u3002<\/span><\/section>\n
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He, Yang-Hui. “Machine-learning mathematical structures.” International Journal of Data Science in the Mathematical Sciences<\/em> 1.01 (2023): 23-47.<\/span><\/section>\n<\/blockquote>\n


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3. \u4eba\u662f\u600e\u4e48\u505a\u6570\u5b66\u7684\uff1f<\/strong><\/strong><\/strong><\/p>\n<\/section>\n

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  • \n

    \u81ea\u4e0b\u800c\u4e0a\u7684\u81ea\u52a8\u5b9a\u7406\u8bc1\u660e<\/strong><\/p>\n<\/li>\n<\/ul>\n


    <\/section>\n
    \u7f57\u7d20-\u6000\u7279\u6d77<\/span>\uff08Russel-Whitehead\uff09<\/span>\u7684\u300a\u6570\u5b66\u539f\u7406\u300b<\/span> (Principia Mathematica\uff0c1910s\uff09<\/span>\u5c06\u6570\u5b66\u5f62\u5f0f\u5316\u3001\u516c\u7406\u5316\u3002\u7136\u800c1930\u5e74\u4ee3\u7684\u65f6\u5019\uff0c\u54e5\u5fb7\u5c14\u7528\u4e0d\u5b8c\u5907\u7406\u8bba<\/span>\uff08incompleteness theorem\uff09<\/span>\uff0c\u56fe\u7075\u7528\u4e0d\u53ef\u5224\u5b9a\u6027<\/span>\uff08undecidability\uff09<\/span>\u6307\u51fa\u4e86\u8fd9\u4e2a\u6846\u67b6\u7684\u7f3a\u9677\u3002<\/span><\/section>\n

    <\/section>\n
    \u4f46\u52a1\u5b9e\u7684\u6570\u5b66\u5bb6\u57fa\u672c\u4e0d\u4f1a\u8003\u8651\u54e5\u5fb7\u5c14\u7684\u4e0d\u5b8c\u5907\u7406\u8bba\uff0c\u6240\u4ee5\u81ea\u52a8\u5b9a\u7406\u8bc1\u660e<\/strong><\/span>\uff08automated theorem proving\uff09<\/span>\u4ecd\u5e7f\u6cdb\u88ab\u6570\u5b66\u5bb6\u6240\u63a5\u53d7\u3002\u81ea\u52a8\u5b9a\u7406\u8bc1\u660e\u76ee\u524d\u6700\u6210\u529f\u7684\u5c31\u662f2013\u5e74\u5f00\u59cb\u7684 Lean\u3002\u5e1d\u56fd\u7406\u5de5\u5927\u5b66\u7684Buzzard\u548cDavenport\u5df2\u7ecf\u628a\u622a\u81f3\u672c\u79d1\u6570\u5b66\u7684\u6240\u6709\u547d\u9898\u548c\u5b83\u4eec\u7684\u8bc1\u660e\u8fc7\u7a0b\u7528 Lean \u8dd1\u8fc7\u4e00\u904d\uff0c\u53d1\u73b0\u6240\u6709\u7684\u547d\u9898\u90fd\u4e92\u76f8\u4e0d\u77db\u76fe\u3002Buzzard\u8ba4\u4e3a\u6570\u5b66\u7814\u7a76\u7684\u672a\u6765\u5fc5\u987b\u4f9d\u8d56\u81ea\u52a8\u5b9a\u7406\u8bc1\u660e\uff0c\u4f46\u53ef\u80fd\u9700\u8981\u4e00\u4e2a\u4e16\u7eaa\u4ee5\u4e0a\u7684\u65f6\u95f4\u624d\u80fd\u5b9e\u73b0\u3002\u800cSzegedy\u8ba4\u4e3a2030\u5e74\u5e94\u8be5\u5c31\u80fd\u7528\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u5b9e\u73b0\u3002\u6211\u4eec\u628a\u8fd9\u79cd\u505a\u6570\u5b66\u7684\u65b9\u5f0f\u53eb\u505a\u201c\u81ea\u4e0b\u800c\u4e0a<\/strong>\u201d<\/strong>\u7684\u6570\u5b66<\/strong>\u3002<\/span><\/section>\n
    \n
    Lean \u9879\u76ee\uff1ahttps:\/\/lean-lang.org\/about\/<\/span><\/section>\n
    \u76f8\u5173\u79d1\u666e\u6587\u7ae0\uff1ahttps:\/\/www.quantamagazine.org\/building-the-mathematical-library-of-the-future-20201001\/<\/span><\/section>\n<\/blockquote>\n
      \n
    • \n

      \u81ea\u4e0a\u800c\u4e0b\u7684\u673a\u5668\u5f15\u5bfc\u4eba\u7c7b\u76f4\u89c9<\/strong><\/p>\n<\/li>\n<\/ul>\n


      <\/section>\n
      \u5386\u53f2\u4e0a\uff0c\u505a\u6570\u5b66\u7814\u7a76\u7684\u65b9\u5f0f\u66f4\u63a5\u8fd1\u4e8e\u81ea\u4e0a\u800c\u4e0b\uff0c\u4e5f\u5c31\u662f\u4f9d\u9760\u76f4\u89c9<\/strong>\u3002\u4e00\u822c\u662f\u5148\u770b\u6570\u636e\uff0c\u901a\u8fc7\u5bf9\u6570\u636e\u7684\u89c2\u5bdf\u5f97\u5230\u731c\u60f3\uff0c\u7136\u540e\u518d\u53bb\u9a8c\u7b97\u6216\u8005\u8bc1\u660e\uff0c\u6700\u540e\u53d1\u8868\u6587\u7ae0\u3002\u6570\u5b66\u5bb6\u548c\u7406\u8bba\u7269\u7406\u5b66\u5bb6\u5199\u6587\u7ae0\u7684\u8fc7\u7a0b\u8ddf\u601d\u8def\u6b63\u597d\u76f8\u53cd\uff0c\u6587\u7ae0\u662f\u4ece\u5b9a\u4e49\u5f00\u59cb\uff0c\u5230\u5f15\u7406\uff0c\u7136\u540e\u5b9a\u7406\u7b49\u7b49\uff0c\u4f46\u505a\u6570\u5b66\u7684\u8fc7\u7a0b\u662f\u5148\u60f3\u5230\u4e00\u4e2a\u4f8b\u5b50\uff0c\u7136\u540e\u53bb\u505a\u8fd9\u4e2a\u4f8b\u5b50\uff0c\u63a5\u7740\u505a\u7b2c\u4e8c\u4e2a\u4f8b\u5b50\u3001\u7b2c\u4e09\u4e2a\u4f8b\u5b50\uff0c\u4e5f\u8bb8\u662f\u5bf9\u7684\uff0c\u5c31\u505a\u4e00\u4e2a\u63a8\u60f3\uff0c\u7136\u540e\u5199\u6587\u7ae0\u3002<\/span><\/section>\n

      <\/span><\/section>\n
      \u4e3e\u4e2a\u4f8b\u5b50\uff0c18\u523019\u4e16\u7eaa\u6700\u4f1f\u5927\u7684\u795e\u7ecf\u7f51\u7edc\u662f\u9ad8\u65af\u7684\u5927\u8111<\/strong>\u3002\u5728\u9ad8\u65af16\u5c81\u7684\u65f6\u5019\u5c31\u5df2\u7ecf\u770b\u51fa\u6765\u5c0f\u4e8ex\u7684\u8d28\u6570\u7684\u4e2a\u6570\u662fx\/log(x)\u3002\u800c\u4eba\u7c7b\u76f4\u523050\u5e74\u540e\u53d1\u660e\u4e86\u590d\u5206\u6790\u624d\u80fd\u8bc1\u660e\u8fd9\u4e2a\u95ee\u9898\u3002\u6240\u4ee5\u4ed6\u7684\u76f4\u89c9\u662f\u4e00\u79cd\u8d85\u4eba\u7684\u795e\u7ecf\u7f51\u7edc\u3002\u867d\u7136\u6211\u4eec\u73b0\u5728\u6ca1\u6709\u9ad8\u65af\u7684\u5927\u8111\uff0c\u4f46\u6211\u4eec\u6709\u4e0a\u5343\u4e07\u4e2a\u795e\u7ecf\u7f51\u7edc\uff0c\u6765\u8fd1\u4f3c\u9ad8\u65af\u7684\u76f4\u89c9\u611f\u3002\u518d\u6bd4\u5982\uff0c20\u4e16\u7eaa Birch \u548c Swinnerton \u63d0\u51fa\u7684 Birch & Swinnerton-Dyer <\/span>(BSD) <\/span>\u731c\u60f3<\/span>\uff08\u76ee\u524d\u8fd8\u662f\u4e2a\u5f00\u653e\u95ee\u9898\uff09<\/span>\uff0c\u662f\u7531\u4ed6\u4eec\u5728\u5251\u6865\u7684\u725b\u987f\u7814\u7a76\u6240<\/span>\uff08Issac Newton Institute\uff09<\/span>\u7684\u5730\u4e0b\u5ba4\u91cc\u901a\u8fc7\u8ba1\u7b97\u5f88\u591a\u5f88\u591a\u7684\u692d\u5706\u66f2\u7ebf\u7684\u79e9<\/span>\uff08rank\uff09<\/span>\u4ea7\u751f\u7684\u3002<\/span><\/section>\n

      <\/span><\/section>\n
      \uff081\uff09\u4e00\u4e2a\u8868\u793a\u8bba\/\u7fa4\u8bba\u7684\u4f8b\u5b50<\/span><\/section>\n

      <\/span><\/section>\n
      \u4e00\u4e2a\u6709\u9650\u7fa4\u662f\u5b8c\u5168\u7531\u5b83\u7684Cayley\u8868\u5b9a\u4e49\u7684\u3002\u90a3\u5355\u7fa4\u7684Cayley\u8868\u8ddf\u975e\u5355\u7fa4\u7684Cayley\u8868\u662f\u4e0d\u662f\u4e0d\u4e00\u6837\u5462\uff1f\u5f53\u65f6\u4f55\u8001\u5e08\u53ca\u4ed6\u7684\u5408\u4f5c\u8005\u53d6\u4e86\u5927\u7ea6\u4e00\u5343\u4e2a\u7fa4\uff0c\u5176\u4e2d\u6709\u7684\u662f\u5355\u7684\u6709\u7684\u662f\u975e\u5355\u7684\u3002\u7136\u540e\u628a\u8fd9\u4e9b\u7fa4\u7684 Cayley \u8868\u753b\u5728\u4e00\u4e2aN\u5e73\u65b9\u7684\u6b27\u6c0f\u7a7a\u95f4\u91cc\uff0c\u5c31\u5f97\u5230\u5f88\u591a\u5f88\u591a\u70b9\uff0c\u518d\u76f8\u52a0<\/span>\uff08\u56e0\u4e3aup to commutation\uff09<\/span>\uff0c\u5c31\u5f97\u5230\u6570\u5341\u4e07\u7684\u6570\u636e\u70b9\uff0c\u5176\u4e2d\u4e00\u4e9b\u88ab\u6807\u8bb0\u4e3a\u5355\u7fa4\uff0c\u53e6\u4e00\u4e9b\u88ab\u6807\u8bb0\u4e3a\u975e\u5355\u7fa4\u3002\u7136\u540e\u7528\u652f\u6301\u5411\u91cf\u673a\u627e\u51fa\u5206\u79bb\u5355\u7fa4\u70b9\u4e0e\u975e\u5355\u7fa4\u70b9\u7684\u8d85\u5e73\u9762\u3002\u8fd9\u9879\u5de5\u4f5c\u53ea\u662f\u4e00\u4e2aproto conjecture\uff0c\u56e0\u4e3a\u8fd9\u6837\u505a\u53ef\u4ee5\u5f97\u5230\u8fd9\u6837\u7684\u7ed3\u679c\uff0c\u4f46\u4e0d\u77e5\u9053\u4e3a\u4ec0\u4e48\u4f1a\u5f97\u5230\u8fd9\u6837\u7684\u7ed3\u679c\uff0c\u800c\u4e14\u4e0d\u80fd\u544a\u8bc9\u4e00\u4e2a\u505a\u8868\u793a\u8bba\u7684\u6570\u5b66\u5bb6\u600e\u4e48\u53bb\u9a8c\u8bc1\u8fd9\u4e2a\u95ee\u9898\u3002\u672a\u6765\u4ed6\u4eec\u5e0c\u671b\u80fd\u628a\u8fd9\u4e2a\u505a\u6210\u4e00\u4e2a\u660e\u786e\u7684\u731c\u60f3\u3002<\/span><\/section>\n
      \n
      He, Yang-Hui, and Minhyong Kim. “Learning algebraic structures: preliminary investigations.” International Journal of Data Science in the Mathematical Sciences<\/em> 1.01 (2023): 3-22.<\/span><\/section>\n<\/blockquote>\n

      <\/section>\n
      \uff082\uff09\u4e00\u4e2a\u7ec4\u5408\u8bba\u7684\u4f8b\u5b50<\/span><\/section>\n

      <\/span><\/section>\n
      \u4e18\u6210\u6850\u5148\u751f\u57282020\u5e74\u66fe\u95ee\u8fc7\u4f55\u8001\u5e08\uff0c\u5982\u679c\u53d6\u5f88\u591a\u7684\u6709\u9650\u56fe\uff0c\u80fd\u4e0d\u80fd\u627e\u5230Ricci-\u5e73\u5766\u56fe\u5f62\uff1f\u4ed6\u7684\u7075\u611f\u662f\u5361\u62c9\u6bd4\u4e18\u6d41\u5f62\u662f\u5e73\u5766\u6d41\u5f62\uff0c\u90a3\u5b83\u662f\u4e0d\u662f\u6709\u4e00\u4e2a\u79bb\u6563\u9886\u57df\u7684\u7c7b\u6bd4\u5462\uff1f\u4f55\u8001\u5e08\u7528\u4e86Wolfram \u7684\u6709\u9650 simple graphs \u6570\u636e\u5e93\uff0c\u7136\u540e\u7528\u5206\u7c7b\u6216\u8005\u56de\u5f52\u6216\u8005\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u53bb\u5b66\u4e60\u8fd9\u4e9b\u56fe\u7684\u90bb\u63a5\u77e9\u9635\uff0c\u770b\u662f\u5426\u80fd\u627e\u51fa Hamilton cycle \u6216\u8005 Euler cycle\u3002\u5176\u4e2d Hamilton cycle \u662f\u4e2aNP\u95ee\u9898\uff0c\u5b83\u662f\u770b\u4e00\u5f20\u56fe\u91cc\u9762\u662f\u4e0d\u662f\u80fd\u627e\u5230\u7ecf\u8fc7\u6240\u6709\u70b9\u5e76\u53ea\u7ecf\u8fc7\u4e00\u6b21\u7684\u4e00\u6761\u95ed\u5408\u8def\u5f84\u3002\u4f55\u8001\u5e08\u53d1\u73b0\u4ece\u7edf\u8ba1\u89d2\u5ea6\uff0c\u795e\u7ecf\u7f51\u7edc\u627e\u5230\u6709 Hamilton cycle \u7684\u56fe\u548c\u6ca1\u6709 Hamilton cycle \u7684\u56fe\uff0c\u53ef\u4ee5\u8fbe\u5230 80% \u7684\u51c6\u786e\u7387\u3002<\/span><\/section>\n

      <\/span><\/section>\n
      \uff083\uff09\u4e00\u4e2a\u6570\u8bba\u7684\u4f8b\u5b50<\/span><\/section>\n

      <\/section>\n
      \u4e4b\u524d\u63d0\u5230\uff0c\u7528\u795e\u7ecf\u7f51\u7edc\u53bb\u9884\u6d4b\u4e0b\u4e00\u4e2a\u8d28\u6570<\/span>\u6216\u8005 Liouville Lamba \u51fd\u6570\u7684\u89c4\u5f8b\u90fd\u662f\u5f88\u96be\u5b9e\u73b0\u7684\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u7528\u4eba\u5de5\u667a\u80fd\u7684\u65b9\u6cd5\u53bb\u786c\u7b97\u6570\u8bba\u95ee\u9898\u662f\u5f88\u96be\u7684\u3002\u4f46\u662f\uff0c\u6211\u4eec\u53ef\u4ee5\u628a\u6570\u8bba\u95ee\u9898\u53d8\u6210\u4e00\u4e2a\u692d\u5706\u66f2\u7ebf\u7684\u95ee\u9898\u3002LMFDB \u662f\u4e00\u4e2a\u5305\u542b1000\u4e07\u5de6\u53f3\u692d\u5706\u66f2\u7ebf\u7684\u6570\u636e\u5e93\u3002\u4f55\u8001\u5e08\u4e0e\u4ed6\u7684\u6570\u8bba\u5408\u4f5c\u8005\u66fe\u5c1d\u8bd5\u4e3a\u7ed9\u5b9a\u7684\u4e00\u4e2a\u692d\u5706\u66f2\u7ebf\u7b97L-\u51fd\u6570\u3002\u692d\u5706\u66f2\u7ebf\u662f\u4e00\u4e2a\u4e8c\u5143\u4e09\u6b21\u591a\u9879\u5f0f\uff0c\u76ee\u7684\u662f\u5f53\u7cfb\u6570\u4e3a\u6574\u6570\u65f6\uff0c\u53bb\u627e\u4e24\u4e2a\u53d8\u91cf\u7684\u6574\u6570\u89e3\u6216\u8005\u6709\u7406\u6570\u89e3\u3002\u5f88\u591a\u6570\u8bba\u95ee\u9898\u90fd\u53ef\u4ee5\u5f52\u6839\u5230\u8fd9\u4e2a\u95ee\u9898\u4e0a\u3002\u76ee\u524d\u6ca1\u6709\u529e\u6cd5\u627e\u6574\u6570\u89e3\uff0c\u4f46\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a\u751f\u6210\u51fd\u6570\u7136\u540e\u5728\u6709\u9650\u57df\u4e0a\u53bb\u6570\u70b9\uff0c\u6211\u4eec\u628a\u8fd9\u4e2a\u751f\u6210\u51fd\u6570\u53eb\u505aL-\u51fd\u6570\u3002<\/span><\/section>\n

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      BSD \u731c\u60f3\u544a\u8bc9\u6211\u4eec\u4e00\u4e2aL-\u51fd\u6570\u7684\u89e3\u6790\u6027\u8d28\uff0c\u5373\u692d\u5706\u66f2\u7ebf\u4e0a\u7684\u6709\u7406\u6570\u70b9\u6709\u591a\u5c11\u3002\u4f55\u8001\u5e08\u4e0d\u7528\u692d\u5706\u66f2\u7ebf\u4e0a\u9762\u7684\u7cfb\u6570\uff0c\u800c\u662f\u7528L-\u51fd\u6570\u4e0a\u9762\u7684\u7cfb\u6570\uff0c\u7528LMFDB\u6570\u636e\u5e93\u4e2d\u7684\u6570\u636e\u53bb\u9a8c\u8bc1\u3002\u4ed6\u53d1\u73b0\u7528\u4e00\u4e9b\u5e38\u7528\u7684\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u53bb\u9884\u6d4b\u79e9\uff0ctorsion \u7b49\u6027\u8d28\u975e\u5e38\u597d\u3002\u6bd4\u5982\uff0c\u5bf9\u4e8e\u5224\u65ad\u692d\u5706\u7684\u79e9\u662f0\u6216\u662f1 \uff0c\u4e5f\u5c31\u662f\u5b83\u6709\u6709\u9650\u591a\u4e2a\u6709\u7406\u6570\u70b9\u8fd8\u662f\u65e0\u9650\u591a\u4e2a\u65e0\u7406\u6570\u70b9\uff0c\u903b\u8f91\u65af\u8482\u56de\u5f52\u5f88\u5feb\u5c31\u80fd\u8fbe\u5230\u51e0\u4e4e100%\u7684\u9884\u6d4b\u51c6\u786e\u7387\u3002<\/span>\uff08Goldfield \u548c Bhargava \u56e0\u4e3a\u8bc1\u660e\u4e86\u692d\u5706\u66f2\u7ebf\u7684\u79e9\u4ece\u7edf\u8ba1\u610f\u4e49\u4e0a\u6765\u8bf4\u670950%\u4e3a0\uff0c50%\u4e3a1\uff0c\u800c\u83b7\u5f97\u4e86\u83f2\u5c14\u5179\u5956\uff09<\/span><\/section>\n

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      \u4ece\u8fd9\u4e48\u591a\u4f8b\u5b50\u91cc\uff0c\u4f55\u8001\u5e08\u60f3\u8868\u8fbe\u7684\u4e3b\u5f20\u662f\uff0c\u5982\u679c\u4f60\u60f3\u53bb\u53d1\u73b0\u65b0\u7684\u6570\u5b66\u95ee\u9898\uff0c\u5c31\u5fc5\u987b\u96c6\u5408\u4eba\u548c\u4eba\u5de5\u667a\u80fd<\/strong>\u2014\u2014<\/strong>\u9700\u8981\u4eba\u5de5\u667a\u80fd\u6765\u5f15\u5bfc\u76f4\u89c9\uff0c\u7136\u540e\u9700\u8981\u4eba\u6765\u63d0\u51fa\u731c\u60f3<\/strong>\u3002\u4ed6\u63d0\u5230\u4e86\u8868\u793a\u7684\u91cd\u8981\u6027\uff0c\u6bd4\u5982\u4ed6\u4eec\u4e4b\u524d\u7528\u692d\u5706\u66f2\u7ebf\u7684 Weierstrass \u7cfb\u6570\u6765\u9884\u6d4bBSD\u731c\u60f3\u7684\u6027\u8d28\uff0c\u5f97\u5230\u7684\u7ed3\u679c\u5f88\u4e0d\u597d\uff0c\u540e\u9762\u5f97\u5230Oliver\u548cLee\u7684\u542f\u53d1\u7528\u03b1(p)\u7cfb\u6570\u9884\u6d4b\u8868\u73b0\u5c31\u5f88\u597d\u3002\u4eba\u7684\u89e3\u8bfb\u4e5f\u5f88\u91cd\u8981\uff0c\u6bd4\u5982\u901a\u8fc7\u89e3\u8bfb\u7528 \u03b1(p) \u7cfb\u6570\u9884\u6d4b BSD \u731c\u60f3\u7684\u6027\u8d28\u7684\u795e\u7ecf\u7f51\u7edc\u548c\u4e3b\u6210\u5206\u5206\u6790\uff0c\u5f97\u5230\u4e86\u4e00\u4e2a\u65b0\u7684\u660e\u786e\u7684\u731c\u60f3\uff0c\u4ed6\u4eec\u628a\u8fd9\u4e2a\u731c\u60f3\u53eb\u505a Murmurations\u3002\u4ed6\u4eec\u60f3\u77e5\u9053\u4e3a\u4ec0\u4e48\u673a\u5668\u5b66\u4e60\u80fd\u4ece\u03b1(p)\u7cfb\u6570\u5f88\u597d\u5730\u9884\u6d4b\u692d\u5706\u66f2\u7ebf\u7684\u79e9\u3002\u901a\u8fc7\u4e3b\u6210\u5206\u5206\u6790\uff0c\u4ed6\u4eec\u53d1\u73b0\u79e9\u4e3a0\u30011\u548c2\u7684\u692d\u5706\u66f2\u7ebf\u7684\u03b1(p)\u7cfb\u6570\u6295\u5c04\u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e0a\u7684\u70b9\u90fd\u662f\u5206\u79bb\u7684\uff0c\u7136\u540e\u4ed6\u4eec\u53bb\u8fdb\u4e00\u6b65\u6316\u6398\u4e3a\u4ec0\u4e48\u8fd9\u4e9b\u70b9\u662f\u5206\u79bb\u7684\u3002<\/span><\/section>\n
      \n
      Alessandretti, Laura, Andrea Baronchelli, and Yang-Hui He. “Machine learning meets number theory: the data science of Birch\u2013Swinnerton-Dyer.” MACHINE LEARNING: IN PURE MATHEMATICS AND THEORETICAL PHYSICS<\/em>. 2023. 1-39.<\/span><\/section>\n

      <\/section>\n
      He, Yang-Hui, et al. “Murmurations of elliptic curves.” arXiv preprint arXiv:2204.10140<\/em> (2022).<\/span><\/section>\n<\/blockquote>\n
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      \u4ed6\u4eec\u53d1\u73b0\uff0cPCA\u5b9e\u9645\u4e0a\u662f\u53d6\u4e86\u03b1(p)\u7cfb\u6570\u7684 vertical \u5e73\u5747\u503c<\/span>\uff08\u5b9a\u4e49\u5982\u4e0b\u56fe\u5de6\u4fa7\u6240\u793a\uff09<\/span>\u3002\u5982\u679c\u5c06\u8fd9\u4e9b vertical \u5e73\u5747\u503c\u753b\u51fa\u6765\uff0c\u4f1a\u53d1\u73b0\u4e0b\u56fe\u53f3\u4fa7\u8fd9\u6837\u7684\u6ce2\u52a8\u6a21\u5f0f\u3002<\/span><\/span><\/section>\n

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      \u5bf9\u4e8e\u89c2\u5bdf\u5230\u7684\u8fd9\u4e2a\u73b0\u8c61\uff0c\u6700\u8fd1\u4f55\u8001\u5e08\u4e0e\u8bb8\u591a\u505aBSD\u731c\u60f3\u7684\u7814\u7a76\u8005\u8fd8\u5728\u63a2\u8ba8\u4e2d\uff0c\u76ee\u524d\u5f97\u5230\u4e86\u4e00\u4e9b\u521d\u6b65\u8bc1\u660e\u7ed3\u679c\uff1a\u6ce2\u52a8\u5728\u6a21\u578b\u5f0f\u91cc\u6536\u655b\u5230\u4e00\u4e2a\u660e\u786e\u7684\u66f2\u7ebf\u3002\u800c\u6a21\u578b\u5f0f\u548c\u692d\u5706\u66f2\u7ebf\u6709\u5173\uff0c\u5982\u679c\u80fd\u8bc1\u660e\u51fa\u5728\u692d\u5706\u66f2\u7ebf\u91cc\u4e5f\u6709\u8fd9\u6837\u7684\u6027\u8d28\uff0c\u800c\u4e14\u80fd\u77e5\u9053\u8fd9\u4e2a\u660e\u786e\u7684\u66f2\u7ebf\u662f\u591a\u5c11\uff0c\u90a3\u5bf9BSD\u731c\u60f3\u5c06\u4f1a\u6709\u4e00\u5b9a\u7684\u8d21\u732e\u3002<\/span><\/section>\n

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