{"id":42988,"date":"2023-06-07T18:18:59","date_gmt":"2023-06-07T10:18:59","guid":{"rendered":"https:\/\/swarma.org\/?p=42988"},"modified":"2023-06-07T18:18:59","modified_gmt":"2023-06-07T10:18:59","slug":"%e5%a4%8d%e6%9d%82%e6%80%a7%e7%9a%84%e8%b7%a8%e5%ad%a6%e7%a7%91%e5%85%b3%e8%81%94%ef%bc%9a%e5%88%86%e6%95%b0%e9%98%b6%e5%be%ae%e7%a7%af%e5%88%86%e3%80%81%e9%87%8d%e6%95%b4%e5%8c%96%e7%be%a4%e4%b8%8e","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=42988","title":{"rendered":"\u590d\u6742\u6027\u7684\u8de8\u5b66\u79d1\u5173\u8054\uff1a\u5206\u6570\u9636\u5fae\u79ef\u5206\u3001\u91cd\u6574\u5316\u7fa4\u4e0e\u673a\u5668\u5b66\u4e60"},"content":{"rendered":"
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Haoyu Niu, YangQuan Chen, Lihong Guo, Bruce J. West<\/span><\/span> <\/span>| \u4f5c\u8005<\/strong><\/span><\/strong><\/p>\n

\u5b9b\u8212<\/span> <\/span>| \u8bd1\u8005<\/span><\/strong><\/span><\/span><\/span><\/strong><\/span><\/span><\/span><\/strong><\/p>\n

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\u8bba\u6587\u9898\u76ee\uff1a<\/span><\/p>\n

A New Triangle: Fractional Calculus, Renormalization Group, and Machine Learning <\/span><\/p>\n

\u8bba\u6587\u94fe\u63a5\uff1a<\/span><\/p>\n

https:\/\/asmedigitalcollection.asme.org\/IDETC-CIE\/proceedings-abstract\/IDETC-CIE2021\/85437\/1128150<\/span><\/p>\n<\/section>\n<\/blockquote>\n


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\u5206\u6570\u9636\u5fae\u79ef\u5206\u662f\u5bf9\u51fd\u6570\u8fdb\u884c\u975e\u6574\u6570\u9636\u79ef\u5206\u548c\u5fae\u5206\u7684\u5b9a\u91cf\u5206\u6790\u3002<\/span><\/strong>\u9636\u6570\u53ef\u4ee5\u662f\u5b9e\u6570\u3001\u590d\u6570\uff0c\u751a\u81f3\u662f\u53d8\u91cf\u7684\u51fd\u6570\u3002\u5173\u4e8e\u975e\u6574\u6570\u9636\u5fae\u5206\u7684\u610f\u4e49\u7684\u7b2c\u4e00\u6b21\u8bb0\u5f55\u51fa\u73b0\u57281695\u5e74\u6d1b\u5fc5\u8fbe<\/span>\uff08Guillaume de l’H\u00f4pital\uff09<\/span>\u5199\u7ed9\u83b1\u5e03\u5c3c\u8328<\/span>\uff08Gottfried Wilhelm Leibniz\uff09<\/span>\u7684\u4e00\u5c01\u4fe1\u4e2d\u3002\u83b1\u5e03\u5c3c\u8328\u4e0e\u725b\u987f\u662f\u540c\u65f6\u4ee3\u4eba\uff0c\u5e76\u4e0e\u4ed6\u5404\u81ea\u72ec\u7acb\u4f46\u5171\u540c\u53d1\u660e\u4e86\u65e0\u7a77\u5c0f\u5fae\u79ef\u5206[1]\u3002\u4ece\u90a3\u65f6\u8d77\uff0c\u8bb8\u591a\u8d21\u732e\u8005\u4e3a\u5206\u6570\u9636\u5bfc\u6570\u548c\u79ef\u5206\u63d0\u4f9b\u4e86\u5b9a\u4e49[2]\uff0c\u51e0\u4e2a\u4e16\u7eaa\u4ee5\u6765\uff0c\u5206\u6570\u9636\u5fae\u79ef\u5206\u7684\u7406\u8bba\u53ca\u5176\u5e94\u7528\u5f97\u5230\u4e86\u5f88\u5927\u7684\u6269\u5c55[3, 4, 5]\u3002<\/span><\/p>\n

\u5728\u8fd1\u51e0\u5341\u5e74\u91cc\uff0c\u5206\u6570\u9636\u52a8\u529b\u5b66\u7684\u6982\u5ff5\u5728\u7edf\u8ba1\u7269\u7406\u548c\u5316\u5b66\u7269\u7406\u9886\u57df\u5f97\u5230\u4e86\u53d1\u5c55\u548c\u5173\u6ce8[6, 7, 8]\u3002\u4f8b\u5982\uff0c\u901a\u8fc7\u4f7f\u7528\u5206\u6570\u9636\u5fae\u5206\u548c\u5206\u6570\u9636\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u6700\u4f18\u56fe\u50cf\u5904\u7406\u5f97\u5230\u4e86\u6539\u8fdb\uff0cChen \u7b49\u4eba\u8fdb\u884c\u4e86\u603b\u7ed3[9, 10, 11]\u3002\u5f02\u5e38\u6269\u6563\u5728[12,13]\u4e2d\u4f7f\u7528\u5206\u6570\u9636\u6269\u6563\u65b9\u7a0b\u8fdb\u884c\u63cf\u8ff0\uff0cMetzler \u7b49\u4eba\u4f7f\u7528\u5206\u6570\u9636\u6717\u4e4b\u4e07\u65b9\u7a0b<\/span>\uff08Langevin equation\uff09<\/span>\u5bf9\u7c98\u5f39\u6027\u6750\u6599\u8fdb\u884c\u5efa\u6a21[14]\u3002<\/span><\/p>\n

\u5206\u6570\u9636\u601d\u7ef4<\/span>\uff08Fractional-order thinkin<\/span>g\uff0cFOT<\/span>\uff09<\/span>\uff0c\u662f\u4f7f\u7528\u5206\u6570\u9636\u5fae\u79ef\u5206\u8fdb\u884c\u601d\u8003\u7684\u4e00\u79cd\u65b9\u5f0f\u3002\u4f8b\u5982\uff0c\u5728\u6574\u6570\u4e4b\u95f4\u6709\u975e\u6574\u6570\u3002\u5728\u903b\u8f910\u548c\u903b\u8f911\u4e4b\u95f4\uff0c\u6709\u6a21\u7cca\u903b\u8f91[15]\u3002\u4e0e\u6574\u6570\u9636\u6837\u6761<\/span>\uff08integer-order splines\uff09<\/span>\u76f8\u6bd4\uff0c\u6709\u5206\u6570\u9636\u6837\u6761[16]\u3002\u5728\u9ad8\u9636\u6574\u6570\u77e9\u4e4b\u95f4\uff0c\u6709\u975e\u6574\u6570\u9636\u77e9\u3002\u8bb8\u591a\u7814\u7a76\u9886\u57df\u90fd\u5305\u542b\u4e86\u5206\u6570\u9636\u601d\u7ef4\uff0c\u4f8b\u5982\uff0c\u81ea\u76f8\u4f3c[17, 18]\u3001\u65e0\u6807\u5ea6\u6216\u6807\u5ea6\u4e0d\u53d8\u6027\u3001\u5e42\u5f8b\u3001\u957f\u7a0b\u76f8\u5173<\/span>\uff08long-range dependence\uff0cLRD\uff09<\/span>[19, 20]\uff0c\u4ee5\u53ca1\/f<\/span>\u03b1<\/span><\/sup><\/span>\u566a\u58f0[21, 22]\u3002<\/span><\/p>\n


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\u91cd\u6574\u5316\u7fa4<\/span>\uff08Renormalization group\uff0cRG\uff09<\/span>\u662f\u4e00\u4e2a\u6982\u5ff5\u6846\u67b6\uff0c\u5305\u542b\u591a\u79cd\u6280\u672f\uff0c\u5982\u5b9e\u7a7a\u95f4\u91cd\u6574\u5316\u7fa4[23]\uff0c\u6cdb\u51fd\u91cd\u6574\u5316\u7fa4[24]\uff0c\u5bc6\u5ea6\u77e9\u9635\u91cd\u6574\u5316\u7fa4<\/span>\uff08DM\u91cd\u6574\u5316\u7fa4\uff09<\/span>[25]\u3002\u91cd\u6574\u5316\u7fa4\u6700\u521d\u662f\u5728\u7c92\u5b50\u7269\u7406\u5b66\u4e2d\u88ab\u53d1\u660e\u7684\u3002Stueckelbe\u7b49\u4eba[26]\u5728\u91cf\u5b50\u573a\u8bba\u4e2d\u9884\u89c1\u5e76\u9996\u6b21\u4ece\u6982\u5ff5\u4e0a\u63d0\u51fa\u4e86\u91cd\u6574\u5316\u7fa4\u3002\u4ed6\u4eec\u6307\u51fa\uff0c\u91cd\u6574\u5316\u7fa4\u5c55\u793a\u4e86\u4e00\u7ec4\u4ece\u88f8\u9879\u5230\u62b5\u6d88\u9879\u7684\u53d8\u6362\u3002\u8fc4\u4eca\u4e3a\u6b62\uff0c\u91cd\u6574\u5316\u7fa4\u5df2\u7ecf\u6269\u5c55\u5230\u8bb8\u591a\u5176\u4ed6\u9886\u57df\uff0c\u5982\u56fa\u4f53\u7269\u7406\u3001\u6d41\u4f53\u529b\u5b66\u548c\u7269\u7406\u5b87\u5b99\u5b66\u3002\u5728\u7406\u8bba\u7269\u7406\u4e2d\uff0c\u91cd\u6574\u5316\u7fa4\u662f\u4e00\u79cd\u5f62\u5f0f\u5316\u7684\u5de5\u5177\uff0c\u5141\u8bb8\u7814\u7a76\u8005\u63a2\u8ba8\u7269\u7406\u7cfb\u7edf\u5728\u4e0d\u540c\u5c3a\u5ea6\u4e0a\u7684\u53d8\u5316\u3002\u5c3a\u5ea6\u4e0a\u7684\u53d8\u5316\u901a\u5e38\u5b9a\u4e49\u4e3a\u5c3a\u5ea6\u53d8\u6362<\/span>\uff08<\/span>scale transformation\uff09<\/span><\/span>\u3002<\/span><\/span><\/p>\n

\u91cd\u6574\u5316\u7fa4\u4e0e<\/span>\u81ea\u76f8\u4f3c\u6027\u5bc6\u5207\u76f8\u5173\u3002\u5728\u91cf\u5b50\u573a\u8bba\u4e2d\uff0c\u8ba1\u7b97\u4e2d\u7684\u53d1\u6563\u90e8\u5206\u53ef\u80fd\u5bfc\u81f4\u975e\u7269\u7406\u6027\u7684\u65e0\u7a77\u5927\u7ed3\u679c\u3002\u4f7f\u7528\u91cd\u6574\u5316\u7fa4\uff0c\u53ef\u4ee5\u5c06\u53d1\u6563\u90e8\u5206\u91cd\u65b0\u5b9a\u4e49\u4e3a\u5c11\u91cf\u53ef\u6d4b\u91cf\u7684\u91cf\uff0c\u4ece\u800c\u5f97\u5230\u6709\u9650\u7684\u7ed3\u679c\u3002\u4f8b\u5982\uff0c\u91cf\u5b50\u573a\u8bba\u7528\u4e8e\u8ba1\u7b97\u91cf\u5b50\u6c34\u5e73\u4e0a\u57fa\u672c\u529b\u7684\u5f71\u54cd\u3002\u5728\u91cf\u5b50\u7535\u52a8\u529b\u5b66\u4e2d\uff0c\u7535\u5b50\u53ef\u4ee5\u4e0d\u65ad\u5730\u53d1\u5c04\u548c\u518d\u5438\u6536\u201c\u865a\u62df\u201d\u5149\u5b50\uff0c\u8fd9\u610f\u5473\u7740\u5176\u603b\u80fd\u91cf\u548c\u603b\u8d28\u91cf\u53ef\u4ee5\u662f\u65e0\u7a77\u5927\u7684\u3002\u7136\u800c\uff0c\u4f7f\u7528\u91cd\u6574\u5316\u7fa4\u7406\u8bba\u53ef\u4ee5\u89e3\u51b3\u53d1\u6563\u95ee\u9898\uff0c\u4e5f\u5c31\u662f\u5c06\u5305\u542b\u8fd9\u4e9b\u865a\u62df\u8fc7\u7a0b\u7684\u201c\u88f8\u201d\u7535\u5b50\u7684\u8d28\u91cf\u5b9a\u4e49\u4e3a\u6d4b\u91cf\u8d28\u91cf\u3002\u53d7\u91cf\u5b50\u7535\u52a8\u529b\u5b66\u7406\u8bba\u6210\u529f\u7684\u542f\u53d1\uff0c\u5176\u4ed6\u4e00\u4e9b\u91cf\u5b50\u573a\u8bba\u4e5f\u5728\u84ec\u52c3\u53d1\u5c55\uff0c\u5982\u7535\u5f31\u7406\u8bba\u548c\u91cf\u5b50\u8272\u52a8\u529b\u5b66\u3002\u7136\u800c\uff0c\u4e00\u4e2a\u80fd\u591f\u6db5\u76d6\u6240\u6709\u57fa\u672c\u529b<\/span>\uff08\u5982\u5f15\u529b\uff09<\/span>\u7684\u53ef\u91cd\u6574\u5316\u7406\u8bba<\/span>\uff08renormalizable theory\uff09<\/span>\u4ecd\u5728\u7814\u7a76\u4e2d\u3002<\/span><\/p>\n

\u9664\u4e86\u4e0a\u8ff0\u672a\u63d0\u53ca\u7684\u4e24\u4e2a\u4e3b\u8981\u7814\u7a76\u9886\u57df\u5916\uff0c\u52a8\u529b\u5b66\u7cfb\u7edf\u7684\u6df7\u6c8c\u884c\u4e3a\u548c\u7edf\u8ba1\u529b\u5b66\u4e2d\u4e34\u754c\u70b9\u7684\u666e\u9002\u6027\u7406\u8bba\uff0c\u5c1a\u4e14\u6ca1\u6709\u4f53\u73b0\u91cd\u6574\u5316\u7fa4\u7684\u5fc5\u7136\u6027\u3002\u8f83\u8fd1\u671f\u7684\u7814\u7a76\u662f\u5c06\u91cd\u6574\u5316\u7fa4\u5e94\u7528\u4e8e\u63cf\u8ff0\u590d\u6742\u52a8\u6001\u7f51\u7edc\u7684\u76f8\u53d8\uff0c\u5728\u4e34\u754c\u70b9\u5904\uff0c\u7269\u7406\u53ef\u89c2\u6d4b\u91cf\u53d8\u5f97\u4e0d\u8fde\u7eed\u3002\u793e\u4f1a\u548c\u751f\u7269\u590d\u6742\u52a8\u6001\u7f51\u7edc\u5c3d\u7ba1\u662f\u6709\u9650\u7ef4\u7684\uff0c\u4f46\u5b83\u4eec\u90fd\u88ab\u8bc1\u660e\u662f\u4f0a\u8f9b\u666e\u9002\u7c7b\u7684\u6210\u5458<\/span>\uff08Ising Universality class\uff09<\/span>\uff0c\u56e0\u6b64\u5728\u8fc7\u53bb\u5e94\u7528\u4e2d\u4e0d\u4f7f\u7528\u70ed\u529b\u5b66\u6781\u9650\u3002\u5173\u4e8e\u8fd9\u4e2a\u95ee\u9898\uff0c\u53ef\u4ee5\u53c2\u8003 West \u7b49\u4eba\u8bba\u6587[27]\u8fdb\u884c\u7684\u66f4\u8be6\u7ec6\u8ba8\u8bba\u3002\u5229\u7528\u8fd9\u4e9b\u601d\u60f3\uff0c\u7814\u7a76\u8005[28]\u5f00\u53d1\u4e86\u4e00\u4e2a\u8ba4\u77e5\u7684\u4e24\u7ea7\u6a21\u578b\uff0c\u5e76\u63d0\u4f9b\u4e86\u6d1e\u89c1\uff0c\u8bf4\u660e\u7ebf\u6027\u903b\u8f91\u601d\u7ef4\u5982\u4f55\u88ab\u6096\u8bba\u7834\u574f\uff0c\u53ea\u80fd\u7528\u5206\u6570\u9636\u601d\u7ef4\u6765\u89e3\u51b3\u3002<\/span><\/p>\n


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\u673a\u5668\u5b66\u4e60<\/span>\uff08Machine Learning\uff0cML\uff09<\/span>\u662f\u8ba9\u8ba1\u7b97\u673a\u80fd\u591f\u4ece\u6570\u636e\u4e2d\u5b66\u4e60\u7684\u79d1\u5b66<\/span>\uff08\u53ca\u201c\u827a\u672f\u201d\uff09<\/span>[29]\u30021997\u5e74\uff0cTom Mitchell \u7ed9\u51fa\u4e86\u673a\u5668\u5b66\u4e60\u4e00\u4e2a\u66f4\u4fa7\u91cd\u4e8e\u5de5\u7a0b\u7684\u5b9a\u4e49[30]\uff1a\u201c\u4e00\u4e2a\u8ba1\u7b97\u673a\u7a0b\u5e8f\u53ef\u4ee5\u4ece\u7ecf\u9a8c E \u4e2d\u5b66\u4e60\u5173\u4e8e\u67d0\u4e2a\u4efb\u52a1 T \u548c\u67d0\u4e2a\u6027\u80fd\u5ea6\u91cf P\uff0c\u5982\u679c\u5b83\u5728\u4efb\u52a1 T \u4e0a\u7684\u6027\u80fd<\/span>\uff08\u901a\u8fc7 P \u6d4b\u91cf\uff09<\/span>\u968f\u7740\u7ecf\u9a8c E \u7684\u63d0\u9ad8\u800c\u6539\u5584\uff0c\u90a3\u4e48\u5b83\u88ab\u8ba4\u4e3a\u4ece\u7ecf\u9a8c E \u4e2d\u5b66\u4e60\u4e86\u3002\u201d 2006\u5e74\uff0cHinton \u7b49\u4eba[31]\u8bad\u7ec3\u4e86\u4e00\u4e2a\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u6765\u8bc6\u522b\u624b\u5199\u6570\u5b57\uff0c\u51c6\u786e\u7387\u8d85\u8fc798%\u3002\u4ece\u90a3\u65f6\u8d77\uff0c\u7814\u7a76\u4eba\u5458\u5bf9\u6df1\u5ea6\u5b66\u4e60<\/span>\uff08DL\uff09<\/span>\u7684\u5174\u8da3\u8d8a\u6765\u8d8a\u6d53\u539a\uff0c\u8fd9\u79cd\u70ed\u60c5\u6269\u5c55\u5230\u673a\u5668\u5b66\u4e60\u7684\u8bb8\u591a\u9886\u57df\uff0c\u5982\u56fe\u50cf\u5904\u7406[32,33]\u3001\u81ea\u7136\u8bed\u8a00\u5904\u7406[34]\uff0c\u751a\u81f3\u7cbe\u51c6\u519c\u4e1a[35, 36, 37]\u3002<\/span><\/p>\n

\u4e3a\u4ec0\u4e48\u6211\u4eec\u9700\u8981\u673a\u5668\u5b66\u4e60\uff1f<\/span>\u603b\u7684\u6765\u8bf4\uff0c\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u901a\u5e38\u53ef\u4ee5\u7b80\u5316\u89e3\u51b3\u65b9\u6848\uff0c\u5e76\u6bd4\u4f20\u7edf\u65b9\u6cd5\u8868\u73b0\u66f4\u597d\uff0c\u800c\u4f20\u7edf\u65b9\u6cd5\u53ef\u80fd\u9700\u8981\u66f4\u591a\u7684\u624b\u52a8\u8c03\u6574\u89c4\u5219\u3002<\/span>\u6b64\u5916\uff0c\u4f20\u7edf\u65b9\u6cd5\u53ef\u80fd\u65e0\u6cd5\u4e3a\u590d\u6742\u73b0\u8c61\u627e\u5230\u6b63\u786e\u7684\u89e3\u51b3\u65b9\u6848\u3002<\/span>\u673a\u5668\u5b66\u4e60\u6280\u672f\u53ef\u4ee5\u5e2e\u52a9\u89e3\u91ca\u8fd9\u79cd\u590d\u6742\u6027\uff0c\u5e76\u66f4\u597d\u5730\u9002\u5e94\u65b0\u6570\u636e\u3002<\/span>\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u53ef\u4ee5\u83b7\u53d6\u5173\u4e8e\u590d\u6742\u95ee\u9898\u548c\u5927\u6570\u636e\u7684\u53ef\u53d8\u6027[38]\u3002<\/span>
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\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7684\u5206\u7c7b\u6709\u8bb8\u591a\u4e0d\u540c\u7684\u7c7b\u578b\u548c\u65b9\u6cd5<\/span>\uff08\u56fe2\uff09<\/span>\u3002\u6839\u636e\u662f\u5426\u5305\u542b\u4eba\u7c7b\u76d1\u7763\uff0c\u673a\u5668\u5b66\u4e60\u53ef\u4ee5\u5206\u4e3a\u6709\u76d1\u7763\u3001\u65e0\u76d1\u7763\u3001\u534a\u76d1\u7763\u548c\u5f3a\u5316\u5b66\u4e60<\/span>\uff08RL\uff09<\/span>\u3002\u6839\u636e\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u662f\u5426\u80fd\u591f\u5728\u7ebf\u5b66\u4e60\uff0c\u5b83\u4eec\u53ef\u4ee5\u5206\u4e3a\u5728\u7ebf\u5b66\u4e60\u548c\u6279\u91cf\u5b66\u4e60\u3002\u6839\u636e\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u662f\u5426\u68c0\u6d4b\u8bad\u7ec3\u6570\u636e\u6a21\u5f0f\u5e76\u521b\u5efa\u9884\u6d4b\u6a21\u578b\uff0c\u673a\u5668\u5b66\u4e60\u53ef\u4ee5\u5206\u4e3a\u57fa\u4e8e\u5b9e\u4f8b\u7684\u5b66\u4e60\u548c\u57fa\u4e8e\u6a21\u578b\u7684\u5b66\u4e60[29]\u3002<\/span><\/p>\n

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\u56fe2. \u6839\u636e\u662f\u5426\u5305\u542b\u4eba\u7c7b\u76d1\u7763\uff0c\u673a\u5668\u5b66\u4e60\u53ef\u4ee5\u5206\u4e3a\u76d1\u7763\u5b66\u4e60\u3001\u65e0\u76d1\u7763\u5b66\u4e60\u3001\u534a\u76d1\u7763\u5b66\u4e60\u548c\u5f3a\u5316\u5b66\u4e60\u3002\u6839\u636e\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u662f\u5426\u80fd\u591f\u5b9e\u65f6\u589e\u91cf\u5b66\u4e60\uff0c\u53ef\u4ee5\u5206\u4e3a\u5728\u7ebf\u5b66\u4e60\u548c\u6279\u91cf\u5b66\u4e60\u3002\u6839\u636e\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u662f\u5426\u80fd\u68c0\u6d4b\u8bad\u7ec3\u6570\u636e\u6a21\u5f0f\u5e76\u521b\u5efa\u9884\u6d4b\u6a21\u578b\uff0c\u673a\u5668\u5b66\u4e60\u53ef\u4ee5\u5206\u4e3a\u57fa\u4e8e\u5b9e\u4f8b\u7684\u5b66\u4e60\u548c\u57fa\u4e8e\u6a21\u578b\u7684\u5b66\u4e60[29]\u3002<\/span><\/section>\n

\u5927\u591a\u6570\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u901a\u8fc7\u6c42\u89e3\u4f9d\u8d56\u4e8e\u4e00\u9636\u5bfc\u6570<\/span>\uff08\u96c5\u53ef\u6bd4\u77e9\u9635\uff09<\/span>\u7684\u4f18\u5316\u95ee\u9898\u6765\u8fdb\u884c\u8bad\u7ec3\uff0c\u8fd9\u4e9b\u5bfc\u6570\u51b3\u5b9a\u4e86\u6743\u91cd\u662f\u589e\u52a0\u8fd8\u662f\u51cf\u5c11\u3002\u4e3a\u4e86\u83b7\u5f97\u5de8\u5927\u7684\u901f\u5ea6\u63d0\u5347\uff0c\u4eba\u4eec\u4f7f\u7528\u66f4\u5feb\u7684\u4f18\u5316\u5668\uff0c\u4ee3\u66ff\u4f20\u7edf\u7684\u68af\u5ea6\u4e0b\u964d<\/span>\uff08Gradient Descent\uff0cGD\uff09<\/span>\u4f18\u5316\u5668\u3002\u4f8b\u5982\uff0c\u6700\u53d7\u6b22\u8fce\u7684\u4f18\u5316\u5668\u6709\u52a8\u91cf\u4f18\u5316[39]\u3001Nesterov\u52a0\u901f\u68af\u5ea6[40]\u3001AdaGrad[41]\u3001RMSProp[42]\u548cAdam\u4f18\u5316[43]\u3002\u4e8c\u9636\u4f18\u5316\u65b9\u6cd5<\/span>\uff08\u6d77\u68ee\u77e9\u9635\uff09<\/span>\u901a\u5e38\u5177\u6709\u66f4\u5feb\u7684\u6536\u655b\u901f\u5ea6\uff0c\u4f46\u8ba1\u7b97\u6210\u672c\u66f4\u9ad8\u3002\u56e0\u6b64\uff0c\u56de\u7b54\u4ee5\u4e0b\u95ee\u9898\u975e\u5e38\u91cd\u8981\uff1a\u4ec0\u4e48\u662f\u66f4\u4f18\u7684\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\uff1f\u5982\u679c\u5bfc\u6570\u662f\u5206\u6570\u9636\u800c\u4e0d\u662f\u6574\u6570\u9636\uff0c\u4f1a\u53d1\u751f\u4ec0\u4e48\uff1f<\/span><\/p>\n

\u5728\u8fd9\u65b9\u9762\u7684\u7814\u7a76\u4ecd\u7136\u6709\u5f85\u6df1\u5165\u3002\u5206\u6570\u9636\u5bfc\u6570\u53ef\u80fd\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u4e3a\u4f18\u5316\u8fc7\u7a0b\u63d0\u4f9b\u65b0\u7684\u89c1\u89e3\u548c\u66f4\u597d\u7684\u6027\u80fd\u3002\u4f7f\u7528\u5206\u6570\u9636\u5bfc\u6570\u7684\u68af\u5ea6\u4f18\u5316\u65b9\u6cd5\u53ef\u80fd\u4f1a\u5bfc\u81f4\u66f4\u6709\u6548\u7684\u5b66\u4e60\u7b97\u6cd5\uff0c\u7279\u522b\u662f\u5728\u5904\u7406\u5177\u6709\u590d\u6742\u7ed3\u6784\u7684\u6570\u636e\u65f6\u3002\u7136\u800c\uff0c\u4e3a\u4e86\u5728\u673a\u5668\u5b66\u4e60\u4e2d\u6210\u529f\u5e94\u7528\u5206\u6570\u9636\u5fae\u79ef\u5206\uff0c\u9700\u8981\u8fdb\u4e00\u6b65\u7814\u7a76\u548c\u5f00\u53d1\u76f8\u5173\u65b9\u6cd5\u3002\u8fd9\u53ef\u80fd\u5305\u62ec\u5bf9\u73b0\u6709\u7b97\u6cd5\u7684\u8c03\u6574\uff0c\u4ee5\u4fbf\u9002\u5e94\u5206\u6570\u9636\u5bfc\u6570\uff0c\u4ee5\u53ca\u5f00\u53d1\u65b0\u7684\u5206\u6570\u9636\u4f18\u5316\u65b9\u6cd5\uff0c\u4ee5\u5728\u673a\u5668\u5b66\u4e60\u4efb\u52a1\u4e2d\u5b9e\u73b0\u66f4\u9ad8\u6548\u7684\u5b66\u4e60\u3002<\/span><\/p>\n


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\u56db\u3001\u5206\u6570\u9636\u5fae\u79ef\u5206\u4e0e\u91cd\u6574\u5316\u7fa4<\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n
 <\/span><\/section>\n

\u5728\u76f8\u53d8\u548c\u4e34\u754c\u73b0\u8c61\u7684\u7814\u7a76\u4e2d\uff0c\u91cd\u6574\u5316\u7fa4\u53ef\u4ee5\u6210\u529f\u5730\u8868\u5f81\u4e34\u754c\u70b9\u9644\u8fd1\u7cfb\u7edf\u7684\u81ea\u76f8\u4f3c\u6027[44]\u3002\u7531\u4e8e\u5206\u6570\u9636\u5fae\u79ef\u5206\u662f\u89e3\u91ca\u81ea\u76f8\u4f3c\u6027\u6216\u590d\u6742\u6027\u7684\u53e6\u4e00\u79cd\u5de5\u5177\uff0c\u56e0\u6b64\u5206\u6570\u9636\u5fae\u79ef\u5206\u548c\u91cd\u6574\u5316\u7fa4\u4e4b\u95f4\u53ef\u80fd\u5b58\u5728\u5f3a\u70c8\u7684\u8054\u7cfb<\/strong>\u3002Guo \u7b49\u4eba[45]\u901a\u8fc7\u8ba8\u8bba\u5e26\u6709\u6807\u5ea6\u5f8b<\/span>\uff08scaling law\uff09<\/span>\u548c\u590d\u6742\u73b0\u8c61\u7684\u91cd\u6574\u5316\u7fa4\uff0c\u4ece\u4e0d\u540c\u5b66\u79d1\u4e2d\u6db5\u76d6\u4e86\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u3001\u6df7\u6c8c\u7406\u8bba\u3001\u5206\u5f62\u7edf\u8ba1\u548c\u5206\u6570\u9636\u5fae\u79ef\u5206\uff0c\u4ee5\u66f4\u597d\u5730\u7406\u89e3\u5b83\u4eec\u4e4b\u95f4\u7684\u5173\u7cfb\u3002<\/span><\/p>\n

\u5e42\u5f8b\u901a\u5e38\u88ab\u63cf\u8ff0\u4e3a\uff1a<\/span><\/p>\n

<\/p>\n

\u5f53k\u4e3a\u8d1f\u5e38\u6570\u65f6\uff0c f(x)\u662f\u4e00\u79cd\u9006\u5e42\u5f8b\u3002\u8fd9\u4e2a\u5e42\u5f8b\u7684\u4e00\u4e2a\u91cd\u8981\u7279\u6027\u662f\u6807\u5ea6\u4e0d\u53d8\u6027[46]\uff0c\u7531\u4ee5\u4e0b\u516c\u5f0f\u786e\u5b9a\uff1a<\/span><\/p>\n

<\/p>\n

\u6ce8\u610f\uff0c\u5f53x\u662f\u65f6\u95f4\u65f6\uff0c\u6807\u5ea6\u63cf\u8ff0\u4e86\u7cfb\u7edf\u52a8\u529b\u5b66\u7684\u6027\u8d28\u3002\u7136\u800c\uff0c\u5f53\u7cfb\u7edf\u662f\u968f\u673a\u7684\u65f6\u5019\uff0c\u6807\u5ea6\u662f\u6982\u7387\u5bc6\u5ea6\u51fd\u6570<\/span>\uff08probability density function\uff0cPDF\uff0c\u6216\u76f8\u5173\u7ed3\u6784\uff09<\/span>\u7684\u6027\u8d28\uff0c\u662f\u7cfb\u7edf\u6574\u4f53\u6027\u8d28\u7684\u7ea6\u675f\u3002<\/span><\/p>\n

\u5f53\u7814\u7a76\u590d\u6742\u6027\u65f6\uff0c\u6211\u4eec\u4f1a\u95ee\u5b83\u610f\u5473\u7740\u4ec0\u4e48\uff1f\u7814\u7a76\u8005\u5728\u4f55\u65f6\u5f00\u59cb\u5c06\u7cfb\u7edf\u3001\u7f51\u7edc\u6216\u73b0\u8c61\u89c6\u4e3a\u201c\u590d\u6742\u201d[47,48]\uff1f\u4f3c\u4e4e\u6211\u4eec\u8fd8\u4e0d\u77e5\u9053\u4e00\u4e2a\u6e05\u6670\u3001\u7edf\u4e00\u7684\u5bf9\u201c\u590d\u6742\u6027\u201d\u7684\u5b9a\u4e49\u3002\u5206\u6570\u9636\u5fae\u79ef\u5206\u548c\u91cd\u6574\u5316\u7fa4\u7684\u8054\u5408\u4f7f\u7528\u53ef\u80fd\u4f1a\u56de\u7b54\u4ee5\u4e0b\u4e24\u4e2a\u95ee\u9898[45]\uff1a<\/span><\/section>\n
1.\u6211\u4eec\u8be5\u5982\u4f55\u63cf\u8ff0\u590d\u6742\u6027\uff1f<\/span><\/section>\n

2.\u4e3a\u4e86\u66f4\u597d\u5730\u7406\u89e3\u73b0\u5b9e\u4e16\u754c\u7684\u590d\u6742\u73b0\u8c61\uff0c\u5e94\u8be5\u4f7f\u7528\u4ec0\u4e48\u65b9\u6cd5\u6765\u5206\u6790\u590d\u6742\u6027\uff1f<\/span><\/p>\n

\u5728\u79d1\u5b66\u754c\u7684\u5f88\u5927\u4e00\u90e8\u5206\u4eba\u4e2d\u8fbe\u6210\u4e86\u5171\u8bc6\uff0c\u5f53\u4e0e\u6240\u7814\u7a76\u8fc7\u7a0b\u76f8\u5173\u7684\u6570\u636e\u5206\u5e03\u662f\u9006\u5e42\u5f8b\u65f6\uff0c\u73b0\u8c61\u5c31\u662f\u590d\u6742\u7684\u3002\u5728West\u548cGrigolini\u7684\u4e66\u4e2d[49]\uff0c\u6709\u4e00\u4e2a\u8868\u683c\u5217\u51fa\u4e86\u8fc7\u53bb\u4e24\u4e2a\u4e16\u7eaa\u53d1\u73b0\u7684\u7ecf\u9a8c\u5e42\u5f8b\u548c\u9006\u5e42\u5f8b\u7684\u6837\u672c\u3002\u4f8b\u5982\uff0c\u5728\u65e0\u6807\u5ea6\u7f51\u7edc\u4e2d\uff0c\u8fde\u901a\u6027\u7684\u5ea6\u5206\u5e03\u9075\u5faa\u9006\u5e42\u5f8b[50,51]\uff1b\u5728\u5305\u542b\u7c89\u7ea2\u566a\u58f0\u7684\u4fe1\u53f7\u5904\u7406\u4e2d\uff0c\u529f\u7387\u8c31\u662f\u9006\u5e42\u5f8b[20]\u3002\u5bf9\u4e8e\u5176\u4ed6\u4f8b\u5b50\uff0c\u5982\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u3001\u81ea\u76f8\u5173\u51fd\u6570<\/span>\uff08auto-correlation function\uff0cACF\uff09<\/span>[52]\u3001\u5f02\u901f\u751f\u957f<\/span>\uff08Y=aXb<\/sup><\/span>\uff09<\/span>[53]\u3001\u5f02\u5e38\u5f1b\u8c6b<\/span>\uff08\u968f\u65f6\u95f4\u6f14\u53d8\uff09<\/span>[54]\u3001\u5f02\u5e38\u6269\u6563<\/span>\uff08\u5747\u65b9\u8017\u6563\u4e0e\u65f6\u95f4\u7684\u5173\u7cfb\uff09<\/span>[13]\u3001\u81ea\u76f8\u4f3c\uff0c\u5b83\u4eec\u90fd\u53ef\u4ee5\u7528\u9006\u5e42\u5f8b\u6765\u63cf\u8ff0\u3002<\/span><\/p>\n

Douglas[55]\u7814\u7a76\u4e86\u8868\u9762\u76f8\u4e92\u4f5c\u7528\u805a\u5408\u7269\u6a21\u578b\uff0c\u5e76\u4f7f\u7528\u5206\u6570\u9636\u5fae\u79ef\u5206\u83b7\u5f97\u4e86\u914d\u5206\u51fd\u6570\u7684\u7cbe\u786e\u89e3\u3002\u6b64\u5916\uff0cDouglas\u9996\u6b21\u89e3\u91ca\u4e86\u91cd\u6574\u5316\u7fa4\u6807\u5ea6\u51fd\u6570\u4e0e\u7cbe\u786e\u6807\u5ea6\u51fd\u6570\u4e4b\u95f4\u7684\u57fa\u672c\u5173\u8054\u3002Qian[56]\u89e3\u91ca\u4e86\u5206\u6570\u5e03\u6717\u8fd0\u52a8<\/span>\uff08fractional Brownian motion\uff0cFBM\uff09<\/span>\u4e0e\u7edf\u8ba1\u7269\u7406\u4e2d\u7684\u91cd\u6574\u5316\u7fa4\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u5e76\u5206\u6790\u4e86\u590d\u6742\u73b0\u8c61\u7684\u7edf\u8ba1\u3001\u51e0\u4f55\u548c\u5206\u5f62\u6027\u8d28\u3002\u4e3a\u4e86\u8bc1\u660e\u5206\u6570\u9636\u5fae\u79ef\u5206\u4e0e\u91cd\u6574\u5316\u7fa4\u4e4b\u95f4\u7684\u8054\u7cfb\u662f\u9006\u5e42\u5f8b\uff0cGuo \u7b49\u4eba[45]\u4e3e\u4e86\u4e00\u4e2a\u7ef4\u5c14\u65af\u7279\u62c9\u65af\u968f\u673a\u6e38\u8d70<\/span>\uff08Weierstrass Random Walk\uff0cWRW\uff09<\/span>[57]\u7684\u4f8b\u5b50\u3002\u7136\u540e\uff0c\u5c06\u91cd\u6574\u5316\u7fa4\u65b9\u6cd5\u5e94\u7528\u4e8e\u6676\u683c\u7ed3\u6784\u51fd\u6570\uff0c\u786e\u5b9a\u4e86\u7ef4\u5c14\u65af\u7279\u62c9\u65af\u968f\u673a\u6e38\u8d70\u7684\u6807\u5ea6\u6027\u8d28\u3002Zaslavsky[58]\u4f7f\u7528\u975e\u53ef\u79ef\u54c8\u5bc6\u987f\u91cf\u7814\u7a76\u6df7\u6c8c\u7c92\u5b50\u52a8\u529b\u5b66\u3002\u4e3a\u4e86\u89e3\u91ca\u5206\u5f62\u65f6\u7a7a\u7cfb\u7edf\u4e2d\u968f\u673a\u6e38\u8d70\u7684\u6574\u4f53\u884c\u4e3a\uff0c\u5bfc\u51fa\u4e86\u5206\u6570Fokker-Planck-Kolmogorov<\/span>\uff08FFPK\uff09<\/span>\u65b9\u7a0b\u3002\u901a\u8fc7\u4f7f\u7528\u91cd\u6574\u5316\u7fa4\u65b9\u6cd5\u6c42\u89e3FFPK\u65b9\u7a0b\u3002<\/span><\/p>\n


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\u4e94\u3001\u5206\u6570\u9636\u5fae\u79ef\u5206\u4e0e\u673a\u5668\u5b66\u4e60<\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n
 <\/span><\/section>\n
\u5982\u524d\u6240\u8ff0\uff0c\u673a\u5668\u5b66\u4e60\u662f\u4e00\u4e2a\u70ed\u95e8\u7684\u7814\u7a76\u8bfe\u9898\uff0c\u5e76\u53ef\u80fd\u4f1a\u5728\u672a\u6765\u4e00\u6bb5\u65f6\u95f4\u5185\u4fdd\u6301\u70ed\u95e8\u3002\u673a\u5668\u5982\u4f55\u9ad8\u6548<\/span>\uff08\u6700\u4f18\u5316\uff09<\/span>\u5730\u5b66\u4e60\u59cb\u7ec8\u5f88\u91cd\u8981\u3002\u5b66\u4e60\u8fc7\u7a0b\u7684\u5173\u952e\u662f\u4f18\u5316\u65b9\u6cd5\u3002\u56e0\u6b64\uff0c\u8bbe\u8ba1\u4e00\u79cd\u9ad8\u6548\u7684\u4f18\u5316\u65b9\u6cd5\u6709\u52a9\u4e8e\u56de\u7b54\u4ee5\u4e0b\u4e09\u4e2a\u95ee\u9898\uff1a<\/span><\/section>\n
1.\u6700\u4f73\u4f18\u5316\u65b9\u5f0f\u662f\u4ec0\u4e48\uff1f<\/span><\/section>\n
2.\u6709\u6ca1\u6709\u66f4\u4f18\u7684\u4f18\u5316\u65b9\u6cd5\uff1f\u591a\u4e2a\u6700\u4f18\u89e3\uff1f<\/span><\/section>\n

3.\u6211\u4eec\u80fd\u5426\u8981\u6c42\u201c\u66f4\u4f18\u7684\u673a\u5668\u5b66\u4e60\u201d\uff0c\u4f8b\u5982\uff0c\u7528\u6700\u5c11\/\u6700\u5c0f\u7684\u6807\u6ce8\u6570\u636e\u8fdb\u884c\u6df1\u5ea6\u5b66\u4e60\uff1f<\/span><\/p>\n

\u5728\u673a\u5668\u5b66\u4e60\u7684\u8fc7\u7a0b\u4e2d\uff0c\u6211\u4eec\u5173\u5fc3\u8fc7\u7a0b\u7684\u901f\u5ea6\u548c\u51c6\u786e\u6027\u3002\u7b97\u6cd5\u5f88\u91cd\u8981\uff0c\u5426\u5219\uff0c\u6570\u636e\u6807\u6ce8\u548c\u5176\u4ed6\u52b3\u52a8\u529b\u53ef\u80fd\u4f1a\u4ea7\u751f\u5f88\u9ad8\u7684\u6210\u672c\u3002\u56e0\u6b64\uff0c\u673a\u5668\u5b66\u4e60\u7684\u5173\u952e\u662f\u6240\u91c7\u7528\u7684\u4f18\u5316\u65b9\u6cd5\u3002\u6536\u655b\u901f\u5ea6\u548c\u5168\u5c40\u641c\u7d22\u662f\u4f18\u5316\u65b9\u6cd5\u7684\u4e24\u4e2a\u91cd\u8981\u90e8\u5206\u3002\u4ece\u7406\u8bba\u4e0a\u8bb2\uff0c\u6709\u4e24\u5927\u7c7b\u4f18\u5316\u65b9\u6cd5\uff0c\u5373\u65e0\u5bfc\u6570\u65b9\u6cd5\u548c\u57fa\u4e8e\u68af\u5ea6\u7684\u65b9\u6cd5\u3002\u5bf9\u4e8e\u65e0\u5bfc\u6570\u65b9\u6cd5\uff0c\u6709\u76f4\u63a5\u641c\u7d22\u6280\u672f\uff0c\u5305\u62ec\u7c92\u5b50\u7fa4\u4f18\u5316<\/span>\uff08PSO\uff09<\/span>[59,60]\u7b49\u3002\u5bf9\u4e8e\u57fa\u4e8e\u68af\u5ea6\u7684\u65b9\u6cd5\uff0c\u6709\u68af\u5ea6\u4e0b\u964d<\/span>\uff08G<\/span>D\uff09<\/span>\u53ca\u5176\u53d8\u79cd\u3002\u5982\u4e0b\u6240\u793a\uff0c\u5f53\u4f7f\u7528\u5206\u6570\u9636\u5fae\u79ef\u5206\u65f6\uff0c\u8fd9\u4e24\u7c7b\u65b9\u6cd5\u90fd\u8868\u73b0\u51fa\u66f4\u597d\u7684\u6027\u80fd<\/strong>\u3002<\/span><\/p>\n

\u5bf9\u4e8e\u65e0\u5bfc\u6570\u65b9\u6cd5<\/strong><\/span>\uff08derivative-free methods\uff09<\/span>\uff0c\u6709\u5355\u4e2a\u4ee3\u7406\u641c\u7d22\u548c\u57fa\u4e8e\u7fa4\u4f53\u7684\u641c\u7d22\u65b9\u6cd5\u3002\u63a2\u7d22\u901a\u5e38\u901a\u8fc7\u968f\u673a\u6027\u6216\u6839\u636e\u4e00\u4e9b\u9884\u5b9a\u4e49\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u4ea7\u751f\u7684\u968f\u673a\u6570\u6765\u5b9e\u73b0\u3002\u5229\u7528\u5c40\u90e8\u4fe1\u606f\u5982\u68af\u5ea6\uff0c\u5728\u5c40\u90e8\u533a\u57df\u8fdb\u884c\u66f4\u5bc6\u96c6\u7684\u641c\u7d22\uff0c\u8fd9\u79cd\u5f3a\u5316\u53ef\u4ee5\u63d0\u9ad8\u6536\u655b\u901f\u7387\u3002\u90a3\u4e48\u95ee\u9898\u6765\u4e86\uff1a\u4ec0\u4e48\u662f\u6700\u4f18\u968f\u673a\u6027\uff1fWei \u7b49\u4eba[61]\u7814\u7a76\u4e86\u57fa\u4e8e\u7fa4\u4f53\u7684\u641c\u7d22\u4e2d\u7684\u6700\u4f18\u968f\u673a\u6027\u3002\u56db\u79cd\u6e10\u8fd1\u91cd\u5c3e\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u5373\u7c73\u5854\u683c-\u7d2f\u592b\u52d2<\/span>\uff08Mittag-Leffler\uff09<\/span>\u5206\u5e03\u3001\u5e15\u7d2f\u6258<\/span>\uff08Pareto\uff09<\/span>\u5206\u5e03\u3001\u97e6\u4f2f<\/span>\uff08Weibull\uff09<\/span>\u5206\u5e03\u548c\u67ef\u897f<\/span>\uff08Cauchy\uff09<\/span>\u5206\u5e03\uff0c\u5df2\u88ab\u7528\u4e8e\u6837\u672c\u8def\u5f84\u5206\u6790\u3002\u6839\u636e\u5b9e\u9a8c\u7ed3\u679c\uff0c\u968f\u673a\u6027\u589e\u5f3a\u7684\u5e03\u8c37\u9e1f\u641c\u7d22<\/span>\uff08cuckoo search\uff09<\/span>\u7b97\u6cd5[62, 63, 64]\u80fd\u591f\u66f4\u6709\u6548\u3001\u66f4\u7a33\u5065\u5730\u8bc6\u522b\u5206\u6570\u9636\u7cfb\u7edf\u7684\u672a\u77e5\u7279\u5b9a\u53c2\u6570\u3002\u968f\u673a\u6027\u589e\u5f3a\u7684\u5e03\u8c37\u9e1f\u641c\u7d22\u7b97\u6cd5\u53ef\u88ab\u89c6\u4e3a\u89e3\u51b3\u5b9e\u9645\u590d\u6742\u4f18\u5316\u95ee\u9898\u7684\u6709\u524d\u9014\u7684\u5de5\u5177\u3002\u539f\u56e0\u662f\u5728\u63a2\u7d22\u8fc7\u7a0b\u4e2d\uff0c\u5e94\u7528\u4e86\u5206\u6570\u9636\u566a\u58f0\u7684\u6700\u4f18\u968f\u673a\u6027\uff0c\u8fd9\u6bd4\u6539\u826f\u7684\u7c92\u5b50\u7fa4\u4f18\u5316\u3001\u5e03\u8c37\u9e1f\u641c\u7d22\u90fd\u66f4\u4f18\u3002\u5206\u6570\u9636\u566a\u58f0\u662f\u6307\u7a33\u5b9a\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570[65]\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u5728\u8ba8\u8bba\u6700\u4f18\u968f\u673a\u6027\u65f6\uff0c\u6211\u4eec\u5728\u8ba8\u8bba\u5206\u6570\u9636\u5fae\u79ef\u5206[66]\uff01<\/span><\/p>\n

\u68af\u5ea6\u4e0b\u964d<\/span><\/strong>\u662f\u4e00\u79cd\u975e\u5e38\u5e38\u7528\u7684\u4f18\u5316\u7b97\u6cd5\uff0c\u5b83\u53ef\u4ee5\u901a\u8fc7\u8fed\u4ee3\u8c03\u6574\u53c2\u6570\u4ee5\u6700\u5c0f\u5316\u4ee3\u4ef7\u51fd\u6570\u6765\u627e\u5230\u6700\u4f18\u89e3\u3002\u968f\u673a\u68af\u5ea6\u4e0b\u964d\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u968f\u673a\u9009\u62e9\u65f6\u95f4\u3002\u56e0\u6b64\uff0c\u4ee3\u4ef7\u51fd\u6570\u4f1a\u4e0a\u4e0b\u6ce2\u52a8\uff0c\u5e73\u5747\u800c\u8a00\u662f\u51cf\u5c11\u7684\uff0c\u8fd9\u6709\u52a9\u4e8e\u4ece\u5c40\u90e8\u6700\u4f18\u89e3\u4e2d\u9003\u8131\u3002\u6709\u65f6\u5019\u566a\u58f0\u4f1a\u88ab\u52a0\u5165\u5230\u68af\u5ea6\u4e0b\u964d\u65b9\u6cd5\u4e2d\uff0c\u800c\u901a\u5e38\u8fd9\u6837\u7684\u566a\u58f0\u9075\u5faa\u6587\u732e\u4e2d\u7684\u9ad8\u65af\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u3002\u6211\u4eec\u95ee\uff1a\u201c\u4e3a\u4ec0\u4e48\u4e0d\u662f\u91cd\u5c3e\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff1f\u201d\u8fd9\u4e2a\u95ee\u9898\u7684\u7b54\u6848\u53ef\u80fd\u5f15\u9886\u672a\u6765\u6709\u8da3\u7684\u7814\u7a76\u3002<\/span><\/p>\n

\u53d1\u5c55\u9ad8\u6548\u5b66\u4e60\u8fc7\u7a0b\u7684\u5173\u952e\u662f\u4f18\u5316\u65b9\u6cd5\u3002\u56e0\u6b64\uff0c\u8bbe\u8ba1\u4e00\u79cd\u9ad8\u6548\u7684\u4f18\u5316\u65b9\u6cd5\u975e\u5e38\u91cd\u8981\u3002\u65e0\u5bfc\u6570\u65b9\u6cd5\u4ee5\u53ca\u57fa\u4e8e\u68af\u5ea6\u7684\u65b9\u6cd5\uff0c\u5982 Nesterov \u52a0\u901f\u68af\u5ea6\u4e0b\u964d<\/span>\uff08NAGD\uff09<\/span>\uff0c\u90fd\u53ef\u4ee5\u4e3a\u8fd9\u4e2a\u7814\u7a76\u9886\u57df\u4f5c\u51fa\u8d21\u732e\u3002\u5bf9\u4e8eNesterov \u52a0\u901f\u68af\u5ea6\u4e0b\u964d\uff0cJordan \u7814\u7a76[67]\u7684\u4e00\u4e2a\u4e3b\u8981\u601d\u60f3\u662f\u5728\u8fde\u7eed\u65f6\u95f4\u57df\u4e2d\u5206\u6790\u8fed\u4ee3\u7b97\u6cd5\u3002\u5bf9\u4e8e\u5fae\u5206\u65b9\u7a0b\uff0c\u53ef\u4ee5\u4f7f\u7528 Lyapunov \u6216\u53d8\u5206\u65b9\u6cd5\u6765\u5206\u6790\u5176\u6027\u8d28\uff0c\u4f8b\u5982\u6536\u655b\u901f\u7387\u4e3a O(1\/t\u00b2)\u3002\u4eba\u4eec\u8fd8\u53ef\u4ee5\u4f7f\u7528\u53d8\u5206\u65b9\u6cd5\u6765\u63a8\u5bfc\u51fa\u4f18\u5316\u65b9\u6cd5\u7684\u4e3b\u5fae\u5206\u65b9\u7a0b\uff0c\u4f8b\u5982\u6700\u5c0f\u4f5c\u7528\u539f\u7406[68]\uff0c\u54c8\u5bc6\u987f\u53d8\u5206\u539f\u7406[69]\u548c\u91cf\u5b50\u529b\u5b66\u8def\u5f84\u79ef\u5206\u65b9\u6cd5[70]\u3002<\/span><\/p>\n

Jordan \u7684\u5de5\u4f5c\u63ed\u793a\u4e86\u53ef\u4ee5\u5c06\u8fed\u4ee3<\/span>\uff08\u4f18\u5316\uff09<\/span>\u7b97\u6cd5\u8f6c\u6362\u4e3a\u5176\u8fde\u7eed\u65f6\u95f4\u6781\u9650\u60c5\u51b5\uff0c\u8fd9\u53ef\u4ee5\u7b80\u5316\u5206\u6790<\/span>\uff08Lyapunov \u65b9\u6cd5\uff09<\/span>\u3002\u53ef\u4ee5\u76f4\u63a5\u8bbe\u8ba1\u4e00\u4e2a\u8fd0\u52a8\u5fae\u5206\u65b9\u7a0b<\/span>\uff08equation of motion \uff0cEOM\uff09<\/span>\uff0c\u7136\u540e\u5c06\u5176\u79bb\u6563\u5316\u4ee5\u5bfc\u51fa\u8fed\u4ee3\u7b97\u6cd5<\/span>\uff08\u53d8\u5206\u65b9\u6cd5\uff09<\/span>\u3002\u5173\u952e\u662f\u627e\u5230\u5408\u9002\u7684 Lyapunov \u6cdb\u51fd\u6765\u5206\u6790\u7a33\u5b9a\u6027\u548c\u6536\u655b\u901f\u7387\u3002Jordan \u63ed\u793a\u7684\u65b0\u5947\u4e8b\u5b9e\u662f\uff0c\u4f18\u5316\u7b97\u6cd5\u53ef\u4ee5\u4f7f\u7528\u62c9\u683c\u6717\u65e5\u529b\u5b66<\/span>\uff08\u6b27\u62c9-\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\uff09<\/span>\u3001\u8fd0\u52a8\u5fae\u5206\u65b9\u7a0b\u7cfb\u7edf\u5730\u5408\u6210\u3002<\/span><\/p>\n

\u53d7 M. Jordan \u5728\u9891\u57df\u7684\u601d\u60f3\u542f\u53d1\uff0c\u7814\u7a76\u8005\u5728[38]\u4e2d\u8bbe\u8ba1\u4e86\u4e00\u4e2a\u8fde\u7eed\u65f6\u95f4\u5206\u6570\u9636\u7cfb\u7edf\u3002\u6b64\u5916\uff0c\u6839\u636e\u5185\u90e8\u6a21\u578b\u539f\u5219<\/span>\uff08internal model principle \uff0cIMP\uff09<\/span>\uff0c\u5728 S \u6216 Z \u53d8\u6362\u57df\u4e2d\u8bbe\u8ba1\u548c\u5206\u6790\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u662f\u53ef\u884c\u7684[38]\u3002\u603b\u7684\u6765\u8bf4\uff0cM. Jordan \u63d0\u51fa\u4e86\u8fd9\u4e2a\u95ee\u9898\uff1a\u201c\u6709\u6ca1\u6709\u6700\u4f18\u7684\u4f18\u5316\u65b9\u6cd5\uff1f\u201d\u6211\u4eec\u7684\u56de\u7b54\u662f\u80af\u5b9a\u7684\uff0c\u901a\u8fc7\u9650\u5236\u52a8\u529b\u5b66\u5206\u6790\u3001\u79bb\u6563\u5316\u548c\u7ed3\u5408\u5176\u4ed6\u968f\u673a\u6027\u7684\u968f\u673a\u68af\u5ea6\u4e0b\u964d\uff0c\u4f8b\u5982\u6717\u4e4b\u4e07\u8fd0\u52a8\u3002\u8fd9\u91cc\u63d0\u51fa\u7684\u95ee\u9898\u662f\uff1a\u201c\u6709\u6ca1\u6709\u66f4\u4f18\u7684\u4f18\u5316\u65b9\u6cd5\uff1f\u201d\u7b54\u6848\u4ecd\u7136\u662f\u80af\u5b9a\u7684\uff0c\u4f46\u9700\u8981\u4f7f\u7528\u5206\u6570\u9636\u5fae\u79ef\u5206\u6765\u4f18\u5316\u968f\u673a\u68af\u5ea6\u4e0b\u964d\u3001\u968f\u673a\u641c\u7d22\u548c\u5185\u90e8\u6a21\u578b\u539f\u5219\u4e2d\u7684\u968f\u673a\u6027\u3002\u6cbf\u7740\u8fd9\u4e2a\u601d\u8def\uff0c\u8fd8\u6709\u66f4\u591a\u7684\u6f5c\u529b\u53ef\u4ee5\u8fdb\u884c\u8fdb\u4e00\u6b65\u7684\u7814\u7a76\u3002<\/span><\/p>\n

\u975e\u5c40\u90e8\u6a21\u578b\u901a\u5e38\u7528\u4e8e\u63cf\u8ff0\u4e0d\u80fd\u901a\u8fc7\u7ecf\u5178\u65b9\u6cd5\u51c6\u786e\u63cf\u8ff0\u7684\u7269\u7406\u7cfb\u7edf\u548c<\/span>\uff08\u6216\uff09<\/span>\u8fc7\u7a0b[71]\u3002\u4f8b\u5982\uff0c\u5206\u6570\u975e\u5c40\u90e8\u9ea6\u514b\u65af\u97e6\u65b9\u7a0b\u548c\u76f8\u5e94\u7684\u5206\u6570\u6ce2\u52a8\u65b9\u7a0b\u5df2\u7ecf\u5728\u7814\u7a76[72]\u4e2d\u5e94\u7528\u4e8e\u5206\u6570\u77e2\u91cf\u5fae\u79ef\u5206[73]\u3002\u975e\u5c40\u90e8\u5fae\u5206\u7b97\u5b50[74]\uff0c\u5305\u62ec\u68af\u5ea6\/\u6d77\u68ee\u77e9\u9635\u7684\u975e\u5c40\u90e8\u6a21\u62df\uff0c\u662f\u8fd9\u4e9b\u975e\u5c40\u90e8\u6a21\u578b\u7684\u5173\u952e\uff0c\u8fd9\u53ef\u80fd\u5728\u4e0d\u4e45\u7684\u5c06\u6765\u4e0e\u5206\u6570\u9636\u5fae\u79ef\u5206\u4e00\u8d77\u6210\u4e3a\u975e\u5e38\u6709\u8da3\u7684\u7814\u7a76\u3002<\/span><\/p>\n


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\u516d\u3001\u91cd\u6574\u5316\u7fa4\u4e0e\u673a\u5668\u5b66\u4e60<\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n
 <\/span><\/section>\n

\u673a\u5668\u5b66\u4e60\u7814\u7a76\u7684\u76ee\u6807\u662f\u4ece\u8bad\u7ec3\u6570\u636e\u4e2d\u5b66\u4e60\u5e76\u63d0\u53d6\u91cd\u8981\u7279\u5f81\u3002\u4f5c\u4e3a\u673a\u5668\u5b66\u4e60\u7684\u4e00\u4e2a\u5b50\u9886\u57df\uff0c\u6df1\u5ea6\u5b66\u4e60\u4f7f\u7528\u591a\u5c42\u8868\u5f81\u76f4\u63a5\u4ece\u8bad\u7ec3\u6570\u636e\u4e2d\u5b66\u4e60\u7279\u5f81\uff0c\u8fd9\u5728\u8bb8\u591a\u7814\u7a76\u4e3b\u9898\u4e2d\u53d6\u5f97\u4e86\u6210\u529f\uff0c\u5982\u7cbe\u51c6\u519c\u4e1a[75, 76, 77]\u548c\u76ee\u6807\u68c0\u6d4b[78, 79, 80]\u3002\u5c3d\u7ba1\u6df1\u5ea6\u5b66\u4e60\u7b97\u6cd5\u53d6\u5f97\u4e86\u6210\u529f\uff0c\u4f46\u5173\u4e8e\u4e3a\u4ec0\u4e48\u673a\u5668\u5b66\u4e60\u6280\u672f\u5728\u7279\u5f81\u5b66\u4e60\u4e0a\u8868\u73b0\u826f\u597d\u7684\u8bc1\u636e\u4ecd\u7136\u5f88\u5c11\u3002<\/span><\/p>\n

\u6700\u8fd1\uff0c\u4e00\u4e9b\u6a2a\u5411\u4ea4\u53c9\u7814\u7a76\u8868\u660e\u91cd\u6574\u5316\u7fa4\u4e0e\u673a\u5668\u5b66\u4e60\u4e4b\u95f4\u5b58\u5728\u5173\u8054[81,82,83]\u3002Jefferson \u6307\u51fa\uff0c\u91cd\u6574\u5316\u7fa4\u4e0e\u673a\u5668\u5b66\u4e60\u4e4b\u95f4\u7684\u8054\u7cfb\u6700\u521d\u662f\u5728\u67d0\u4e9b\u6676\u683c\u6a21\u578b<\/span>\uff08lattice models\uff09<\/span>\u7684\u80cc\u666f\u4e0b\u5efa\u7acb\u7684[84]\u3002\u62bd\u53d6\u91cd\u6574\u5316\u7fa4\uff0c\u4e0e\u62bd\u53d6\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc<\/span>\uff08\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\uff09<\/span>\u7ed3\u6784\uff0c\u8868\u9762\u4e0a\u5177\u6709\u76f8\u4f3c\u6027\uff0c\u5373\u7ed3\u6784\u7279\u5f81\u88ab\u8fb9\u7f18\u5316\u5728\u9690\u85cf\u81ea\u7531\u5ea6\u4e2d\u3002\u4f8b\u5982\uff0cMehta \u7b49\u4eba\u63d0\u51fa\uff0c\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u53ef\u80fd\u4f7f\u7528\u7c7b\u4f3c\u4e8e\u5e7f\u4e49\u91cd\u6574\u5316\u7fa4\u7684\u65b9\u6848\u4ece\u6570\u636e\u4e2d\u5b66\u4e60\u7279\u5f81[82]\u3002\u5728\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u4f4e\u7ea7\u7279\u5f81\u88ab\u8f93\u5165\u5230\u5177\u6709\u66f4\u62bd\u8c61\u9ad8\u7ea7\u7279\u5f81\u7684\u66f4\u9ad8\u5c42\u3002\u5728\u7279\u5f81\u63d0\u53d6\u8fc7\u7a0b\u4e2d\uff0c\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u5b66\u4f1a\u5ffd\u7565\u65e0\u5173\u7279\u5f81\uff0c\u4fdd\u7559\u76f8\u5173\u7279\u5f81\u3002<\/strong>\u8fd9\u79cd\u8fde\u7eed\u7c97\u7c92\u5316\u8fc7\u7a0b\u7c7b\u4f3c\u4e8e\u91cd\u6574\u5316\u7fa4\uff0c\u5b83\u901a\u8fc7\u6574\u5408\u77ed\u8ddd\u79bb\u81ea\u7531\u5ea6\u6765\u63d0\u53d6\u7269\u7406\u7cfb\u7edf\u7684\u76f8\u5173\u7279\u5f81\uff0c\u4ee5\u63cf\u8ff0\u5927\u5c3a\u5ea6\u73b0\u8c61<\/strong>\u3002\u4e3a\u4e86\u8bf4\u660e\u4ed6\u4eec\u7684\u89c2\u70b9\uff0c\u4ed6\u4eec\u57fa\u4e8e\u53d7\u9650\u73bb\u5c14\u5179\u66fc\u673a\u6784\u5efa\u4e86\u53d8\u5206\u91cd\u6574\u5316\u7fa4\u548c\u6df1\u5ea6\u5b66\u4e60\u67b6\u6784\u4e4b\u95f4\u7684\u6620\u5c04\u3002<\/span><\/p>\n

Lin \u7b49\u4eba[83]\u63a2\u8ba8\u4e86\u5982\u4f55\u5c06\u7269\u7406\u6027\u8d28<\/span>\uff08\u5982\u5bf9\u79f0\u6027\u3001\u5c40\u90e8\u6027\u3001\u53ef\u7ec4\u5408\u6027\u548c\u591a\u9879\u5f0f\u5bf9\u6570\u6982\u7387\uff09<\/span>\u8f6c\u5316\u4e3a\u795e\u7ecf\u7f51\u7edc\u3002\u4ed6\u4eec\u8ba4\u4e3a\uff0c\u5f53\u7edf\u8ba1\u8fc7\u7a0b\u5c5e\u4e8e\u7269\u7406\u548c\u673a\u5668\u5b66\u4e60\u4e2d\u5e38\u89c1\u7684\u67d0\u79cd\u5c42\u6b21\u7ed3\u6784\u5f62\u5f0f\u65f6\uff0c\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u53ef\u4ee5\u6bd4\u6d45\u5c42\u7f51\u7edc\u8868\u73b0\u5f97\u66f4\u597d\u3002\u4f8b\u5982\uff0c\u7269\u7406\u5b66\u7684\u4e00\u4e2a\u539f\u5219\u662f\u5c40\u90e8\u6027\uff0c\u5373\u4e8b\u7269\u53ea\u76f4\u63a5\u5f71\u54cd\u5176\u7d27\u90bb\u533a\u57df\u3002\u5f53\u5728\u673a\u5668\u5b66\u4e60\u5e94\u7528\u4e2d\u8003\u8651\u5c40\u90e8\u6027\u65f6\uff0c\u4e00\u7ec4\u5c40\u90e8\u968f\u673a\u53d8\u91cf\u7684\u4efb\u610f\u53d8\u6362\u5c06\u5bfc\u81f4\u975e\u5c40\u90e8\u96c6\u5408[83]\u3002\u7136\u800c\uff0c\u5c40\u90e8\u6027\u53ef\u4ee5\u4fdd\u7559\u3002\u4f8b\u5982\uff0c\u5728\u7b80\u5355\u7684\u5757\u81ea\u65cb<\/span>\uff08block-spin\uff09<\/span>\u91cd\u6574\u5316\u7fa4\u4e2d\uff0c\u81ea\u65cb\u88ab\u5206\u7ec4\u6210\u5757\uff0c\u7136\u540e\u88ab\u89c6\u4e3a\u968f\u673a\u53d8\u91cf\u3002\u6839\u636e\u9ad8\u7cbe\u5ea6\uff0c\u8fd9\u4e9b\u5757\u4ec5\u4e0e\u5176\u6700\u8fd1\u90bb\u76f8\u4e92\u8026\u5408\u3002\u8fd9\u79cd\u5c40\u90e8\u6027\u901a\u5e38\u88ab\u751f\u7269\u548c\u4eba\u5de5\u89c6\u89c9\u7cfb\u7edf\u6240\u5229\u7528\u3002\u7136\u800c\uff0c\u8f83\u4f4e\u5c42\u53ea<\/span>\u6267\u884c\u8f83\u5c40\u90e8\u7684\u64cd\u4f5c\u3002<\/span><\/p>\n

\u5728\u7269\u7406\u7cfb\u7edf\u4e2d\uff0c\u901a\u7528\u6027\u8d28\u5927\u591a\u6570\u65f6\u5019\u53ef\u4ee5\u51b3\u5b9a\u7269\u7406\u7279\u6027\uff0c\u8fd9\u53ef\u4ee5\u901a\u8fc7\u91cd\u6574\u5316\u7fa4\u7a0b\u5e8f\u63ed\u793a\u3002\u7136\u800c\uff0c\u5173\u952e\u7684\u81ea\u7531\u5ea6\u53ef\u80fd\u4e0d\u5bb9\u6613\u627e\u51fa\u3002\u56e0\u6b64\uff0cKoch-Janusz \u7b49\u4eba\u63d0\u51fa\u4e86\u4e00\u79cd\u57fa\u4e8e\u6a21\u578b\u65e0\u5173\u7684\u3001\u4fe1\u606f\u8bba\u7279\u5f81\u7684\u4eba\u5de5\u795e\u7ecf\u7f51\u7edc\uff0c\u7528\u4e8e\u5b9e\u7a7a\u95f4\u91cd\u6574\u5316\u7fa4\u8fc7\u7a0b[85]\u3002\u57fa\u4e8e\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\uff0c\u8be5\u6a21\u578b\u53ef\u4ee5\u8bc6\u522b\u7a7a\u95f4\u533a\u57df\u5185\u7684\u7269\u7406\u76f8\u5173\u81ea\u7531\u5ea6\uff0c\u5e76\u8fed\u4ee3\u5730\u6267\u884c\u91cd\u6574\u5316\u7fa4\u7c97\u7c92\u5316\u6b65\u9aa4\u3002\u8f93\u5165\u6570\u636e\u9075\u5faa\u73bb\u5c14\u5179\u66fc\u5206\u5e03\uff0c\u5e76\u4e14\u4e0d\u63d0\u4f9b\u5173\u4e8e\u7cfb\u7edf\u7684\u5fae\u89c2\u7ec6\u8282\u7684\u8fdb\u4e00\u6b65\u77e5\u8bc6\u3002\u795e\u7ecf\u7f51\u7edc\u7684\u53c2\u6570\u901a\u8fc7\u57fa\u4e8e\u8bc4\u4f30\u7a7a\u95f4\u5206\u79bb\u533a\u57df\u4e4b\u95f4\u7684\u5b9e\u7a7a\u95f4\u4e92\u4fe1\u606f<\/span>\uff08real-space mutual information\uff0cRSMI\uff09<\/span>\u7684\u8bad\u7ec3\u7b97\u6cd5\u6765\u4f18\u5316\u3002\u4e8c\u7ef4\u7ecf\u5178\u7edf\u8ba1\u7269\u7406\u4e2d\u7684\u4f0a\u8f9b\u6a21\u578b\u548c\u4e8c\u805a\u4f53\u6a21\u578b\u88ab\u7528\u4e8e\u65b9\u6cd5\u9a8c\u8bc1\u3002<\/span><\/p>\n


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\u4e03\u3001\u65b0\u7684\u5206\u6570\u9636\u5fae\u79ef\u5206-\u91cd\u6574\u5316\u7fa4-\u673a\u5668\u5b66\u4e60<\/strong><\/strong><\/p>\n

\u4e09\u5143\u5173\u7cfb\u53ca\u65b0\u673a\u9047<\/strong><\/strong><\/p>\n<\/section>\n

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\u603b\u7684\u6765\u8bf4\uff0c\u4f5c\u8005\u5728\u672c\u6587\u4e2d\u8ba8\u8bba\u4e86\u5206\u6570\u9636\u5fae\u79ef\u5206\u3001\u91cd\u6574\u5316\u7fa4\u548c\u673a\u5668\u5b66\u4e60\u4e4b\u95f4\u7684\u8054\u7cfb\u3002\u8bc1\u660e\u4e86\u5206\u6570\u9636\u5fae\u79ef\u5206\u3001\u91cd\u6574\u5316\u7fa4\u548c\u673a\u5668\u5b66\u4e60\u4e4b\u95f4\u5b58\u5728\u4e00\u79cd\u65b0\u7684\u4e09\u5143\u5173\u7cfb\u3002\u672c\u8282\u5c06\u5c55\u793a\u5173\u4e8e\u628a\u5206\u6570\u9636\u5fae\u79ef\u5206\u4e0e\u91cd\u6574\u5316\u7fa4\u548c\u7269\u7406\u77e5\u8bc6\u8054\u7cfb\u8d77\u6765\u7684\u672a\u6765\u7814\u7a76\u673a\u4f1a\u3002<\/span><\/p>\n

1. \u6cdb\u5316<\/span><\/strong><\/span><\/p>\n

\u901a\u8fc7\u5206\u6570\u9636\u5fae\u79ef\u5206\u3001\u91cd\u6574\u5316\u7fa4\u548c\u673a\u5668\u5b66\u4e60\uff0c\u53ef\u4ee5\u63d0\u9ad8\u6cdb\u5316\u6027\u80fd<\/span>\uff08generalization\uff09<\/span>\u3002\u673a\u5668\u5b66\u4e60\u6a21\u578b\u5177\u6709\u5f88\u5927\u7684\u6f5c\u529b\u53ef\u4ee5\u5f97\u5230\u63d0\u5347\uff0c\u5c24\u5176\u662f\u5c06\u6cdb\u5316\u90e8\u5206\u4e0e\u5206\u6570\u9636\u5fae\u79ef\u5206\u548c\u91cd\u6574\u5316\u7fa4\u65b9\u6cd5\u8054\u7cfb\u8d77\u6765\u3002\u4f8b\u5982\uff0cMehta \u7b49\u4eba\u63d0\u51fa\uff0c\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u53ef\u80fd\u91c7\u7528\u4e86\u4e00\u79cd\u7c7b\u4f3c\u4e8e\u5e7f\u4e49\u91cd\u6574\u5316\u7fa4\u7684\u65b9\u6848\u6765\u4ece\u6570\u636e\u4e2d\u5b66\u4e60\u7279\u5f81[82]\u3002\u6574\u4e2a\u590d\u6742\u7cfb\u7edf\u901a\u5e38\u6709\u8bb8\u591a\u5c0f\u7ec4\u4ef6\uff0c\u5b83\u4eec\u53ef\u4ee5\u4e0e\u9644\u8fd1\u7684\u7ec4\u4ef6\u548c\u73af\u5883\u76f8\u4e92\u4f5c\u7528\uff0c\u4f7f\u5f97\u5b83\u4eec\u7684\u884c\u4e3a\u96be\u4ee5\u9884\u6d4b\u3002\u56e0\u6b64\uff0c\u96c6\u5408\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u52a8\u529b\u5b66\u4e0d\u80fd\u7528\u76f8\u7a7a\u95f4\u4e2d\u7684\u6807\u51c6\u504f\u5fae\u5206\u65b9\u7a0b\u63cf\u8ff0\u3002\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\u65b9\u7a0b\u88ab\u786e\u5b9a\u4e3a\u4f20\u7edf\u76f8\u7a7a\u95f4\u65b9\u7a0b\u7684\u5206\u6570\u9636\u6cdb\u5316[45]\u3002<\/span><\/p>\n

2.\u8bad\u7ec3\u6570\u636e\u96c6\u7684\u53ef\u53d8\u6027\u548c\u591a\u6837\u6027<\/span><\/strong><\/span><\/p>\n

\u53ef\u53d8\u6027<\/span>\uff08variability\uff09<\/span>\u662f\u8bad\u7ec3\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u8fc7\u7a0b\u4e2d\u8ba8\u8bba\u7684\u6700\u5173\u952e\u7684\u7279\u5f81\u3002\u53ef\u53d8\u6027\u53ef\u4ee5\u6307\u8bad\u7ec3\u6570\u636e\u96c6\u7684\u51e0\u4e2a\u5c5e\u6027\u3002\u9996\u5148\uff0c\u9700\u8981\u4f7f\u7528\u5f02\u5e38\u548c\u79bb\u7fa4\u503c\u68c0\u6d4b\u65b9\u6cd5\u4e86\u89e3\u6570\u636e\u4e2d\u7684\u4e0d\u4e00\u81f4\u6027\u6570\u91cf\u3002\u5176\u6b21\uff0c\u53ef\u53d8\u6027\u4e5f\u53ef\u4ee5\u6307\u591a\u6837\u6027<\/span>\uff08diversity\uff09<\/span>[86, 87]\uff0c\u8fd9\u662f\u7531\u4e8e\u4e0d\u540c\u6570\u636e\u7c7b\u578b\u548c\u6765\u6e90\u4ea7\u751f\u7684\uff0c\u4f8b\u5982\u5065\u5eb7\u6216\u4e0d\u5065\u5eb7[88, 89]\u3002\u5982\u524d\u6240\u8ff0\uff0c\u5bf9\u4e8e\u91cd\u6574\u5316\u7fa4\u65b9\u6cd5\uff0c\u5728\u91cf\u5b50\u573a\u8bba\u4e2d\uff0c\u8ba1\u7b97\u7684\u53d1\u6563\u90e8\u5206\u53ef\u80fd\u5bfc\u81f4\u65e0\u610f\u4e49\u7684\u65e0\u7a77\u5927\u7814\u7a76\u7ed3\u679c\u3002\u901a\u8fc7\u5e94\u7528\u91cd\u6574\u5316\u7fa4\uff0c\u53d1\u6563\u90e8\u5206\u53ef\u4ee5\u5438\u9644\u5230\u8f83\u5c11\u7684\u53ef\u6d4b\u91cf\u91cf\u4e2d\uff0c\u4ea7\u751f\u6709\u9650\u7684\u7ed3\u679c\u3002\u91cd\u6574\u5316\u7fa4\u548c\u5206\u6570\u9636\u5fae\u79ef\u5206\u6280\u672f\u53ef\u4ee5\u589e\u5f3a\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7684\u8bad\u7ec3\u6570\u636e\u591a\u6837\u6027\u3002\u53cd\u8fc7\u6765\uff0c\u5728\u76f8\u540c\u6027\u80fd\u8981\u6c42\u4e0b\uff0c\u6211\u4eec\u53ef\u4ee5\u671f\u671b\u201c\u5c0f\u6570\u636e\u201d\u800c\u4e0d\u662f\u201c\u5927\u6570\u636e\u201d<\/strong>\u3002<\/span><\/p>\n

3.\u52a0\u901f\u5b66\u4e60<\/span><\/strong><\/span><\/p>\n

\u5728\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u4f4e\u7ea7\u7279\u5f81\u88ab\u8f93\u5165\u5230\u5177\u6709\u66f4\u62bd\u8c61\u9ad8\u7ea7\u7279\u5f81\u7684\u66f4\u9ad8\u5c42\u795e\u7ecf\u7f51\u7edc\u4e2d\u3002\u5728\u7279\u5f81\u63d0\u53d6\u8fc7\u7a0b\u4e2d\uff0c\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u53ef\u4ee5\u5b66\u4f1a\u5ffd\u7565\u65e0\u5173\u7279\u5f81\uff0c\u4fdd\u7559\u76f8\u5173\u7279\u5f81\uff0c\u4ece\u800c\u4f7f\u5b66\u4e60\u8fc7\u7a0b\u66f4\u5feb\u3001\u66f4\u9ad8\u6548\u3002\u5982\u524d\u6240\u8ff0\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u7c7b\u4f3c\u4e8e\u91cd\u6574\u5316\u7fa4\uff0c\u5b83\u901a\u8fc7\u6574\u5408\u77ed\u8ddd\u79bb\u81ea\u7531\u5ea6\u6765\u63d0\u53d6\u7269\u7406\u7cfb\u7edf\u7684\u76f8\u5173\u7279\u5f81\uff0c\u4ee5\u63cf\u8ff0\u5927\u5c3a\u5ea6\u73b0\u8c61\u3002\u4e3a\u4e86\u8bf4\u660e\u8fd9\u4e2a\u60f3\u6cd5\uff0cMehta \u7b49\u4eba[82]\u6839\u636e\u53d7\u9650\u73bb\u5c14\u5179\u66fc\u673a\u6784\u5efa\u4e86\u53d8\u5206\u91cd\u6574\u5316\u7fa4\u548c\u6df1\u5ea6\u5b66\u4e60\u67b6\u6784\u4e4b\u95f4\u7684\u6620\u5c04\u3002\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u53ef\u4ee5\u4ece\u91cd\u6574\u5316\u7fa4\u548c\u5206\u6570\u9636\u5fae\u79ef\u5206\u4e2d\u53d7\u76ca\u3002<\/span><\/p>\n

4.\u6700\u4f18\u968f\u673a\u6027<\/span><\/strong><\/span><\/p>\n

\u968f\u673a\u6027\u5728\u63a2\u7d22\u548c\u5e94\u7528\u65b9\u9762\u90fd\u8d77\u7740\u91cd\u8981\u4f5c\u7528\u3002\u5177\u6709\u826f\u597d\u968f\u673a\u5206\u914d\u6743\u91cd\u7684\u795e\u7ecf\u7f51\u7edc\u67b6\u6784\u53ef\u4ee5\u8f7b\u677e\u80dc\u8fc7\u5177\u6709\u7cbe\u7ec6\u8c03\u6574\u6743\u91cd\u7684\u66f4\u7cdf\u7cd5\u7684\u67b6\u6784[90]\u3002\u56e0\u6b64\uff0c\u4f7f\u7528\u91cd\u6574\u5316\u7fa4\u548c\u5206\u6570\u9636\u5fae\u79ef\u5206\u65b9\u6cd5\u8ba8\u8bba\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u7684\u6700\u4f18\u968f\u673a\u6027\u81f3\u5173\u91cd\u8981\u3002\u5bf9\u4e8e\u5206\u6570\u9636\u5fae\u79ef\u5206\uff0c\u968f\u673a\u6027\u6709\u4e30\u5bcc\u7684\u5f62\u5f0f[91]\uff0c\u4f8b\u5982\u91cd\u5c3e\u6027\uff0c\u5b83\u5bf9\u5e94\u4e8e\u5206\u6570\u9636\u4e3b\u65b9\u7a0b[92]\u3002\u5728\u6982\u7387\u8bba\u4e2d\uff0c\u91cd\u5c3e\u5206\u5e03<\/span>\uff08heavy-tailed distributions\uff09<\/span>\u662f\u5176\u5c3e\u90e8\u4e0d\u5448\u6307\u6570\u8870\u51cf\u7684\u5206\u5e03[93]\u3002\u56e0\u6b64\uff0c\u5b83\u4eec\u7684\u5c3e\u90e8\u6bd4\u6307\u6570\u5206\u5e03\u7684\u5c3e\u90e8\u6709\u66f4\u9ad8\u7684\u6743\u91cd\u3002\u91cd\u5c3e\u5206\u5e03\u5e7f\u6cdb\u7528\u4e8e\u4e0d\u540c\u5b66\u79d1\u7684\u5efa\u6a21\uff0c\u5982\u91d1\u878d[94]\u3001\u4fdd\u9669[95]\u548c\u533b\u5b66[96]\u3002<\/span><\/p>\n

\u5728[83]\u4e2d\uff0cLin \u7b49\u4eba\u63a2\u8ba8\u4e86\u5982\u4f55\u5c06\u7269\u7406\u6027\u8d28\u4e0e\u91cd\u6574\u5316\u7fa4\u65b9\u6cd5\u7ed3\u5408\u8d77\u6765\uff0c\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\u8fdb\u884c\u8f6c\u6362\u3002\u5728[85]\u4e2d\uff0cKoch-Janusz \u7b49\u4eba\u63d0\u51fa\u4e86\u4e00\u79cd\u57fa\u4e8e\u5b9e\u7a7a\u95f4\u91cd\u6574\u5316\u7fa4\u8fc7\u7a0b\u7684\u6a21\u578b\u65e0\u5173\u7684\u3001\u4fe1\u606f\u8bba\u7279\u5f81\u7684\u4eba\u5de5\u795e\u7ecf\u7f51\u7edc\u3002\u57fa\u4e8e\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\uff0c\u8be5\u6a21\u578b\u53ef\u4ee5\u8bc6\u522b\u7a7a\u95f4\u533a\u57df\u4e2d\u7684\u7269\u7406\u76f8\u5173\u81ea\u7531\u5ea6\uff0c\u5e76\u8fed\u4ee3\u5730\u6267\u884c\u91cd\u6574\u5316\u7fa4\u7c97\u7c92\u5316\u6b65\u9aa4[85]\u3002\u6211\u4eec\u76f8\u4fe1\uff0c\u8fd9\u4e9b\u793a\u4f8b\u53ef\u4ee5\u4e3a\u4f7f\u7528\u5206\u6570\u9636\u5fae\u79ef\u5206\u3001\u91cd\u6574\u5316\u7fa4\u548c\u673a\u5668\u5b66\u4e60\u4e00\u8d77\u7814\u7a76\u6700\u4f18\u968f\u673a\u6027\u6253\u5f00\u65b0\u7684\u89c6\u91ce\u3002<\/span><\/p>\n


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\u516b\u3001\u603b\u7ed3<\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n

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\u5206\u6570\u9636\u5fae\u79ef\u5206\u3001\u91cd\u6574\u5316\u7fa4\u548c\u673a\u5668\u5b66\u4e60\u662f\u4e09\u4e2a\u6700\u5177\u751f\u4ea7\u529b\u7684\u7814\u7a76\u9886\u57df\uff0c\u53ef\u80fd\u5728\u4e0d\u4e45\u7684\u5c06\u6765\u4ecd\u7136\u5982\u6b64\u3002\u5173\u4e8e\u8fd9\u4e09\u4e2a\u4e3b\u9898\u7684\u7814\u7a76\u5df2\u7ecf\u8fdb\u884c\u5f97\u76f8\u5f53\u5e7f\u6cdb\u3002\u7136\u800c\uff0c\u5f88\u5c11\u6709\u7814\u7a76\u63a2\u8ba8\u4e86\u5206\u6570\u9636\u5fae\u79ef\u5206\u3001\u91cd\u6574\u5316\u7fa4\u548c\u673a\u5668\u5b66\u4e60\u4e4b\u95f4\u7684\u4e09\u5143\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u4f5c\u8005\u5728\u672c\u6587\u4e2d\u9610\u8ff0\u4e86\u4e09\u5143\u5173\u7cfb\u4e2d\u7684\u5173\u952e\u8054\u7cfb\u3002<\/span><\/p>\n

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\u5bf9\u4e8e\u5206\u6570\u9636\u5fae\u79ef\u5206\u548c\u673a\u5668\u5b66\u4e60\uff0c\u63cf\u8ff0\u4e86\u5927\u6570\u636e\u3001\u53ef\u53d8\u6027\u3001\u4f18\u5316\u548c\u975e\u5c40\u90e8\u6a21\u578b\u4e2d\u7684\u5173\u952e\u8054\u7cfb\u3002\u4f5c\u8005\u4ecb\u7ecd\u4e86\u65e0\u5bfc\u6570\u4f18\u5316\u65b9\u6cd5\u548c\u57fa\u4e8e\u68af\u5ea6\u7684\u4f18\u5316\u65b9\u6cd5\uff0c\u5e76\u89e3\u91ca\u4e86\u5206\u6570\u9636\u5fae\u79ef\u5206\u5982\u4f55\u4e3a\u8fd9\u4e9b\u7814\u7a76\u9886\u57df\u505a\u51fa\u8d21\u732e\u3002\u5206\u6570\u9636\u52a8\u529b\u5b66\u54cd\u5e94\u4e86\u66f4\u9ad8\u7ea7\u7684\u8868\u5f81\uff0c\u4ee5\u6355\u6349\u4ee5\u524d\u88ab\u5e73\u6ed1\u5904\u7406\u7684\u8fc7\u5c0f\u6216\u8fc7\u5927\u5c3a\u5ea6\u7684\u7ed3\u6784\uff0c\u8fdb\u800c\u5e94\u5bf9\u6211\u4eec\u6240\u9762\u5bf9\u7684\u66f4\u590d\u6742\u7684\u4e16\u754c\u3002\u5047\u8bbe\u4e00\u4e2a\u5e0c\u671b\u83b7\u5f97\u6bd4\u4f7f\u7528\u6574\u6570\u9636\u5fae\u79ef\u5206\u65b9\u6cd5\u53ef\u80fd\u8fbe\u5230\u7684\u6700\u4f73\u7ed3\u679c\u8fd8\u8981\u597d\u7684\u7ed3\u679c\uff0c\u6216\u8005\u5e0c\u671b\u201c\u66f4\u4f18\u5316\u201d\uff0c\u6211\u4eec\u4e3b\u5f20\u5e94\u7528\u5206\u6570\u9636\u601d\u7ef4\u3002\u5728\u5927\u6570\u636e\u65f6\u4ee3\uff0c\u51b3\u7b56\u548c\u63a7\u5236\u9700\u8981\u5206\u6570\u9636\u5fae\u79ef\u5206\uff0c\u4f8b\u5982\u5206\u6570\u9636\u4fe1\u53f7\u3001\u7cfb\u7edf\u548c\u63a7\u5236\u3002<\/span><\/p>\n

\u6700\u540e\uff0c\u89e3\u91ca\u4e86\u91cd\u6574\u5316\u7fa4\u548c\u673a\u5668\u5b66\u4e60\u4e4b\u95f4\u7684\u5173\u8054\u3002\u8ba8\u8bba\u4e86\u4e92\u4fe1\u606f\u3001\u7279\u5f81\u63d0\u53d6\u548c\u5c40\u90e8\u6027\u3002\u8bb8\u591a\u4ea4\u53c9\u7814\u7a76\u8868\u660e\u91cd\u6574\u5316\u7fa4\u548c\u673a\u5668\u5b66\u4e60\u4e4b\u95f4\u5b58\u5728\u8054\u7cfb\u3002\u91cd\u6574\u5316\u7fa4\u4e0e\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\u76f8\u4f3c\uff0c\u5373\u7279\u5f81\u8fb9\u7f18\u5316\u4e8e\u9690\u85cf\u81ea\u7531\u5ea6\u4e2d\u3002\u673a\u5668\u5b66\u4e60\u7684\u672a\u6765\u5e94\u8be5\u662f\u7269\u7406\u542f\u53d1\u5f0f\u7684\u3001\u79d1\u5b66\u7684<\/span>\uff08\u5d4c\u5165\u56e0\u679c\u5173\u7cfb\u6216\u53d1\u73b0\u56e0\u679c\u5173\u7cfb\uff09<\/span>\u4ee5\u53ca\u6d89\u53ca\u4f7f\u7528\u5206\u6570\u9636\u5fae\u79ef\u5206\u548c\u91cd\u6574\u5316\u7fa4\uff0c\u4f7f\u5efa\u6a21\u66f4\u63a5\u8fd1\u771f\u5b9e\u672c\u8d28\u3002<\/span><\/span><\/em><\/p>\n


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\u53c2\u8003\u6587\u732e<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n

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    Vale \u0301rio, D., Machado, J., and Kiryakova, V., 2014. \u201cSome pioneers of the applications of fractional calculus\u201d. Frac- tional Calculus and Applied Analysis, 17(2), pp. 552\u2013578.<\/section>\n<\/li>\n
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    Abel, N., 1823. \u201cSolution of a couple of problems by means of definite integrals\u201d. Magazin for Naturvidenskaberne, 2(55), p. 2.<\/section>\n<\/li>\n
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    Podlubny, I., Magin, R. L., and Trymorush, I., 2017. \u201cNiels Henrik Abel and the birth of fractional calculus\u201d. Frac- tional Calculus and Applied Analysis, 20(5), pp. 1068\u2013 1075.<\/section>\n<\/li>\n
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    Ross, B., 1977. \u201cThe development of fractional calculus 1695\u20131900\u201d. Historia Mathematica, 4(1), pp. 75\u201389.<\/section>\n<\/li>\n
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    Pramukkul, P., Svenkeson, A., Grigolini, P., Bologna, M., and West, B., 2013. \u201cComplexity and the fractional calcu- lus\u201d. Advances in Mathematical Physics, 2013, pp. 1\u20137.<\/section>\n<\/li>\n
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    \u56e0\u679c\u3001\u6d8c\u73b0\u4e0e\u673a\u5668\u5b66\u4e60\uff1a\u56e0\u679c\u6d8c\u73b0\u8bfb\u4e66\u4f1a\u7b2c\u4e8c\u5b63\u542f\u52a8<\/a><\/p>\n


    <\/p>\n


    <\/p>\n

    \n
    \n
    \n
    \n

    \u96c6\u667a\u5b66\u56ed\u6700\u65b0AI\u8bfe\u7a0b\uff0c<\/strong><\/p>\n

    \u5f20\u6c5f\u6559\u6388\u4eb2\u6388\uff1a<\/strong>\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u6280\u672f\u57fa\u7840<\/strong><\/p>\n

    \u2014\u2014\u4ece\u53ef\u5fae\u5206\u7f16\u7a0b\u5230\u56e0\u679c\u63a8\u7406<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

    \n

    <\/span><\/section>\n<\/section>\n

    <\/section>\n
    <\/span><\/section>\n
    \u81ea1956\u5e74\u201c\u4eba\u5de5\u667a\u80fd\u201d\u8bde\u751f\u4e8e\u8fbe\u7279\u8305\u65af\u4f1a\u8bae\u4ee5\u6765\uff0c\u5df2\u7ecf\u7ecf\u5386\u4e86\u4ece\u65e9\u671f\u7684\u4ee5\u7b26\u53f7\u63a8\u7406\u4e3a\u4e3b\u4f53\u7684\u7b2c\u4e00\u4ee3\u4eba\u5de5\u667a\u80fd\uff0c\u548c\u4ee5\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u3001\u673a\u5668\u5b66\u4e60\u4e3a\u4e3b\u4f53\u7684\u7b2c\u4e8c\u4ee3\u4eba\u5de5\u667a\u80fd\u3002ChatGPT\u7684\u6a2a\u7a7a\u51fa\u4e16\u3001\u751f\u6210\u5f0fAI\u7684\u666e\u53ca\u3001AI for Science\u7b49\u65b0\u9886\u57df\u7684\u7a81\u7834\uff0c\u6807\u5fd7\u7740\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u7684\u547c\u4e4b\u6b32\u51fa\u3002\u53ef\u5fae\u5206\u7f16\u7a0b\u3001\u795e\u7ecf\u5fae\u5206\u65b9\u7a0b\u3001\u81ea\u76d1\u7763\u5b66\u4e60\u3001\u751f\u6210\u5f0f\u6a21\u578b\u3001Transformer\u3001\u57fa\u4e8e\u56fe\u7f51\u7edc\u7684\u5b66\u4e60\u4e0e\u63a8\u7406\u3001\u56e0\u679c\u8868\u5f81\u4e0e\u56e0\u679c\u63a8\u65ad\uff0c\u57fa\u4e8e\u4e16\u754c\u6a21\u578b\u7684\u5f3a\u5316\u5b66\u4e60\u2026\u2026\uff0c\u6240\u6709\u8fd9\u4e9b\u8131\u80ce\u4e8e\u524d\u4e24\u4ee3\u4eba\u5de5\u667a\u80fd\u7684\u6280\u672f\u8981\u7d20\u5f88\u6709\u53ef\u80fd\u5c06\u6784\u6210\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u7684\u7406\u8bba\u4e0e\u6280\u672f\u7684\u57fa\u7840\u3002<\/span><\/section>\n


    <\/p>\n

    \u672c\u8bfe\u7a0b\u8bd5\u56fe\u7cfb\u7edf\u68b3\u7406\u4ece\u673a\u5668\u5b66\u4e60\u5230\u5927\u8bed\u8a00\u6a21\u578b\uff0c\u4ece\u56fe\u795e\u7ecf\u7f51\u7edc\u5230\u56e0\u679c\u63a8\u7406\u7b49\u4e00\u7cfb\u5217\u53ef\u80fd\u6210\u4e3a\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u57fa\u7840\u7684\u6280\u672f\u8981\u7d20\uff0c\u4e3a\u7814\u7a76\u8005\u6216\u5b66\u751f\u5728\u751f\u6210\u5f0fAI\u3001\u5927\u6a21\u578b\u3001AI for Science\u7b49\u76f8\u5173\u9886\u57df\u7684\u5b66\u4e60\u548c\u7814\u7a76\u5de5\u4f5c\u5960\u5b9a\u57fa\u7840\u3002<\/span><\/p>\n


    <\/section>\n
    <\/span><\/section>\n
    <\/section>\n
    https:\/\/campus.swarma.org\/course\/5084?from=wechat<\/span><\/section>\n
    <\/span><\/section>\n


    <\/p>\n


    <\/p>\n<\/section>\n

    \n

    \u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/p>\n

    1. <\/strong>George Ellis \u7ecf\u5178\u8bba\u6587\uff1a\u81ea\u4e0a\u800c\u4e0b\u7684\u56e0\u679c\u4e0e\u6d8c\u73b0<\/a><\/strong><\/span><\/section>\n
    2. <\/strong>\u56e0\u679c\u4e00\u81f4\u6027\uff1a\u56e0\u679c\u4e0e\u591a\u5c3a\u5ea6\u7684\u78b0\u649e<\/a><\/strong><\/span><\/section>\n
    <\/a><\/section>\n
    3.<\/strong> \u5f53\u81ea\u7136\u8bed\u8a00\u5904\u7406\u9047\u4e0a\u91cf\u5b50\u8ba1\u7b97\u4e0e\u8303\u7574\u8bba\uff1a\u4eba\u5de5\u667a\u80fd\u5f00\u542f\u66f4\u5927\u60f3\u8c61\u7a7a\u95f4<\/strong><\/a><\/section>\n
    4. \u300a\u5f20\u6c5f\u00b7\u590d\u6742\u79d1\u5b66\u524d\u6cbf27\u8bb2\u300b\u5b8c\u6574\u4e0a\u7ebf\uff01<\/strong><\/a><\/strong><\/strong><\/strong><\/section>\n
    5. \u6210\u4e3a\u96c6\u667aVIP\uff0c\u89e3\u9501\u5168\u7ad9\u8bfe\u7a0b\uff0f\u8bfb\u4e66\u4f1a<\/span><\/strong><\/a><\/strong><\/strong><\/strong><\/strong><\/section>\n
    6. \u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/strong><\/a><\/strong><\/strong><\/strong><\/section>\n


    <\/p>\n


    <\/p>\n<\/section>\n

    \u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfb\u4e66\u4f1a<\/span><\/strong><\/span><\/strong><\/section>\n

    <\/mp-style-type><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

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