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space&#xff09;\u2014\u2014\u4e00\u4e2a\u4ee5\u7cfb\u7edf\u6240\u6709\u53d8\u91cf\u4e3a\u5750\u6807\u7684\u62bd\u8c61\u7a7a\u95f4\u2014\u2014\u5e76\u53d1\u73b0\u5176\u4e2d\u5b58\u5728\u7740\u590d\u6742\u7684\u7f20\u7ed3\u7ed3\u6784&#xff1a;\u7a33\u5b9a\u7684\u5468\u671f\u8f68\u9053\u3001\u4e0d\u7a33\u5b9a\u7684\u978d\u70b9\u3001\u4ee5\u53ca\u5f7c\u6b64\u65e0\u9650\u903c\u8fd1\u5374\u53c8\u6c38\u4e0d\u76f8\u4ea4\u7684&#034;\u540c\u5bbf\u8f68\u9053&#034;\u3002\u8fd9\u4e00\u8f6c\u5411&#xff0c;\u88ab\u540e\u4e16\u516c\u8ba4\u4e3a\u6df7\u6c8c\u7406\u8bba&#xff08;chaos theory&#xff09;\u4e0e\u73b0\u4ee3\u52a8\u529b\u7cfb\u7edf\u7406\u8bba\u7684\u771f\u6b63\u8d77\u70b9\u3002<\/p>\n<p>\u503c\u5f97\u6ce8\u610f\u7684\u662f&#xff0c;\u5e9e\u52a0\u83b1\u7684\u5931\u8d25\u5e76\u975e\u56e0\u4e3a\u4ed6&#034;\u6570\u5b66\u4e0d\u591f\u597d&#034;&#xff0c;\u800c\u662f\u56e0\u4e3a\u771f\u5b9e\u7cfb\u7edf\u672c\u8eab\u5c31\u662f\u90a3\u6837\u6784\u9020\u7684\u2014\u2014\u6df7\u6c8c\u5185\u751f\u4e8e\u786e\u5b9a\u6027\u7684\u725b\u987f\u65b9\u7a0b&#xff0c;\u800c\u975e\u5916\u751f\u4e8e\u6211\u4eec\u7684\u65e0\u77e5\u3002\u8fd9\u4e00\u70b9&#xff0c;\u6b63\u662f\u672c\u6587\u5168\u90e8\u8bba\u8bc1\u7684\u51fa\u53d1\u70b9&#xff1a;\u590d\u6742\u7cfb\u7edf\u4e4b\u6240\u4ee5\u96be\u4ee5\u88ab\u5b8c\u5168\u7406\u89e3&#xff0c;\u6839\u6e90\u4e0d\u5728\u4e8e\u4eba\u7c7b\u7b97\u529b\u6216\u6280\u5de7\u7684\u4e0d\u8db3&#xff0c;\u800c\u5728\u4e8e\u7406\u89e3\u6d3b\u52a8\u6240\u9762\u5bf9\u7684\u5bf9\u8c61&#xff08;\u771f\u5b9e\u4e16\u754c&#xff09;\u5728\u7ed3\u6784\u4e0a\u5c31\u62d2\u7edd\u88ab\u5b8c\u6574\u5730\u88c5\u8fdb\u4efb\u4f55\u5355\u4e00\u6a21\u578b\u3002<\/p>\n<h3>1.3 \u6838\u5fc3\u547d\u9898&#xff1a;T-\u03a0-O \u6846\u67b6\u7684\u63d0\u51fa<\/h3>\n<p>\u672c\u6587\u7684\u51fa\u53d1\u70b9&#xff0c;\u6b63\u662f\u628a\u5e9e\u52a0\u83b1\u7684\u8fd9\u573a\u8f6c\u5411&#xff0c;\u63d0\u70bc\u6210\u4e00\u5957\u53ef\u4ee5\u666e\u904d\u64cd\u4f5c\u7684\u7406\u89e3\u65b9\u6cd5\u8bba\u3002\u6211\u4eec\u5c06\u5176\u547d\u540d\u4e3a T-\u03a0-O \u6846\u67b6\u3002\u5176\u4e2d\u4e09\u4e2a\u7b26\u53f7\u7684\u542b\u4e49\u5982\u4e0b&#xff1a;<\/p>\n<ul>\n<li>T&#xff08;the real Thing&#xff09;\u662f\u5b87\u5b99\u771f\u5b9e\u7cfb\u7edf\u672c\u8eab&#xff0c;\u56ca\u62ec\u6240\u6709\u81ea\u7531\u5ea6\u3001\u6240\u6709\u521d\u59cb\u6761\u4ef6\u3001\u6240\u6709\u76f8\u4e92\u4f5c\u7528&#xff1b;<\/li>\n<li>\u03a0&#xff08;Projection&#xff09;\u662f\u4eba\u7c7b\u4e3a\u903c\u8fd1 T \u800c\u6784\u9020\u7684\u4efb\u610f\u6570\u5b66\u6216\u8ba4\u77e5\u6295\u5f71&#xff0c;\u5b83\u5fc5\u7136\u4fdd\u7559\u4e00\u90e8\u5206\u7ed3\u6784\u3001\u4e22\u5931\u4e00\u90e8\u5206\u4fe1\u606f&#xff1b;<\/li>\n<li>O&#xff08;Order&#xff09;\u662f\u5728\u67d0\u4e2a \u03a0 \u4e4b\u4e0b\u4ecd\u7136\u6210\u7acb\u3001\u53ef\u88ab\u8ba1\u7b97\u4e0e\u68c0\u9a8c\u7684\u79e9\u5e8f\u3002<\/li>\n<\/ul>\n<p>\u6211\u4eec\u60f3\u8bba\u8bc1\u7684\u662f&#xff1a;\u5e9e\u52a0\u83b1\u7684\u8f6c\u5411\u5e76\u975e\u6743\u5b9c\u4e4b\u8ba1&#xff0c;\u800c\u662f\u7406\u89e3\u590d\u6742\u4e16\u754c\u7684\u672c\u8d28\u65b9\u5f0f\u3002\u7406\u89e3\u590d\u6742\u7cfb\u7edf&#xff0c;\u4e0d\u662f\u590d\u73b0 T \u7684\u6bcf\u4e00\u4e2a\u7ec6\u8282&#xff0c;\u800c\u662f\u5728\u6070\u5f53\u7684 \u03a0 \u4e2d\u63d0\u53d6\u51fa\u5bf9\u6211\u4eec\u6709\u7528\u7684 O&#xff1b;\u6df7\u6c8c\u4e0d\u662f\u79e9\u5e8f\u7684\u7f3a\u5e2d&#xff0c;\u800c\u53ea\u662f\u63d0\u793a\u6211\u4eec\u2014\u2014\u5728\u5f53\u524d\u5c42\u6b21\u4e0a\u627e\u4e0d\u5230\u7b80\u5355\u7684 O&#xff0c;\u9700\u8981\u6362\u4e00\u4e2a\u6295\u5f71\u3001\u6216\u6362\u4e00\u4e2a\u5c3a\u5ea6\u53bb\u91cd\u65b0\u5bfb\u627e\u3002\u672c\u6587\u7684\u76ee\u6807&#xff0c;\u4e0d\u662f\u518d\u6dfb\u4e00\u6761\u5177\u4f53\u516c\u5f0f&#xff0c;\u800c\u662f\u6f84\u6e05&#034;\u7406\u89e3&#034;\u8fd9\u4ef6\u4e8b\u672c\u8eab\u7684\u8fd0\u4f5c\u673a\u7406&#xff0c;\u5e76\u628a\u8fd9\u79cd\u673a\u7406\u4ea4\u4ed8\u7ed9\u5f53\u4ee3\u6b63\u5728\u5d1b\u8d77\u7684\u4eba\u5de5\u667a\u80fd\u8fd9\u4e00\u65b0\u5f15\u64ce\u3002<\/p>\n<h3>1.4 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\u7684\u6709\u635f\u6295\u5f71&#xff0c;\u5e76\u7ed9\u51fa&#034;\u597d\u7684\u6295\u5f71&#034;\u7684\u5224\u522b\u6807\u51c6\u4e0e&#034;\u79e9\u5e8f&#034;\u7684\u5f62\u5f0f\u5316\u5b9a\u4e49\u3002\u7b2c\u4e09\u6761\u662f\u5386\u53f2\u4e0e\u65f6\u4ee3\u4e3b\u7ebf&#xff08;\u7b2c 5\u30016 \u7ae0&#xff09;&#xff0c;\u5b83\u4ee5 N \u4f53\u95ee\u9898\u4e3a\u539f\u578b\u91cd\u8ff0\u5e9e\u52a0\u83b1\u7684\u8303\u5f0f\u8f6c\u5411&#xff0c;\u5e76\u8bba\u8bc1\u4eba\u5de5\u667a\u80fd\u6b63\u662f\u8fd9\u4e00\u7406\u89e3\u5faa\u73af\u7684\u65b0\u5f15\u64ce\u3002\u5728\u6b64\u57fa\u7840\u4e4b\u4e0a&#xff0c;\u7b2c 7 \u7ae0\u628a\u601d\u8fa8\u843d\u683c\u4e3a\u53ef\u64cd\u4f5c\u7684\u516d\u6b65\u65b9\u6cd5\u8bba\u5e76\u5728\u591a\u4e2a\u9886\u57df\u793a\u8303&#xff0c;\u7b2c 8 \u7ae0\u5bf9\u6846\u67b6\u81ea\u8eab\u4f5c\u51fa\u8bda\u5b9e\u7684\u6279\u5224\u6027\u53cd\u601d\u4e0e\u54f2\u5b66\u5bf9\u8bdd&#xff0c;\u7b2c 9 \u7ae0\u7ed9\u51fa\u7ed3\u8bba\u3001\u6838\u5fc3\u547d\u9898\u51dd\u7ec3\u4e0e\u5f00\u653e\u95ee\u9898\u3002<\/p>\n<p>\u9700\u8981\u9884\u5148\u8bf4\u660e\u672c\u6587\u7684\u4e00\u4e2a\u65b9\u6cd5\u8bba\u81ea\u89c9&#xff1a;T-\u03a0-O \u6846\u67b6\u672c\u8eab&#xff0c;\u4e5f\u53ea\u662f\u5bf9&#034;\u7406\u89e3\u6d3b\u52a8&#034;\u7684\u4e00\u4e2a\u6295\u5f71&#xff08;\u8fd9\u4e00\u70b9\u5c06\u5728\u7b2c 8 \u7ae0\u4e13\u95e8\u8ba8\u8bba&#xff09;\u3002\u56e0\u6b64&#xff0c;\u672c\u6587\u4e0d\u5ba3\u79f0\u81ea\u5df1\u7ed9\u51fa\u4e86\u7406\u89e3\u590d\u6742\u7cfb\u7edf\u7684&#034;\u7ec8\u6781\u7b54\u6848&#034;&#xff0c;\u800c\u53ea\u63d0\u4f9b\u4e00\u9762\u8ba9&#034;\u7406\u89e3&#034;\u8fd9\u4ef6\u4e8b\u53d8\u5f97\u53ef\u5ba1\u89c6\u3001\u53ef\u7ea0\u9519\u3001\u53ef\u4f20\u627f\u7684\u955c\u5b50\u3002\u8fd9\u6b63\u662f\u6846\u67b6\u4fdd\u6301\u81ea\u6d3d\u3001\u907f\u514d\u6ed1\u5411\u6559\u6761\u4e3b\u4e49\u7684\u5173\u952e\u3002<\/p>\n<h2>\u7b2c 2 \u7ae0 \u771f\u5b9e\u7cfb\u7edf T&#xff1a;\u4e0d\u53ef\u88ab\u5b8c\u5168\u5bb9\u7eb3\u7684\u672c\u4f53<\/h2>\n<h3>2.1 T \u7684\u65e0\u9650\u7ef4\u672c\u6027\u4e0e\u5f62\u5f0f\u5316<\/h3>\n<p>\u5728\u7b2c\u4e00\u7ae0\u4e2d&#xff0c;T \u88ab\u76f4\u89c9\u5730\u5b9a\u4e49\u4e3a&#034;\u771f\u5b9e\u7cfb\u7edf\u672c\u8eab&#034;\u3002\u672c\u7ae0\u8981\u505a\u7684\u7b2c\u4e00\u4ef6\u4e8b&#xff0c;\u662f\u628a\u8fd9\u4e2a\u76f4\u89c9\u7f6e\u4e8e\u53ef\u9760\u7684\u57fa\u77f3\u4e4b\u4e0a&#xff0c;\u4f7f\u5176\u6210\u4e3a\u4e00\u4e2a\u53ef\u88ab\u9648\u8ff0\u3001\u53ef\u88ab\u8ba8\u8bba\u7684\u6570\u5b66\u5bf9\u8c61\u3002<\/p>\n<p>\u5728\u672c\u6846\u67b6\u4e2d&#xff0c;T \u662f\u4e00\u4e2a\u672c\u4f53\u8bba\u5360\u4f4d\u7b26&#xff0c;\u6307\u4ee3\u771f\u5b9e\u7cfb\u7edf\u672c\u8eab\u2014\u2014\u5b83\u4e0d\u4f9d\u8d56\u4e8e\u4efb\u4f55\u89c2\u6d4b\u8005\u3001\u4efb\u4f55\u7406\u8bba\u3001\u4efb\u4f55\u6a21\u578b\u800c\u72ec\u7acb\u5b58\u5728&#xff08;\u81f3\u5c11\u5728\u672c\u4f53\u8bba\u610f\u4e49\u4e0a\u6211\u4eec\u5047\u8bbe\u5b83\u5b58\u5728&#xff09;\u3002T \u5305\u542b\u4e86\u6784\u6210\u7cfb\u7edf\u7684\u5168\u90e8\u81ea\u7531\u5ea6&#xff1a;\u6bcf\u4e00\u4e2a\u7c92\u5b50\u7684\u4f4d\u7f6e\u4e0e\u52a8\u91cf\u3001\u6bcf\u4e00\u6b21\u76f8\u4e92\u4f5c\u7528\u7684\u77ac\u65f6\u5f3a\u5ea6\u3001\u6bcf\u4e00\u5904\u8fb9\u754c\u6761\u4ef6\u7684\u5fae\u89c2\u6da8\u843d\u3002\u5bf9\u4e8e\u590d\u6742\u7cfb\u7edf&#xff0c;T \u51e0\u4e4e\u662f\u65e0\u9650\u7ef4\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u7528\u51e0\u4e2a\u5bf9\u7167\u6765\u4f53\u4f1a\u8fd9\u79cd\u7ef4\u5ea6\u707e\u96be&#xff1a;\u4e00\u7247\u5df4\u638c\u5927\u7684\u7a7a\u6c14\u4e2d&#xff0c;\u5305\u542b\u7740\u7ea6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         10<\/p>\n<p>         22<\/p>\n<p>       10^{22}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">22<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e2a\u5206\u5b50&#xff0c;\u6bcf\u4e00\u4e2a\u5206\u5b50\u90fd\u6709\u4e09\u7ef4\u4f4d\u7f6e\u4e0e\u4e09\u7ef4\u901f\u5ea6&#xff0c;\u4e8e\u662f\u8fd9\u7247\u7a7a\u6c14\u5728\u7ecf\u5178\u63cf\u8ff0\u4e0b\u5c31\u6709\u7ea6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        6<\/p>\n<p>        \u00d7<\/p>\n<p>         10<\/p>\n<p>         22<\/p>\n<p>       6\\\\times10^{22}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">6<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u00d7<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">22<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e2a\u81ea\u7531\u5ea6&#xff1b;\u4eba\u8111\u4e2d\u7ea6\u6709 860 \u4ebf\u4e2a\u795e\u7ecf\u5143&#xff0c;\u6bcf\u4e2a\u795e\u7ecf\u5143\u53c8\u4e0e\u4e0a\u5343\u4e2a\u90bb\u5c45\u5f62\u6210\u7a81\u89e6\u8fde\u63a5&#xff0c;\u5176\u72b6\u6001\u7a7a\u95f4\u7684\u7ef4\u5ea6\u8fdc\u8d85\u4efb\u4f55\u53ef\u679a\u4e3e\u7684\u4e0a\u754c&#xff1b;\u4e00\u4e2a\u73b0\u4ee3\u7ecf\u6d4e\u4f53\u7684\u72b6\u6001&#xff0c;\u7531\u65e0\u6570\u4e2a\u4f53\u7684\u504f\u597d\u3001\u4fe1\u5ff5\u3001\u5951\u7ea6\u4e0e\u9884\u671f\u5171\u540c\u51b3\u5b9a&#xff1b;\u800c N \u4f53\u95ee\u9898\u4e00\u65e6\u8fdb\u5165&#034;\u65e0\u6570\u5929\u4f53\u5f15\u529b\u8026\u5408&#034;\u7684\u6781\u9650&#xff0c;\u5176\u81ea\u7531\u5ea6\u540c\u6837\u8d8b\u4e8e\u65e0\u7a77\u3002<\/p>\n<p>\u4e3a\u4e86\u628a\u4e0a\u8ff0\u76f4\u89c9\u5f62\u5f0f\u5316&#xff0c;\u6211\u4eec\u628a T \u8868\u793a\u4e3a\u4e00\u4e2a\u72b6\u6001\u7a7a\u95f4\u4e0a\u7684\u52a8\u529b\u7cfb\u7edf\u3002\u8bbe\u72b6\u6001\u7a7a\u95f4\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        X<\/p>\n<p>       \\\\mathcal{X}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><\/span><\/span><\/span><\/span>&#xff08;\u53ef\u4ee5\u662f\u6709\u9650\u7ef4\u6d41\u5f62&#xff0c;\u4e5f\u53ef\u4ee5\u662f\u65e0\u9650\u7ef4\u51fd\u6570\u7a7a\u95f4\u6216\u6d4b\u5ea6\u7a7a\u95f4&#xff09;&#xff0c;T \u7531\u4e09\u5143\u7ec4 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        (<\/p>\n<p>        X<\/p>\n<p>        ,<\/p>\n<p>         \u03a6<\/p>\n<p>         t<\/p>\n<p>        ,<\/p>\n<p>         \u03bc<\/p>\n<p>         0<\/p>\n<p>        )<\/p>\n<p>       (\\\\mathcal{X}, \\\\Phi^{t}, \\\\mu_0)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0436em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u63cf\u8ff0&#xff0c;\u5176\u4e2d&#xff1a;<\/p>\n<ul>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         X<\/p>\n<p>        \\\\mathcal{X}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><\/span><\/span><\/span><\/span> \u662f\u7cfb\u7edf\u7684\u5168\u90e8\u53ef\u80fd\u72b6\u6001\u6784\u6210\u7684\u96c6\u5408&#xff08;&#034;\u65e0\u9650\u7ef4&#034;\u5728\u6b64\u4f53\u73b0\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         dim<\/p>\n<p>         \u2061<\/p>\n<p>         X<\/p>\n<p>        \\\\dim\\\\mathcal{X}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mop\">dim<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><\/span><\/span><\/span><\/span> \u6781\u5927\u751a\u81f3\u65e0\u7a77&#xff09;&#xff1b;<\/li>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>          \u03a6<\/p>\n<p>          t<\/p>\n<p>         :<\/p>\n<p>         X<\/p>\n<p>         \u2192<\/p>\n<p>         X<\/p>\n<p>        \\\\Phi^{t}:\\\\mathcal{X}\\\\to\\\\mathcal{X}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7936em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2192<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><\/span><\/span><\/span><\/span> \u662f\u65f6\u95f4\u6f14\u5316\u7b97\u5b50&#xff0c;\u6ee1\u8db3\u534a\u7fa4\u6027\u8d28 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          \u03a6<\/p>\n<p>           t<\/p>\n<p>           1<\/p>\n<p>         \u2218<\/p>\n<p>          \u03a6<\/p>\n<p>           t<\/p>\n<p>           2<\/p>\n<p>         &#061;<\/p>\n<p>          \u03a6<\/p>\n<p>            t<\/p>\n<p>            1<\/p>\n<p>           &#043;<\/p>\n<p>            t<\/p>\n<p>            2<\/p>\n<p>        \\\\Phi^{t_1}\\\\circ\\\\Phi^{t_2}&#061;\\\\Phi^{t_1&#043;t_2}<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7936em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3173em\"><span class=\"\" style=\"top: -2.357em;margin-left: 0em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2218<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7936em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3173em\"><span class=\"\" style=\"top: -2.357em;margin-left: 0em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7936em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3173em\"><span class=\"\" style=\"top: -2.357em;margin-left: 0em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mbin mtight\">&#043;<\/span><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3173em\"><span class=\"\" style=\"top: -2.357em;margin-left: 0em;margin-right: 0.0714em\"><span class=\"pstrut\" style=\"height: 2.5em\"><\/span><span class=\"sizing reset-size3 size1 mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.143em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff1b;<\/li>\n<li><span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>          \u03bc<\/p>\n<p>          0<\/p>\n<p>        \\\\mu_0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03bc<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u662f\u521d\u59cb\u72b6\u6001\u7684\u5206\u5e03&#xff08;\u5b83\u5bf9\u5e94\u540e\u6587\u5c06\u8ba8\u8bba\u7684&#034;\u63cf\u8ff0\u6027\u4e0d\u53ef\u5b8c\u5907&#034;\u2014\u2014\u6211\u4eec\u6c38\u8fdc\u53ea\u80fd\u4ee5\u5206\u5e03\u800c\u975e\u5355\u70b9\u53bb\u903c\u8fd1\u771f\u5b9e\u521d\u59cb\u6001&#xff09;\u3002<\/li>\n<\/ul>\n<p>\u5728\u6b64\u5f62\u5f0f\u5316\u4e0b&#xff0c;&#034;\u4e8c\u4f53\u95ee\u9898&#034;\u5bf9\u5e94\u4e00\u4e2a\u53ef\u79ef\u7cfb\u7edf&#xff1a;\u5b58\u5728\u8db3\u591f\u591a\u7684\u72ec\u7acb\u5b88\u6052\u91cf&#xff0c;\u4f7f\u5f97 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03a6<\/p>\n<p>         t<\/p>\n<p>       \\\\Phi^{t}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7936em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u53ef\u88ab\u663e\u5f0f\u5199\u51fa&#xff1b;&#034;\u4e09\u4f53\u53ca\u4ee5\u4e0a&#034;\u5219\u5bf9\u5e94\u4e00\u4e2a\u5178\u578b\u6df7\u6c8c\u7cfb\u7edf&#xff1a;\u4e0d\u5b58\u5728\u8fd9\u6837\u4e00\u7ec4\u5b8c\u5907\u7684\u5b88\u6052\u91cf&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03a6<\/p>\n<p>         t<\/p>\n<p>       \\\\Phi^{t}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7936em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5728\u76f8\u7a7a\u95f4\u4e2d\u8868\u73b0\u51fa\u5bf9\u521d\u59cb\u6761\u4ef6\u7684\u6307\u6570\u654f\u611f\u3002\u5f62\u5f0f\u5316\u7684\u4ef7\u503c\u5728\u4e8e&#xff1a;\u5b83\u7acb\u523b\u628a&#034;\u54ea\u4e9b\u4e3b\u5f20\u662f\u5b9a\u4e49\u3001\u54ea\u4e9b\u662f\u5b9a\u7406\u3001\u54ea\u4e9b\u4ecd\u5c5e\u542f\u53d1\u5f0f&#034;\u8fd9\u4e00\u542b\u7cca\u7684\u95ee\u9898&#xff0c;\u53d8\u6210\u4e86\u53ef\u5224\u522b\u7684\u5bf9\u8c61\u3002<\/p>\n<p>\u7531\u6b64\u786e\u7acb\u672c\u7ae0\u3001\u4e5f\u662f\u5168\u6587\u7684\u7b2c\u4e00\u4e2a\u5173\u952e\u8bba\u70b9&#xff1a;\u4eba\u7c7b\u6240\u80fd\u5199\u4e0b\u7684\u4efb\u4f55\u7406\u8bba\u3001\u65b9\u7a0b\u3001\u6570\u636e\u96c6\u6216\u6a21\u578b&#xff0c;\u90fd\u4e0d\u53ef\u80fd\u5b8c\u6574\u5bb9\u7eb3 T\u3002 \u8fd9\u4e0d\u662f\u5de5\u7a0b\u4e0a\u7684&#034;\u7b97\u4e0d\u52a8&#034;&#xff0c;\u800c\u662f\u539f\u5219\u4e0a&#034;\u88c5\u4e0d\u4e0b&#034;\u3002T \u7684\u65e0\u9650\u7ef4\u672c\u6027&#xff0c;\u51b3\u5b9a\u4e86\u4efb\u4f55\u5bf9 T \u7684\u903c\u8fd1\u90fd\u5fc5\u7136\u662f\u4e00\u79cd\u538b\u7f29\u2014\u2014\u800c\u538b\u7f29&#xff0c;\u5728\u6570\u5b66\u4e0a\u5c31\u610f\u5473\u7740\u6295\u5f71\u3002\u8fd9\u4e00\u70b9\u5c06\u5728\u7b2c 3 \u7ae0\u88ab\u4e25\u683c\u5316\u3002\u6b64\u5904\u6211\u4eec\u53ea\u9700\u786e\u7acb&#xff1a;T \u662f\u7406\u89e3\u7684\u8d77\u70b9&#xff0c;\u4e5f\u662f\u7406\u89e3\u7684\u754c\u9650&#xff1b;\u5b83\u65e2\u662f\u6240\u6709\u77e5\u8bc6\u7684\u6765\u6e90&#xff0c;\u53c8\u662f\u6240\u6709\u77e5\u8bc6\u6c38\u8fdc\u65e0\u6cd5\u5b8c\u5168\u62b5\u8fbe\u7684\u5f7c\u5cb8\u3002<\/p>\n<h3>2.2 \u4e09\u91cd\u4e0d\u53ef\u5b8c\u5907\u6027&#xff1a;\u63cf\u8ff0\u6027\u3001\u52a8\u529b\u5b66\u4e0e\u8ba1\u7b97<\/h3>\n<p>&#034;\u4e0d\u53ef\u5b8c\u5907\u6027&#034;\u662f\u7406\u89e3 T \u7684\u6838\u5fc3\u6982\u5ff5&#xff0c;\u4f46\u5b83\u5728\u4e0d\u540c\u5c42\u6b21\u4e0a\u6709\u4e0d\u540c\u7684\u542b\u4e49&#xff0c;\u9700\u8981\u5c0f\u5fc3\u533a\u5206\u3002\u672c\u7ae0\u628a\u5b83\u5206\u89e3\u4e3a\u4e09\u91cd\u76f8\u4e92\u72ec\u7acb\u3001\u5f7c\u6b64\u52a0\u5f3a\u7684\u6765\u6e90\u3002<\/p>\n<p>\u7b2c\u4e00\u91cd&#xff1a;\u63cf\u8ff0\u6027\u4e0d\u53ef\u5b8c\u5907\u3002 \u6211\u4eec\u5bf9 T \u7684\u89c2\u6d4b\u6c38\u8fdc\u662f\u4e0d\u5b8c\u5907\u7684\u3002\u4efb\u4f55\u6d4b\u91cf\u90fd\u6709\u566a\u58f0\u4e0e\u5206\u8fa8\u7387\u4e0b\u9650&#xff0c;\u4efb\u4f55\u6570\u636e\u96c6\u90fd\u53ea\u662f T \u5728\u65f6\u7a7a\u91c7\u6837\u7f51\u683c\u4e0a\u7684\u4e00\u4e2a\u7a00\u758f\u5207\u7247\u3002\u4ee5\u6570\u503c\u5929\u6c14\u9884\u62a5\u4e3a\u4f8b&#xff0c;\u5373\u4fbf\u6211\u4eec\u62e5\u6709\u5168\u7403\u6570\u5343\u4e2a\u63a2\u7a7a\u7ad9\u3001\u536b\u661f\u4e0e\u6d6e\u6807&#xff0c;\u6240\u5f97\u5230\u7684\u521d\u59cb\u573a\u4ecd\u7136\u662f\u771f\u5b9e\u5927\u6c14\u7684\u4e00\u4e2a\u9ad8\u5ea6\u7c97\u7cd9\u7684\u63d2\u503c\u8fd1\u4f3c&#xff1b;\u4ee5\u7ecf\u6d4e\u7684\u5b8f\u89c2\u7edf\u8ba1\u4e3a\u4f8b&#xff0c;GDP\u3001CPI \u8fd9\u7c7b\u603b\u91cf\u6307\u6807\u672c\u8eab\u5c31\u662f\u5bf9\u88ab\u7814\u7a76\u5bf9\u8c61\u7684\u53c8\u4e00\u5c42\u6295\u5f71&#xff0c;\u800c\u975e T \u672c\u8eab\u3002\u63cf\u8ff0\u6027\u4e0d\u53ef\u5b8c\u5907\u544a\u8bc9\u6211\u4eec&#xff1a;\u6211\u4eec\u8fde&#034;\u770b\u6e05 T \u7684\u8d77\u70b9&#034;\u90fd\u505a\u4e0d\u5230\u3002<\/p>\n<p>\u7b2c\u4e8c\u91cd&#xff1a;\u52a8\u529b\u5b66\u4e0d\u53ef\u5b8c\u5907\u3002 \u5373\u4fbf\u6211\u4eec\u62e5\u6709\u5b8c\u6574\u7684\u521d\u59cb\u6761\u4ef6&#xff0c;\u590d\u6742\u7cfb\u7edf\u7684\u6f14\u5316\u672c\u8eab\u4e5f\u53ef\u80fd\u5728\u957f\u65f6\u6807\u4e0a\u4e0d\u53ef\u9884\u6d4b\u3002N \u4f53\u95ee\u9898\u4e2d&#xff0c;\u4e09\u4f53\u53ca\u4ee5\u4e0a\u7cfb\u7edf\u7684\u8f68\u9053\u5177\u6709\u6b63\u7684 Lyapunov \u6307\u6570&#xff0c;\u521d\u59cb\u8bef\u5dee\u968f\u65f6\u95f4\u6307\u6570\u589e\u957f&#xff0c;\u4f7f\u5f97\u4efb\u4f55\u6709\u9650\u7cbe\u5ea6\u7684\u521d\u59cb\u6761\u4ef6\u90fd\u65e0\u6cd5\u652f\u6491\u957f\u671f\u7cbe\u786e\u9884\u8a00\u2014\u2014\u8fd9\u5c31\u662f\u6df7\u6c8c\u3002\u5927\u6c14\u3001\u6d41\u4f53\u3001\u6c14\u5019\u3001\u4e43\u81f3\u67d0\u4e9b\u795e\u7ecf\u7f51\u7edc\u8bad\u7ec3\u52a8\u529b\u5b66&#xff0c;\u90fd\u5177\u6709\u7c7b\u4f3c\u7684\u7ed3\u6784&#xff1a;\u77ed\u671f\u53ef\u9884\u6d4b&#xff0c;\u957f\u671f\u4ec5\u5728\u7edf\u8ba1\u610f\u4e49\u4e0a\u53ef\u63cf\u8ff0\u3002\u52a8\u529b\u5b66\u4e0d\u53ef\u5b8c\u5907\u544a\u8bc9\u6211\u4eec&#xff1a;\u5373\u4fbf\u770b\u6e05\u4e86\u8d77\u70b9&#xff0c;\u6211\u4eec\u4e5f\u63a8\u4e0d\u5230\u7ec8\u70b9\u3002<\/p>\n<p>\u7b2c\u4e09\u91cd&#xff1a;\u8ba1\u7b97\u6027\u4e0d\u53ef\u5b8c\u5907\u3002 \u5373\u4fbf\u6211\u4eec\u53ea\u60f3&#034;\u6a21\u62df&#034;T&#xff0c;\u4e5f\u9762\u4e34\u6765\u81ea\u8ba1\u7b97\u7406\u8bba\u7684\u66f4\u786c\u7ea6\u675f\u2014\u2014\u8ba1\u7b97\u4e0d\u53ef\u7ea6\u6027&#xff08;computational irreducibility&#xff09;\u3002\u56fe\u7075&#xff08;A. M. Turing, 1936&#xff09;\u8bc1\u660e&#xff1a;\u4e0d\u5b58\u5728\u4e00\u4e2a\u901a\u7528\u7b97\u6cd5&#xff0c;\u80fd\u5224\u5b9a\u4efb\u610f\u7ed9\u5b9a\u7a0b\u5e8f\u662f\u5426\u4f1a\u505c\u673a&#xff08;\u505c\u673a\u95ee\u9898\u4e0d\u53ef\u5224\u5b9a&#xff09;&#xff0c;\u8fd9\u63ed\u793a\u4e86\u4e00\u4e2a\u6839\u672c\u4e8b\u5b9e&#xff1a;\u5b58\u5728\u5b9a\u4e49\u826f\u597d\u3001\u5b8c\u5168\u786e\u5b9a\u7684\u6570\u5b66\u5bf9\u8c61&#xff0c;\u5176\u67d0\u4e9b\u6027\u8d28\u5728\u539f\u5219\u4e0a\u662f\u4e0d\u53ef\u8ba1\u7b97\u7684\u3002\u6c83\u5c14\u592b\u52d2\u59c6&#xff08;S. Wolfram, 2002&#xff09;\u5728\u300a\u4e00\u79cd\u65b0\u79d1\u5b66\u300b\u4e2d\u7cfb\u7edf\u7814\u7a76\u4e86\u5143\u80de\u81ea\u52a8\u673a&#xff0c;\u53d1\u73b0\u5373\u4f7f\u89c4\u5219\u6781\u7b80&#xff08;\u5982\u89c4\u5219 110&#xff09;&#xff0c;\u4e5f\u80fd\u4ea7\u751f\u56fe\u7075\u901a\u7528\u7684\u8ba1\u7b97\u2014\u2014\u8fd9\u610f\u5473\u7740\u67d0\u4e9b\u7cfb\u7edf\u7684\u957f\u671f\u884c\u4e3a\u7b49\u4ef7\u4e8e\u6c42\u89e3\u505c\u673a\u95ee\u9898&#xff0c;\u5728\u539f\u5219\u4e0a\u4e0d\u53ef\u88ab\u4efb\u4f55\u6709\u9650\u7a0b\u5e8f\u52a0\u901f\u9884\u6d4b\u3002Chaitin \u7684\u7b97\u6cd5\u4fe1\u606f\u8bba\u8fdb\u4e00\u6b65\u8868\u660e&#xff1a;\u4e00\u4e2a\u5e8f\u5217\u7684&#034;\u590d\u6742\u5ea6&#034;&#xff08;\u5176\u6700\u77ed\u63cf\u8ff0\u957f\u5ea6&#xff09;\u672c\u8eab\u662f\u4e0d\u53ef\u8ba1\u7b97\u7684\u3002\u628a\u8fd9\u4e00\u7cbe\u795e\u8fc1\u79fb\u5230\u52a8\u529b\u7cfb\u7edf&#xff1a;\u5bf9\u4e8e\u4e00\u822c\u7684\u6df7\u6c8c\u7cfb\u7edf&#xff0c;\u5224\u5b9a&#034;\u7ed9\u5b9a\u521d\u59cb\u6761\u4ef6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         x<\/p>\n<p>         0<\/p>\n<p>       x_0<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5806em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">x<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff0c;\u8f68\u8ff9\u662f\u5426\u6700\u7ec8\u8fdb\u5165\u67d0\u533a\u57df&#034;\u5728\u591a\u6570\u60c5\u5f62\u4e0b\u4e5f\u662f\u4e0d\u53ef\u5224\u5b9a\u7684\u3002\u6362\u8a00\u4e4b&#xff0c;\u5373\u4fbf T \u5b8c\u5168\u786e\u5b9a\u3001\u03a0 \u5df2\u9009\u5b9a\u3001O \u5df2\u88ab\u5f62\u5f0f\u5316&#xff0c;\u67d0\u4e9b\u5173\u4e8e O \u7684\u95ee\u9898\u672c\u8eab\u5c31\u6ca1\u6709\u7b97\u6cd5\u7b54\u6848\u3002\u8fd9\u628a\u4e0d\u53ef\u5b8c\u5907\u6027\u4ece\u8fd1\u4f3c\u8bef\u5dee\u7684\u5c42\u7ea7&#xff0c;\u63d0\u5347\u5230\u4e86\u8ba1\u7b97\u590d\u6742\u5ea6\u7684\u5c42\u7ea7\u3002<\/p>\n<p>\u8fd9\u4e09\u91cd\u6765\u6e90\u53e0\u52a0&#xff0c;\u63ed\u793a\u4e86\u4e00\u4e2a\u5bf9&#034;\u7406\u89e3&#034;\u81f3\u5173\u91cd\u8981\u7684\u7ed3\u8bba&#xff1a;\u590d\u6742\u7cfb\u7edf\u7684\u4e0d\u53ef\u5b8c\u5907\u6027&#xff0c;\u4e0d\u662f\u56e0\u4e3a\u6211\u4eec\u65e0\u77e5&#xff0c;\u800c\u662f\u56e0\u4e3a\u5b83\u4eec\u672c\u8eab\u5c31\u662f\u90a3\u6837\u6784\u9020\u7684\u3002 \u4e00\u4e2a\u4e09\u4f53\u7cfb\u7edf\u5e76\u4e0d\u9700\u8981&#034;\u9690\u85cf\u53d8\u91cf&#034;\u6765\u5236\u9020\u4e0d\u53ef\u9884\u6d4b\u6027&#xff1b;\u786e\u5b9a\u6027\u7684\u725b\u987f\u65b9\u7a0b\u672c\u8eab&#xff0c;\u5c31\u8db3\u4ee5\u4ea7\u751f\u5bf9\u521d\u59cb\u6761\u4ef6\u7684\u6307\u6570\u654f\u611f\u3002\u6362\u8a00\u4e4b&#xff0c;\u4e0d\u53ef\u9884\u6d4b\u6027\u5185\u751f\u4e8e T \u7684\u52a8\u529b\u5b66\u7ed3\u6784&#xff0c;\u800c\u975e\u5916\u751f\u4e8e\u6211\u4eec\u7684\u8ba4\u77e5\u7f3a\u9677\u3002\u8fd9\u4e00\u8ba4\u8bc6&#xff0c;\u662f\u628a&#034;\u5b8c\u5168\u7406\u89e3&#034;\u4ece\u76ee\u6807\u6e05\u5355\u4e2d\u79fb\u9664\u7684\u6700\u5f3a\u786c\u7406\u7531&#xff0c;\u4e5f\u4e3a\u540e\u6587\u8bba\u8bc1&#034;\u597d\u7684\u7406\u89e3\u4e0d\u9700\u8981\u590d\u73b0 T&#034;\u94fa\u5e73\u4e86\u9053\u8def\u3002<\/p>\n<h3>2.3 \u590d\u6742\u7cfb\u7edf\u7684\u7c7b\u578b\u5b66&#xff1a;\u56db\u7c7b\u4e0d\u53ef\u5b8c\u5907\u6027<\/h3>\n<p>\u5e76\u975e\u6240\u6709&#034;\u590d\u6742\u7cfb\u7edf&#034;\u90fd\u56e0\u540c\u4e00\u539f\u56e0\u800c\u590d\u6742\u3002\u6309 T \u4e0d\u53ef\u5b8c\u5907\u6027\u7684\u4e3b\u8981\u6765\u6e90&#xff0c;\u53ef\u628a\u590d\u6742\u7cfb\u7edf\u5206\u4e3a\u56db\u7c7b&#xff0c;\u6bcf\u7c7b\u7684&#034;\u96be&#034;\u5904\u4e0d\u540c&#xff0c;\u6240\u9700\u7684 \u03a0 \u4e5f\u4e0d\u540c\u3002\u5efa\u7acb\u8fd9\u4e00\u7c7b\u578b\u5b66&#xff0c;\u662f\u4e3a\u4e86\u8ba9&#034;\u6362\u6295\u5f71&#034;\u4e0d\u518d\u662f\u4e00\u53e5\u7a7a\u8bdd&#xff1a;\u9762\u5bf9\u4e0d\u540c\u7c7b\u578b T&#xff0c;\u6211\u4eec\u6709\u4e00\u5f20&#034;\u8be5\u8bd5\u54ea\u4e9b \u03a0&#034;\u7684\u8349\u56fe&#xff0c;\u800c\u975e\u76f2\u76ee\u5c1d\u8bd5\u3002<\/p>\n<p>\u7b2c\u4e00\u7c7b&#xff1a;\u6df7\u6c8c\u786e\u5b9a\u578b\u3002 \u5982 N \u4f53\u3001\u53cc\u6446\u3001\u5929\u6c14\u3002\u5176 T \u5b8c\u5168\u786e\u5b9a&#xff0c;\u96be\u70b9\u5728\u4e8e\u5bf9\u521d\u59cb\u6761\u4ef6\u7684\u6307\u6570\u654f\u611f&#xff08;\u52a8\u529b\u5b66\u6765\u6e90&#xff0c;\u89c1 2.2 \u8282&#xff09;\u3002\u5408\u9002 \u03a0 \u662f\u76f8\u7a7a\u95f4\u51e0\u4f55\u4e0e\u7edf\u8ba1\u5206\u5e03\u3002<\/p>\n<p>\u7b2c\u4e8c\u7c7b&#xff1a;\u4e34\u754c\u81ea\u7ec4\u7ec7\u578b\u3002 \u5982\u6c99\u5806\u6a21\u578b\u3001\u5730\u9707\u3001\u795e\u7ecf\u7f51\u7edc\u4e34\u754c\u6001\u3002\u5176 T \u5728\u76f8\u53d8\u70b9\u9644\u8fd1\u5bf9\u5fae\u89c2\u7ec6\u8282\u4e0d\u654f\u611f&#xff0c;\u5374\u5448\u73b0\u5e42\u5f8b\u4e0e\u666e\u9002\u7c7b&#xff08;\u89c1 4.5 \u8282&#xff09;\u3002\u5408\u9002 \u03a0 \u662f\u91cd\u6574\u5316\u4e0e\u6807\u5ea6\u5206\u6790\u3002<\/p>\n<p>\u7b2c\u4e09\u7c7b&#xff1a;\u9002\u5e94\u6f14\u5316\u578b\u3002 \u5982\u751f\u6001\u7cfb\u7edf\u3001\u7ecf\u6d4e\u4f53\u3001\u514d\u75ab\u7cfb\u7edf\u3002\u5176 T \u7684&#034;\u89c4\u5219&#034;\u672c\u8eab\u968f\u7cfb\u7edf\u6f14\u5316\u800c\u6539\u53d8&#xff08;\u4e3b\u4f53\u5728\u9002\u5e94\u73af\u5883&#xff0c;\u73af\u5883\u53c8\u88ab\u4e3b\u4f53\u6539\u53d8&#xff09;\u3002\u8fd9\u662f\u6bd4\u6df7\u6c8c\u66f4\u6df1\u7684\u4e0d\u53ef\u5b8c\u5907\u6027&#xff1a;\u8fde\u6f14\u5316\u7b97\u5b50 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03a6<\/p>\n<p>         t<\/p>\n<p>       \\\\Phi^{t}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7936em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u90fd\u5728\u53d8\u3002\u5408\u9002 \u03a0 \u5fc5\u987b\u5305\u542b&#034;\u9002\u5e94\u6027&#034;\u7ef4\u5ea6&#xff08;agent-based \u6a21\u578b&#xff09;\u3002<\/p>\n<p>\u7b2c\u56db\u7c7b&#xff1a;\u793e\u4f1a\u7b26\u53f7\u578b\u3002 \u5982\u8bed\u8a00\u3001\u6587\u5316\u3001\u5e02\u573a\u53d9\u4e8b\u3002\u5176 T \u5305\u542b\u610f\u4e49\u4e0e\u4fe1\u5ff5&#xff0c;\u96be\u4ee5\u751a\u81f3\u65e0\u6cd5\u5b8c\u5168\u5f62\u5f0f\u5316\u3002\u5408\u9002 \u03a0 \u5f80\u5f80\u662f\u5b9a\u6027\u7684&#xff08;\u8bdd\u8bed\u5206\u6790\u3001\u5236\u5ea6\u53d9\u4e8b&#xff09;&#xff0c;\u4e14\u5fc5\u987b\u627f\u8ba4\u6295\u5f71\u7684\u8bed\u4e49\u635f\u5931\u6781\u5927\u3002<\/p>\n<p>\u7c7b\u578b\u5b66\u628a&#034;\u6295\u5f71\u591a\u5143\u4e3b\u4e49&#034;&#xff08;\u89c1 3.5 \u8282&#xff09;\u843d\u6210\u4e86\u53ef\u64cd\u4f5c\u7684\u5206\u7c7b\u5b66&#xff1a;\u9762\u5bf9\u6df7\u6c8c\u786e\u5b9a\u578b&#xff0c;\u6211\u4eec\u7528\u51e0\u4f55\u4e0e\u7edf\u8ba1&#xff1b;\u9762\u5bf9\u9002\u5e94\u6f14\u5316\u578b&#xff0c;\u6211\u4eec\u7528\u57fa\u4e8e\u4e3b\u4f53\u7684\u4eff\u771f&#xff1b;\u9762\u5bf9\u793e\u4f1a\u7b26\u53f7\u578b&#xff0c;\u6211\u4eec\u5766\u627f\u8bed\u4e49\u635f\u5931\u3002\u5b83\u4e5f\u4e3a\u7b2c 8 \u7ae0\u8ba8\u8bba\u6846\u67b6\u7684\u9002\u7528\u8fb9\u754c\u57cb\u4e0b\u4f0f\u7b14\u2014\u2014\u5bf9\u4e8e\u7b2c\u56db\u7c7b\u7cfb\u7edf&#xff0c;\u6846\u67b6\u80fd\u63d0\u4f9b\u7684\u662f\u7eaa\u5f8b\u800c\u975e\u7b54\u6848\u3002<\/p>\n<h3>2.4 \u5bf9&#034;\u5b8c\u5168\u7406\u89e3&#034;\u8bef\u533a\u7684\u54f2\u5b66\u5426\u51b3\u4e0e\u76ee\u6807\u91cd\u6784<\/h3>\n<p>\u4e3a\u4ec0\u4e48&#034;\u5b8c\u5168\u7406\u89e3\u590d\u6742\u7cfb\u7edf&#034;\u4f1a\u6210\u4e3a\u4e00\u4ee3\u4ee3\u5b66\u8005&#xff08;\u5305\u62ec\u65e9\u671f\u5e9e\u52a0\u83b1\u672c\u4eba&#xff09;\u7684\u6267\u5ff5&#xff1f;\u5176\u54f2\u5b66\u6839\u6e90&#xff0c;\u53ef\u4ee5\u8ffd\u6eaf\u81f3\u7ecf\u5178\u529b\u5b66\u7684\u51b3\u5b9a\u8bba\u56fe\u666f&#xff0c;\u5c24\u5176\u662f\u62c9\u666e\u62c9\u65af&#xff08;Pierre-Simon Laplace&#xff09;\u5728 1814 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2.2 \u8282&#xff0c;\u6211\u4eec\u53ef\u4ee5\u5bf9&#034;\u5b8c\u5168\u7406\u89e3\u590d\u6742\u7cfb\u7edf&#034;\u4f5c\u51fa\u6700\u7ec8\u5426\u51b3&#xff0c;\u4e14\u8fd9\u4e00\u5426\u51b3\u6709\u4e09\u91cd\u72ec\u7acb\u6765\u6e90&#xff0c;\u5f7c\u6b64\u52a0\u5f3a&#xff1a;<\/p>\n<li>\u7ef4\u5ea6\u6765\u6e90&#xff1a;T \u7684\u65e0\u9650\u7ef4\u4f7f\u4efb\u4f55\u6a21\u578b\u5fc5\u7136\u662f\u6709\u635f\u6295\u5f71&#xff08;2.1 \u8282&#xff09;&#xff1b;<\/li>\n<li>\u52a8\u529b\u5b66\u6765\u6e90&#xff1a;\u6df7\u6c8c\u4f7f\u957f\u671f\u7cbe\u786e\u8f68\u9053\u5728\u539f\u7406\u4e0a\u4e0d\u53ef\u9884\u6d4b&#xff08;2.2 \u8282&#xff09;&#xff1b;<\/li>\n<li>\u53ef\u8ba1\u7b97\u6027\u6765\u6e90&#xff1a;\u5373\u4fbf\u5ffd\u7565\u524d\u4e24\u8005&#xff0c;\u67d0\u4e9b\u5173\u4e8e\u7cfb\u7edf\u7684\u95ee\u9898\u5728\u7b97\u6cd5\u4e0a\u6839\u672c\u4e0d\u53ef\u5224\u5b9a&#xff08;2.2 \u8282&#xff09;\u3002<\/li>\n<p>\u8fd9\u4e09\u91cd\u6765\u6e90\u5171\u540c\u8bf4\u660e&#xff1a;\u201c\u5b8c\u5168\u7406\u89e3&#034;\u4e0d\u53ea\u662f\u4e00\u4e2a\u96be\u4ee5\u8fbe\u5230\u7684\u76ee\u6807&#xff0c;\u800c\u662f\u4e00\u4e2a\u5728\u539f\u5219\u4e0a\u4e0d\u6210\u7acb\u7684\u76ee\u6807\u3002\u7ea0\u9519\u7684\u5173\u952e&#xff0c;\u4e0d\u662f\u653e\u5f03\u7406\u89e3&#xff0c;\u800c\u662f\u91cd\u65b0\u5b9a\u4e49\u7406\u89e3&#xff1a;\u4ece&#034;\u590d\u73b0 T \u7684\u6bcf\u4e00\u6761\u8f68\u8ff9&#034;\u9000\u56de\u5230&#034;\u5728\u6295\u5f71 \u03a0 \u4e2d\u63d0\u53d6\u7a33\u5b9a\u6210\u7acb\u7684 O\u201d\u3002\u8fd9\u4e00\u9000&#xff0c;\u4e0d\u662f\u9000\u7f29&#xff0c;\u800c\u662f\u6e05\u9192\u3002<\/p>\n<p>\u5982\u679c\u628a\u76ee\u6807\u4ece&#034;\u590d\u73b0 T&#034;\u6539\u5199\u4e3a&#034;\u5728 T \u4e2d\u63d0\u53d6\u5bf9\u6211\u4eec\u6709\u7528\u7684\u79e9\u5e8f&#034;&#xff0c;\u6574\u4e2a\u7814\u7a76\u7eb2\u9886\u5c31\u8c41\u7136\u5f00\u6717\u3002\u8bf7\u6ce8\u610f\u8fd9\u91cc\u7684\u4ef7\u503c\u8f6c\u5411&#xff1a;\u7406\u89e3\u7684\u76ee\u7684&#xff0c;\u4ece\u6765\u4e0d\u662f\u4e3a\u4e86\u5728\u5934\u8111\u4e2d\u88c5\u4e0b\u6574\u4e2a T&#xff0c;\u800c\u662f\u4e3a\u4e86\u884c\u52a8\u4e0e\u51b3\u7b56\u3002 \u6211\u4eec\u7814\u7a76\u5929\u6c14&#xff0c;\u662f\u4e3a\u4e86\u51b3\u5b9a\u660e\u5929\u662f\u5426\u5e26\u4f1e\u3001\u53f0\u98ce\u8def\u5f84\u5982\u4f55\u758f\u6563&#xff1b;\u6211\u4eec\u7814\u7a76\u7ecf\u6d4e&#xff0c;\u662f\u4e3a\u4e86\u914d\u7f6e\u8d44\u6e90\u3001\u89c4\u907f\u98ce\u9669&#xff1b;\u6211\u4eec\u7814\u7a76 N \u4f53&#xff0c;\u662f\u4e3a\u4e86\u89c4\u5212\u822a\u5929\u5668\u8f68\u9053\u3001\u7406\u89e3\u661f\u7cfb\u6f14\u5316\u3002\u6240\u6709\u8fd9\u4e9b\u76ee\u7684&#xff0c;\u90fd\u53ea\u9700\u8981\u67d0\u4e00\u5c42\u6b21\u3001\u67d0\u4e00\u6295\u5f71\u4e0b\u7684 O&#xff0c;\u800c\u65e0\u9700 T \u7684\u5168\u90e8\u3002\u56e0\u6b64&#xff0c;\u672c\u6587\u7684\u7b2c\u4e00\u4e2a\u5b9e\u8df5\u6027\u7ed3\u8bba\u662f&#xff1a;\u7406\u89e3\u590d\u6742\u7cfb\u7edf&#xff0c;\u4e0d\u662f\u8ffd\u6c42 T \u7684\u5b8c\u5907\u590d\u73b0&#xff0c;\u800c\u662f\u8ffd\u6c42\u5728\u6070\u5f53\u7684 \u03a0 \u4e4b\u4e0b&#xff0c;\u627e\u5230\u8db3\u591f\u652f\u6491\u51b3\u7b56\u7684\u53ef\u8ba1\u7b97 O\u3002 \u8fd9\u4e00\u5b9a\u8c03&#xff0c;\u76f4\u63a5\u5f15\u51fa\u4e86\u4e0b\u4e00\u7ae0\u5bf9&#034;\u6295\u5f71 \u03a0&#034;\u7684\u7cfb\u7edf\u8003\u5bdf\u3002\u4ece\u8fd9\u4e00\u7ae0\u8d77&#xff0c;\u5168\u6587\u7684\u91cd\u5fc3\u5c06\u4e0d\u518d\u7ea0\u7f20\u4e8e&#034;\u6211\u4eec\u4e3a\u4f55\u4e0d\u80fd\u590d\u73b0 T&#034;&#xff0c;\u800c\u662f\u8f6c\u5411\u4e00\u4e2a\u66f4\u5177\u5efa\u8bbe\u6027\u7684\u95ee\u9898\u2014\u2014\u65e2\u7136\u5fc5\u987b\u6295\u5f71&#xff0c;\u6211\u4eec\u8be5\u5982\u4f55\u6295\u5f71\u3001\u53c8\u8be5\u5728\u6295\u5f71\u4e2d\u5bfb\u4ec0\u4e48\u6837\u7684\u79e9\u5e8f\u3002<\/p>\n<h2>\u7b2c 3 \u7ae0 \u6295\u5f71 \u03a0&#xff1a;\u4e00\u5207\u8ba4\u77e5\u6a21\u578b\u7684\u6709\u635f\u6620\u5c04<\/h2>\n<h3>3.1 \u6295\u5f71\u7684\u6570\u5b66\u672c\u8d28<\/h3>\n<p>\u4e0a\u4e00\u7ae0\u8bba\u8bc1\u4e86 T \u4e0d\u53ef\u88ab\u5b8c\u5168\u5bb9\u7eb3&#xff0c;\u5e76\u7531\u6b64\u6307\u51fa&#xff1a;\u4efb\u4f55\u5bf9 T \u7684\u903c\u8fd1\u90fd\u5fc5\u7136\u662f\u4e00\u79cd\u538b\u7f29&#xff0c;\u800c\u538b\u7f29\u5728\u6570\u5b66\u4e0a\u610f\u5473\u7740\u6295\u5f71\u3002\u672c\u7ae0\u8981\u628a&#034;\u6295\u5f71&#034;\u4ece\u4e00\u4e2a\u76f4\u89c9\u8bcd\u6c47&#xff0c;\u53d8\u6210\u6846\u67b6\u4e2d\u53ef\u7cbe\u786e\u64cd\u4f5c\u7684\u6838\u5fc3\u6982\u5ff5\u3002<\/p>\n<p>&#034;\u6295\u5f71&#034;\u5728\u672c\u6587\u4e2d\u9996\u5148\u662f\u4e00\u4e2a\u6709\u4e25\u683c\u6570\u5b66\u542b\u4e49\u7684\u6982\u5ff5&#xff0c;\u7136\u540e\u624d\u662f\u4e00\u4e2a\u8ba4\u77e5\u9690\u55bb\u3002\u5728\u7ebf\u6027\u4ee3\u6570\u91cc&#xff0c;\u628a\u4e00\u4e2a\u9ad8\u7ef4\u5411\u91cf <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        x<\/p>\n<p>        \u2208<\/p>\n<p>         R<\/p>\n<p>         n<\/p>\n<p>       \\\\mathbf{x}\\\\in\\\\mathbb{R}^{n}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5782em;vertical-align: -0.0391em\"><\/span><span class=\"mord mathbf\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6889em\"><\/span><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u6295\u5f71\u5230\u4e00\u4e2a\u4f4e\u7ef4\u5b50\u7a7a\u95f4 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        S<\/p>\n<p>        \u2282<\/p>\n<p>         R<\/p>\n<p>         n<\/p>\n<p>       \\\\mathcal{S}\\\\subset\\\\mathbb{R}^{n}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7224em;vertical-align: -0.0391em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.075em\">S<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2282<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6889em\"><\/span><span class=\"mord\"><span class=\"mord mathbb\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6644em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">n<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0a&#xff0c;\u5f97\u5230 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         x<\/p>\n<p>         ^<\/p>\n<p>        &#061;<\/p>\n<p>        P<\/p>\n<p>        x<\/p>\n<p>       \\\\mathbf{\\\\hat{x}} &#061; P\\\\mathbf{x}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7079em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7079em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathbf\">x<\/span><\/span><span class=\"\" style=\"top: -3.0134em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.2875em\"><span class=\"mord mathbf\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"mord mathbf\">x<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5176\u4e2d <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        P<\/p>\n<p>       P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span> \u662f\u5230 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        S<\/p>\n<p>       \\\\mathcal{S}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.075em\">S<\/span><\/span><\/span><\/span><\/span> \u7684\u6b63\u4ea4\u6295\u5f71\u77e9\u9635&#xff0c;\u6ee1\u8db3 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>         2<\/p>\n<p>        &#061;<\/p>\n<p>        P<\/p>\n<p>       P^{2}&#061;P<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><\/span><\/span><\/span><\/span>\u3002\u6295\u5f71\u7b97\u5b50\u662f\u5e42\u7b49\u7684&#xff1a;\u4f60\u6295\u5f71\u4e24\u6b21&#xff0c;\u7ed3\u679c\u4e0d\u53d8&#xff1b;\u5b83\u4e5f\u662f\u4fe1\u606f\u6709\u635f\u7684&#xff1a;\u4e00\u65e6\u843d\u5230\u4f4e\u7ef4\u5b50\u7a7a\u95f4&#xff0c;\u5782\u76f4\u4e8e\u8be5\u5b50\u7a7a\u95f4\u7684\u5206\u91cf\u5c31\u6c38\u4e45\u4e22\u5931\u4e86\u3002<\/p>\n<p>\u8fd9\u4e2a\u7b80\u5355\u7684\u4e8b\u5b9e&#xff0c;\u6070\u597d\u523b\u753b\u4e86\u4eba\u7c7b\u8ba4\u77e5\u7684\u6839\u672c\u5904\u5883\u3002\u6211\u4eec\u5bf9 T \u7684\u4efb\u4f55\u63cf\u8ff0&#xff0c;\u90fd\u662f\u628a\u65e0\u9650\u7ef4\u7684\u771f\u5b9e&#xff0c;\u6620\u5c04\u5230\u4e00\u4e2a\u6211\u4eec&#034;\u88c5\u5f97\u4e0b\u3001\u7b97\u5f97\u52a8\u3001\u8bf4\u5f97\u6e05&#034;\u7684\u4f4e\u7ef4\u8bed\u4e49\u5b50\u7a7a\u95f4\u4e0a\u3002\u5730\u56fe\u662f\u57ce\u5e02\u7684\u6295\u5f71&#xff08;\u4e22\u5931\u4e86\u6d77\u62d4\u3001\u6750\u8d28\u3001\u6c14\u5473&#xff09;&#xff1b;\u725b\u987f\u529b\u5b66\u662f\u771f\u5b9e\u8fd0\u52a8\u7684\u6295\u5f71&#xff08;\u5728\u5b8f\u89c2\u4f4e\u901f\u4e0b\u4e22\u5931\u4e86\u76f8\u5bf9\u8bba\u4e0e\u91cf\u5b50\u6548\u5e94&#xff09;&#xff1b;\u8d22\u52a1\u62a5\u8868\u662f\u516c\u53f8\u7ecf\u8425\u72b6\u51b5\u7684\u6295\u5f71&#xff08;\u628a\u65e0\u6570\u7b14\u4ea4\u6613\u538b\u7f29\u6210\u51e0\u4e2a\u79d1\u76ee\u4f59\u989d&#xff09;\u3002\u6bcf\u4e00\u6b21\u6295\u5f71\u90fd\u4fdd\u7559\u4e86\u4e00\u90e8\u5206\u7ed3\u6784&#xff0c;\u4e5f\u5fc5\u7136\u4e22\u5931\u4e86\u4e00\u90e8\u5206\u4fe1\u606f\u2014\u2014\u8fd9\u662f\u6570\u5b66\u7684\u94c1\u5f8b&#xff0c;\u4e0d\u662f\u6211\u4eec\u7684\u5931\u8bef\u3002<\/p>\n<p>\u4f46\u6295\u5f71\u4e4b\u6240\u4ee5\u6709\u7528&#xff0c;\u6070\u6070\u5728\u4e8e\u5b83\u7684&#034;\u6709\u635f&#034;\u3002\u5982\u679c\u4e00\u5f20\u5730\u56fe\u628a\u57ce\u5e02\u7684\u6240\u6709\u5206\u5b50\u90fd\u753b\u51fa\u6765&#xff0c;\u5b83\u5c31\u4e0d\u518d\u662f\u5730\u56fe&#xff0c;\u800c\u662f\u4e00\u5806\u65e0\u6cd5\u5bfc\u822a\u7684\u566a\u58f0&#xff1b;\u5982\u679c\u4e00\u5957\u7406\u8bba\u628a T \u7684\u6240\u6709\u81ea\u7531\u5ea6\u90fd\u5199\u8fdb\u6765&#xff0c;\u5b83\u5c31\u4e0d\u518d\u662f\u7406\u8bba&#xff0c;\u800c\u662f\u4e00\u4e2a\u4e0d\u53ef\u5904\u7406\u7684\u91cd\u6f14\u3002\u6295\u5f71\u7684\u4ef7\u503c&#xff0c;\u5728\u4e8e\u5b83\u7528\u53ef\u63a7\u7684\u4fe1\u606f\u635f\u5931&#xff0c;\u6362\u53d6\u4e86\u5bf9\u6211\u4eec\u6240\u5173\u5fc3\u7ed3\u6784\u7684\u51f8\u663e\u3002\u8fd9\u6b63\u662f\u672c\u7ae0\u8981\u8bba\u8bc1\u7684\u6838\u5fc3&#xff1a;\u7406\u89e3\u590d\u6742\u7cfb\u7edf&#xff0c;\u4e0d\u662f\u5bfb\u627e&#034;\u552f\u4e00\u6b63\u786e\u7684\u6a21\u578b&#034;&#xff0c;\u800c\u662f\u5bfb\u627e&#034;\u9002\u5408\u5f53\u524d\u95ee\u9898\u7684\u6295\u5f71&#034;\u3002<\/p>\n<h3>3.2 \u6295\u5f71\u7684\u5f62\u5f0f\u5316&#xff1a;\u53ef\u6d4b\u6620\u5c04\u4e0e\u4fe1\u606f\u635f\u5931<\/h3>\n<p>\u4e3a\u4f7f&#034;\u6295\u5f71&#034;\u7ecf\u5f97\u8d77\u4e25\u683c\u7684\u5b66\u672f\u68c0\u9a8c&#xff0c;\u6211\u4eec\u628a\u5b83\u5f62\u5f0f\u5316\u4e3a\u4e00\u4e2a\u53ef\u6d4b\u6620\u5c04 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03c0<\/p>\n<p>        :<\/p>\n<p>        X<\/p>\n<p>        \u2192<\/p>\n<p>        Y<\/p>\n<p>       \\\\pi:\\\\mathcal{X}\\\\to\\\\mathcal{Y}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2192<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7805em;vertical-align: -0.0972em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.0822em\">Y<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5176\u4e2d <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        X<\/p>\n<p>       \\\\mathcal{X}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><\/span><\/span><\/span><\/span> \u662f T \u7684\u72b6\u6001\u7a7a\u95f4&#xff08;2.1 \u8282&#xff09;&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Y<\/p>\n<p>       \\\\mathcal{Y}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7805em;vertical-align: -0.0972em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.0822em\">Y<\/span><\/span><\/span><\/span><\/span> \u662f\u4f4e\u7ef4\u7684&#034;\u8ba4\u77e5\u5b50\u7a7a\u95f4&#034;&#xff08;\u53ef\u4ee5\u662f\u4e00\u4e2a\u6b27\u6c0f\u7a7a\u95f4\u3001\u4e00\u4e2a\u56fe\u3001\u4e00\u4e2a\u7b26\u53f7\u96c6\u5408&#xff0c;\u751a\u81f3\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u7684\u9690\u5c42&#xff09;\u3002\u03a0 \u7684\u4f5c\u7528&#xff0c;\u662f\u628a T \u7684\u72b6\u6001 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        x<\/p>\n<p>        \u2208<\/p>\n<p>        X<\/p>\n<p>       x\\\\in\\\\mathcal{X}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.5782em;vertical-align: -0.0391em\"><\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><\/span><\/span><\/span><\/span> \u6620\u5c04\u4e3a\u6211\u4eec\u5bf9\u5b83\u7684&#034;\u7b80\u5316\u63cf\u8ff0&#034; <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03c0<\/p>\n<p>        (<\/p>\n<p>        x<\/p>\n<p>        )<\/p>\n<p>        \u2208<\/p>\n<p>        Y<\/p>\n<p>       \\\\pi(x)\\\\in\\\\mathcal{Y}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">x<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2208<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7805em;vertical-align: -0.0972em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.0822em\">Y<\/span><\/span><\/span><\/span><\/span>\u3002<\/p>\n<p>\u6295\u5f71\u7684\u4fe1\u606f\u635f\u5931\u53ef\u88ab\u4e92\u4fe1\u606f&#xff08;mutual information&#xff09;\u6216\u6761\u4ef6\u71b5\u4e25\u683c\u5ea6\u91cf\u3002\u8bb0 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        X<\/p>\n<p>       X<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">X<\/span><\/span><\/span><\/span><\/span> \u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        X<\/p>\n<p>       \\\\mathcal{X}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.1464em\">X<\/span><\/span><\/span><\/span><\/span> \u4e0a\u7684\u968f\u673a\u53d8\u91cf&#xff08;\u670d\u4ece T \u7684\u5206\u5e03&#xff09;&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Y<\/p>\n<p>        &#061;<\/p>\n<p>        \u03c0<\/p>\n<p>        (<\/p>\n<p>        X<\/p>\n<p>        )<\/p>\n<p>       Y&#061;\\\\pi(X)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">Y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">X<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u4e3a\u5176\u6295\u5f71\u3002\u6295\u5f71\u4e22\u5931\u7684\u4fe1\u606f\u91cf\u53ef\u5199\u4e3a&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>          I<\/p>\n<p>          loss<\/p>\n<p>         &#061;<\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         X<\/p>\n<p>         )<\/p>\n<p>         \u2212<\/p>\n<p>         I<\/p>\n<p>         (<\/p>\n<p>         X<\/p>\n<p>         ;<\/p>\n<p>         Y<\/p>\n<p>         )<\/p>\n<p>         &#061;<\/p>\n<p>         H<\/p>\n<p>         (<\/p>\n<p>         X<\/p>\n<p>         \u2223<\/p>\n<p>         Y<\/p>\n<p>         )<\/p>\n<p>         ,<\/p>\n<p>        I_{\\\\text{loss}} &#061; H(X) &#8211; I(X; Y) &#061; H(X\\\\mid Y),<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3361em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0785em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord mtight\">loss<\/span><\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">X<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">X<\/span><span class=\"mpunct\">;<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">Y<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">X<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">Y<\/span><span class=\"mclose\">)<\/span><span class=\"mpunct\">,<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u5373\u7ed9\u5b9a\u6295\u5f71 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Y<\/p>\n<p>       Y<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">Y<\/span><\/span><\/span><\/span><\/span> \u540e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        X<\/p>\n<p>       X<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">X<\/span><\/span><\/span><\/span><\/span> \u7684\u5269\u4f59\u4e0d\u786e\u5b9a\u6027\u3002\u4e00\u4e2a&#034;\u597d\u7684\u6295\u5f71&#034;&#xff08;3.5 \u8282\u5c06\u7ed9\u51fa\u6807\u51c6&#xff09;\u7684\u7279\u5f81&#xff0c;\u6b63\u662f&#xff1a;\u5728\u6240\u5173\u5fc3\u7684\u4efb\u52a1\u4e0a&#xff0c;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        I<\/p>\n<p>        (<\/p>\n<p>        X<\/p>\n<p>        ;<\/p>\n<p>        Y<\/p>\n<p>        )<\/p>\n<p>       I(X;Y)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">I<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">X<\/span><span class=\"mpunct\">;<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">Y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u8db3\u591f\u5927&#xff08;\u4fdd\u7559\u4e86\u5173\u952e\u7ed3\u6784&#xff09;&#xff0c;\u800c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        H<\/p>\n<p>        (<\/p>\n<p>        X<\/p>\n<p>        \u2223<\/p>\n<p>        Y<\/p>\n<p>        )<\/p>\n<p>       H(X\\\\mid Y)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0813em\">H<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0785em\">X<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2223<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.2222em\">Y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u867d\u4e0d\u4e3a\u96f6\u4f46\u53ef\u63a7&#xff08;\u77e5\u9053\u4e22\u4e86\u4ec0\u4e48&#xff09;\u3002<\/p>\n<p>\u8fd9\u4e0e\u4fe1\u606f\u8bba\u4e2d\u7684**\u7387\u5931\u771f\u7406\u8bba&#xff08;rate\u2013distortion theory&#xff09;**\u5b8c\u5168\u540c\u6784&#xff1a;\u6295\u5f71\u672c\u8d28\u4e0a\u662f\u4e00\u6b21\u6709\u635f\u538b\u7f29&#xff0c;\u800c&#034;\u9002\u5408\u95ee\u9898\u7684\u6295\u5f71&#034;\u5c31\u662f\u5728\u8be5\u4efb\u52a1\u5931\u771f\u7ea6\u675f\u4e0b\u7684\u6700\u4f18\u538b\u7f29\u3002\u5f62\u5f0f\u5316\u8fd8\u63ed\u793a\u4e86\u4e00\u4e2a\u5e38\u88ab\u8bef\u89e3\u7684\u70b9&#xff1a;\u82e5\u6709\u4eba\u4e3b\u5f20&#034;\u5b58\u5728\u4e0d\u4f9d\u8d56\u4efb\u4f55 \u03a0 \u7684\u7edd\u5bf9 O&#034;&#xff0c;\u90a3\u4ed6\u5728\u903b\u8f91\u4e0a\u5fc5\u987b\u5148\u628a\u8be5&#034;\u7edd\u5bf9 O&#034;\u7528\u67d0\u79cd\u7b26\u53f7\u7cfb\u7edf\u8868\u8fbe\u51fa\u6765\u2014\u2014\u800c\u8fd9\u4e00\u8868\u8fbe\u884c\u4e3a\u672c\u8eab&#xff0c;\u5c31\u5df2\u7ecf\u662f\u4e00\u4e2a \u03a0\u3002\u56e0\u6b64&#xff0c;\u201c\u7edd\u5bf9 O&#034;\u5728\u53ef\u9648\u8ff0\u7684\u8303\u56f4\u5185\u5fc5\u7136\u574d\u7f29\u4e3a&#034;\u67d0 \u03a0 \u4e0b\u7684 O\u201d\u3002T-\u03a0-O \u6846\u67b6\u5728\u5f62\u5f0f\u5316\u610f\u4e49\u4e0b\u662f\u81ea\u6d3d\u7684&#xff1a;\u5b83\u4e0d\u5141\u8bb8\u81ea\u5df1\u88ab\u6084\u6084\u7ed5\u8fc7&#xff0c;\u4e5f\u4e0d\u5141\u8bb8&#034;\u7edd\u5bf9\u7684\u3001\u65e0\u6295\u5f71\u7684\u79e9\u5e8f&#034;\u6210\u4e3a\u4e00\u4e2a\u5408\u6cd5\u7684\u79d1\u5b66\u4e3b\u5f20\u3002\u8fd9\u6070\u6070\u662f\u5176\u4e25\u8c28\u6027\u7684\u6765\u6e90&#xff0c;\u800c\u975e\u7f3a\u9677\u3002<\/p>\n<h3>3.3 \u4eba\u7c7b\u8ba4\u77e5\u6a21\u578b\u7684\u6295\u5f71\u8c31\u7cfb<\/h3>\n<p>\u628a&#034;\u6295\u5f71&#034;\u5f62\u5f0f\u5316\u4e4b\u540e&#xff0c;\u4e00\u4e2a\u60ca\u4eba\u7684\u7ed3\u8bba\u6d6e\u73b0\u51fa\u6765&#xff1a;\u4eba\u7c7b\u5386\u53f2\u4e0a\u6700\u91cd\u8981\u7684\u8ba4\u77e5\u6a21\u578b&#xff0c;\u5728\u672c\u4f53\u8bba\u5730\u4f4d\u4e0a\u662f\u540c\u6784\u7684\u2014\u2014\u5b83\u4eec\u90fd\u662f \u03a0\u3002\u6211\u4eec\u53ef\u4ee5\u628a\u5b83\u4eec\u6392\u6210\u4e00\u6761&#034;\u6295\u5f71\u8c31\u7cfb&#034;&#xff0c;\u4ee5\u8bf4\u660e \u03a0 \u7684\u666e\u904d\u6027\u4e0e\u591a\u6837\u6027\u3002<\/p>\n<p>\u529b\u5b66\u6295\u5f71&#xff1a;\u725b\u987f\u4e0e\u7231\u56e0\u65af\u5766\u3002 \u725b\u987f\u529b\u5b66\u662f\u4f4e\u901f\u5b8f\u89c2\u4e16\u754c\u7684\u6295\u5f71&#xff0c;\u5b83\u628a\u771f\u5b9e\u8fd0\u52a8\u538b\u7f29\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        F<\/p>\n<p>        &#061;<\/p>\n<p>        m<\/p>\n<p>        a<\/p>\n<p>       F&#061;ma<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">F<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">ma<\/span><\/span><\/span><\/span><\/span> \u4e0e\u4e07\u6709\u5f15\u529b\u5b9a\u5f8b&#xff0c;\u5728\u8fd9\u4e00\u6295\u5f71\u4e0b\u692d\u5706\u8f68\u9053\u3001\u80fd\u91cf\u5b88\u6052\u7b49 O \u6781\u4e3a\u5b8c\u6574\u3002\u4f46\u5f53\u901f\u5ea6\u63a5\u8fd1\u5149\u901f\u6216\u5f15\u529b\u6781\u5f3a\u65f6&#xff0c;\u725b\u987f\u6295\u5f71\u5f00\u59cb\u5931\u771f&#xff0c;\u4e8e\u662f\u7231\u56e0\u65af\u5766\u7684\u5e7f\u4e49\u76f8\u5bf9\u8bba\u4f5c\u4e3a\u4e00\u4e2a\u66f4&#034;\u9ad8\u4fdd\u771f&#034;\u7684\u6295\u5f71\u63a5\u7ba1\u4e86\u76f8\u5e94\u5c3a\u5ea6\u3002\u4e24\u8005\u90fd\u4e0d\u662f T&#xff0c;\u800c\u662f T \u5728\u4e0d\u540c\u5c3a\u5ea6\u4e0a\u7684\u4e0d\u540c\u6295\u5f71\u3002<\/p>\n<p>\u8fde\u7eed\u7edf\u6295\u5f71&#xff1a;\u6d41\u4f53\u529b\u5b66\u3002 \u771f\u5b9e\u6d41\u4f53\u662f\u7ea6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         10<\/p>\n<p>         23<\/p>\n<p>       10^{23}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">23<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e2a\u5206\u5b50\u7684\u79bb\u6563\u78b0\u649e\u3002\u6d41\u4f53\u529b\u5b66\u505a\u4e86\u4e00\u4e2a\u5927\u80c6\u7684\u6295\u5f71&#xff1a;\u628a\u5206\u5b50\u8fd0\u52a8\u5e73\u5747\u5316\u4e3a\u5bc6\u5ea6 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03c1<\/p>\n<p>       \\\\rho<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">\u03c1<\/span><\/span><\/span><\/span><\/span>\u3001\u901f\u5ea6\u573a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        u<\/p>\n<p>       \\\\mathbf{u}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4444em\"><\/span><span class=\"mord mathbf\">u<\/span><\/span><\/span><\/span><\/span>\u3001\u538b\u5f3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        p<\/p>\n<p>       p<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.625em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\">p<\/span><\/span><\/span><\/span><\/span> \u7b49\u5b8f\u89c2\u8fde\u7eed\u573a&#xff0c;\u5e76\u5199\u51fa Navier\u2013Stokes \u65b9\u7a0b\u3002\u8fd9\u4e00\u6295\u5f71\u4e22\u5931\u4e86\u5206\u5b50\u5c42\u9762\u7684\u5168\u90e8\u7ec6\u8282&#xff0c;\u5374\u8ba9\u6211\u4eec\u53ef\u4ee5\u8bbe\u8ba1\u98de\u673a\u673a\u7ffc\u3001\u9884\u6d4b\u6d0b\u6d41\u3002\u5b83\u51f8\u663e\u7684&#xff0c;\u6b63\u662f\u5de5\u7a0b\u5173\u5fc3\u7684\u7ed3\u6784\u3002<\/p>\n<p>\u6570\u636e\u6295\u5f71&#xff1a;\u795e\u7ecf\u7f51\u7edc\u3002 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\u03a0\u3002 \u628a\u5b83\u4eec\u5206\u6210&#034;\u771f\u7406&#034;\u4e0e&#034;\u8fd1\u4f3c&#034;\u662f\u8bef\u5bfc&#xff1b;\u66f4\u8bda\u5b9e\u7684\u5206\u7c7b\u662f&#034;\u9002\u5408\u4ec0\u4e48\u95ee\u9898\u7684\u6295\u5f71&#034;\u3002<\/p>\n<h3>3.4 \u5de5\u7a0b\u5316\u7684\u6295\u5f71&#xff1a;\u4ece PCA \u5230\u81ea\u7f16\u7801\u5668<\/h3>\n<p>3.1\u20133.3 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\u8fd9\u662f\u4e00\u4e2a\u7ebf\u6027\u6295\u5f71&#xff1a;\u627e\u4e00\u7ec4\u6b63\u4ea4\u65b9\u5411&#xff0c;\u4f7f\u6570\u636e\u5728\u8be5\u65b9\u5411\u6295\u5f71\u7684\u65b9\u5dee\u6700\u5927\u3002\u5b83\u5bf9\u5e94&#034;\u597d\u7684\u6295\u5f71&#034;\u6807\u51c6\u4e00&#xff08;\u51f8\u663e\u65b9\u5dee\u6700\u5927\u7684\u7ed3\u6784&#xff09;\u4e0e\u6807\u51c6\u56db&#xff08;\u5bf9\u9ad8\u65af\u6270\u52a8\u7a33\u5b9a&#xff09;&#xff0c;\u4f46\u56e0\u5176\u7ebf\u6027&#xff0c;\u5bf9\u6d41\u5f62\u7ed3\u6784\u65e0\u80fd\u4e3a\u529b\u2014\u2014\u8fd9\u662f\u5b83\u4f5c\u4e3a \u03a0 \u7684\u56fa\u6709\u76f2\u533a\u3002<\/p>\n<p>\u6d41\u5f62\u5b66\u4e60&#xff08;t-SNE\u3001UMAP&#xff09;\u3002 \u5b83\u4eec\u5047\u8bbe\u9ad8\u7ef4\u6570\u636e\u5206\u5e03\u5728\u4f4e\u7ef4\u6d41\u5f62\u4e0a&#xff0c;\u505a\u975e\u7ebf\u6027\u6295\u5f71\u4ee5\u4fdd\u7559\u5c40\u90e8\u90bb\u57df\u7ed3\u6784\u3002\u5b83\u4eec\u51f8\u663e\u4e86&#034;\u7c07\u7ed3\u6784&#034;\u8fd9\u4e00 O&#xff0c;\u5374\u727a\u7272\u4e86\u5168\u5c40\u8ddd\u79bb&#xff08;t-SNE \u7684\u8ddd\u79bb\u4e0d\u53ef\u8de8\u7c07\u6bd4\u8f83&#xff09;\u2014\u2014\u8fd9\u662f\u6807\u51c6\u4e09&#xff08;\u53ef\u8ba1\u7b97 O&#xff09;\u4e0e\u6807\u51c6\u4e8c&#xff08;\u4fe1\u606f\u635f\u5931\u53ef\u63a7&#xff09;\u4e4b\u95f4\u7684\u5178\u578b\u6743\u8861\u3002<\/p>\n<p>\u81ea\u7f16\u7801\u5668&#xff08;Autoencoder&#xff09;\u3002 \u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u88ab\u8bad\u7ec3\u628a\u8f93\u5165\u538b\u7f29\u518d\u91cd\u5efa&#xff0c;\u74f6\u9888\u5c42\u5373&#034;\u6570\u636e\u9a71\u52a8\u7684\u6295\u5f71&#034;\u3002\u5b83\u4e0e PCA \u540c\u6784&#xff08;\u7ebf\u6027\u81ea\u7f16\u7801\u5668&#061;PCA&#xff09;&#xff0c;\u4f46\u56e0\u975e\u7ebf\u6027\u800c\u66f4\u5f3a&#xff1b;\u4ee3\u4ef7\u662f\u6807\u51c6\u4e8c\u7684\u98ce\u9669\u2014\u2014\u6211\u4eec\u5e38\u4e0d\u77e5\u9053\u74f6\u9888\u5c42\u4e22\u6389\u4e86\u4ec0\u4e48&#xff0c;\u8fd9\u6b63\u662f\u53ef\u89e3\u91ca\u6027\u7814\u7a76\u7684\u7528\u6b66\u4e4b\u5730&#xff08;\u89c1 6.4 \u8282&#xff09;\u3002<\/p>\n<p>\u4e09\u4f8b\u8fde\u8d77\u6765\u8bf4\u660e&#xff1a;\u5f53\u6211\u4eec\u5728\u672c\u7ae0\u8c08&#034;\u6295\u5f71&#034;&#xff0c;\u8c08\u7684\u4e0d\u662f\u4e00\u4e2a\u6bd4\u55bb&#xff0c;\u800c\u662f\u4e00\u7ec4\u53ef\u88ab\u7cbe\u786e\u5b9e\u73b0\u3001\u53ef\u88ab\u5b9a\u91cf\u8bc4\u4f30\u3001\u4e14\u5fc5\u7136\u4f34\u968f\u6743\u8861\u7684\u64cd\u4f5c\u3002\u7406\u89e3\u8fd9\u4e9b\u6743\u8861&#xff0c;\u662f\u907f\u514d\u628a\u67d0\u4e00\u5de5\u7a0b\u6295\u5f71\u8bef\u5f53\u6210 T \u7684\u7b2c\u4e00\u6b65\u3002<\/p>\n<h3>3.5 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T&#034;&#xff0c;\u53ea\u8981\u6c42\u6295\u5f71\u5728\u5176\u76ee\u6807\u4e0a&#034;\u591f\u7528\u3001\u53ef\u63a7\u3001\u53ef\u68c0\u9a8c&#034;\u3002\u5b83\u628a&#034;\u8fd1\u4f3c&#034;\u4ece\u8d2c\u4e49\u8bcd\u53d8\u6210\u4e86\u4e2d\u6027\u751a\u81f3\u8912\u4e49\u7684\u65b9\u6cd5\u8bba\u9009\u62e9\u3002<\/p>\n<p>\u57fa\u4e8e\u8fd9\u56db\u6761\u6807\u51c6&#xff0c;\u4e00\u4e2a\u5e38\u89c1\u7684\u8bef\u89e3\u5e94\u5f53\u88ab\u7834\u9664&#xff1a;\u9762\u5bf9\u540c\u4e00\u4e2a\u7cfb\u7edf&#xff0c;\u5e76\u4e0d\u5b58\u5728\u552f\u4e00\u4e00\u4e2a&#034;\u6700\u63a5\u8fd1\u771f\u7406&#034;\u7684\u6a21\u578b&#xff0c;\u5176\u4ed6\u6a21\u578b\u53ea\u662f\u5b83\u7684\u7c97\u7cd9\u8fd1\u4f3c\u3002\u672c\u6846\u67b6\u53cd\u5bf9\u8fd9\u79cd\u5355\u4e00\u771f\u7406\u89c2&#xff0c;\u4e3b\u5f20\u6295\u5f71\u591a\u5143\u4e3b\u4e49&#xff08;projection pluralism&#xff09;\u3002\u7406\u7531\u5f88\u76f4\u63a5&#xff1a;\u540c\u4e00 T&#xff0c;\u5728\u4e0d\u540c\u95ee\u9898\u3001\u4e0d\u540c\u5c3a\u5ea6\u3001\u4e0d\u540c\u51b3\u7b56\u76ee\u6807\u4e0b&#xff0c;\u9700\u8981\u4e0d\u540c\u7684 \u03a0\u3002\u4ee5&#034;\u4e00\u676f\u6c34&#034;\u4e3a\u4f8b&#xff1a;\u82e5\u95ee&#034;\u5b83\u4f55\u65f6\u6cb8\u817e&#034;&#xff0c;\u70ed\u529b\u5b66\u6295\u5f71&#xff08;\u6e29\u5ea6\u3001\u538b\u5f3a\u3001\u76f8\u53d8&#xff09;\u7ed9\u51fa\u6700\u5e72\u51c0\u7684 O&#xff1b;\u82e5\u95ee&#034;\u6c34\u5206\u5b50\u5982\u4f55\u6392\u5217&#034;&#xff0c;\u7edf\u8ba1\u529b\u5b66\u6295\u5f71&#xff08;\u914d\u5206\u51fd\u6570\u3001\u6c22\u952e\u7f51\u7edc&#xff09;\u624d\u662f\u5408\u9002\u7684 \u03a0&#xff1b;\u82e5\u95ee&#034;\u5b83\u5982\u4f55\u6d41\u52a8&#034;&#xff0c;\u6d41\u4f53\u529b\u5b66\u6295\u5f71&#xff08;Navier\u2013Stokes&#xff09;\u624d\u662f\u7b54\u6848&#xff1b;\u82e5\u95ee&#034;\u5b83\u5982\u4f55\u88ab\u559d\u4e0b&#034;&#xff0c;\u5219\u4e00\u4e2a\u5b8f\u89c2\u521a\u4f53\u6295\u5f71\u8db3\u77e3\u3002\u8fd9\u56db\u4e2a\u6a21\u578b\u5f7c\u6b64\u77db\u76fe\u5417&#xff1f;\u5728\u5404\u81ea\u7684\u6295\u5f71\u5b50\u7a7a\u95f4\u5185&#xff0c;\u5b83\u4eec\u5404\u81ea\u81ea\u6d3d&#xff0c;\u5e76\u5404\u81ea\u7ed9\u51fa\u53ef\u88ab\u5b9e\u9a8c\u9a8c\u8bc1\u7684 O\u3002\u5b83\u4eec\u4e0d\u662f&#034;\u771f\u7406 vs \u8fd1\u4f3c&#034;\u7684\u5c42\u7ea7\u5173\u7cfb&#xff0c;\u800c\u662f\u540c\u4e00 T \u5728\u4e0d\u540c\u8bed\u4e49\u5b50\u7a7a\u95f4\u4e0a\u7684\u5e76\u5217\u6295\u5f71\u3002\u76f2\u76ee\u8ffd\u6c42&#034;\u552f\u4e00\u6b63\u786e\u6a21\u578b&#034;&#xff0c;\u53cd\u800c\u4f1a\u8ba9\u6211\u4eec\u5728\u9519\u8bef\u7684\u5b50\u7a7a\u95f4\u91cc\u5f92\u52b3\u5730\u5bfb\u627e\u5e76\u4e0d\u5b58\u5728\u7684 O\u3002\u6295\u5f71\u591a\u5143\u4e3b\u4e49\u7684\u5b9e\u8df5\u542b\u4e49\u662f&#xff1a;\u5f53\u67d0\u4e2a\u95ee\u9898\u957f\u671f\u5361\u4f4f\u65f6&#xff0c;\u7b2c\u4e00\u53cd\u5e94\u4e0d\u5e94\u662f&#034;\u628a\u6a21\u578b\u505a\u5f97\u66f4\u7cbe\u786e&#034;&#xff0c;\u800c\u5e94\u662f&#034;\u6362\u4e00\u4e2a\u6295\u5f71&#034;\u3002 \u8fd9\u6b63\u662f\u5e9e\u52a0\u83b1\u7559\u7ed9\u6211\u4eec\u7684\u6838\u5fc3\u65b9\u6cd5\u8bba\u9057\u4ea7&#xff0c;\u4e5f\u662f\u8fde\u63a5\u672c\u7ae0\u4e0e\u7b2c 5 \u7ae0\u5386\u53f2\u53d9\u4e8b\u7684\u6865\u6881\u3002<\/p>\n<h3>3.6 \u9519\u8bef\u6295\u5f71\u7684\u4ee3\u4ef7<\/h3>\n<p>\u7136\u800c&#xff0c;\u591a\u5143\u4e3b\u4e49\u7edd\u4e0d\u7b49\u4e8e&#034;\u968f\u4fbf\u9009\u4e2a\u6a21\u578b&#034;\u3002\u82e5\u636e\u6b64\u8d70\u5411\u53e6\u4e00\u6781\u7aef&#xff0c;\u5c31\u4f1a\u5760\u5165\u66f4\u6df1\u7684\u9677\u9631\u2014\u2014\u5728\u9519\u8bef \u03a0 \u4e0a\u5bfb\u627e\u4e0d\u5b58\u5728\u7684 O&#xff0c;\u4ee3\u4ef7\u6781\u9ad8\u3002\u6211\u4eec\u5217\u4e3e\u4e09\u4e2a\u771f\u5b9e\u8bef\u7528&#xff0c;\u4ee5\u8bf4\u660e\u6295\u5f71\u9009\u62e9\u9519\u8bef\u4e0d\u53ea\u662f&#034;\u7cbe\u5ea6\u5dee\u4e00\u70b9&#034;&#xff0c;\u800c\u662f\u4f1a\u5236\u9020\u51fa\u6839\u672c\u4e0d\u5b58\u5728\u7684 O&#xff0c;\u8fdb\u800c\u8bef\u5bfc\u51b3\u7b56&#xff1a;<\/p>\n<ul>\n<li>\u7528\u7ebf\u6027\u6a21\u578b\u62df\u5408\u672c\u8d28\u975e\u7ebf\u6027\u7cfb\u7edf\u3002 \u628a\u6d1b\u4f26\u5179\u5438\u5f15\u5b50\u7528\u7ebf\u6027\u56de\u5f52\u53bb\u63cf\u8ff0&#xff0c;\u4f1a\u5f97\u5230&#034;\u7cfb\u7edf\u8fd1\u4f3c\u7a33\u5b9a&#034;\u7684\u865a\u5047 O\u2014\u2014\u56e0\u4e3a\u7ebf\u6027 \u03a0 \u6839\u672c\u4e0d\u643a\u5e26\u6df7\u6c8c\u7ed3\u6784&#xff0c;\u540e\u679c\u662f\u5728\u8be5 \u03a0 \u4e0b\u5b8c\u5168\u6f0f\u5224\u98ce\u9669\u3002<\/li>\n<li>\u7528\u5747\u503c\u63cf\u8ff0\u591a\u5cf0\u5206\u5e03\u3002 \u628a\u53cc\u5cf0\u5206\u5e03&#xff08;\u5982\u4e24\u79cd\u987e\u5ba2\u7fa4\u4f53\u3001\u4e24\u79cd\u6c14\u5019\u6001&#xff09;\u7528\u5355\u4e00\u5747\u503c\u6982\u62ec&#xff0c;\u4f1a\u5f97\u51fa&#034;\u7cfb\u7edf\u5904\u4e8e\u4e2d\u95f4\u72b6\u6001&#034;\u7684\u9519\u89c9 O&#xff0c;\u800c\u771f\u5b9e T \u51e0\u4e4e\u4ece\u4e0d\u5728\u4e2d\u95f4\u3002\u8fd9\u662f&#034;\u9519\u8bef\u7684\u805a\u5408\u5c42\u6b21&#034;\u5bfc\u81f4\u7684 O \u5e7b\u89c9\u3002<\/li>\n<li>\u628a\u76f8\u5173\u5f53\u56e0\u679c\u3002 \u5728\u6d41\u884c\u75c5\u5b66\u6216\u91d1\u878d\u4e2d&#xff0c;\u7528\u76f8\u5173\u6027\u6295\u5f71&#xff08;\u4e00\u4e2a \u03a0&#xff09;\u53d1\u73b0\u7684&#034;\u5171\u53d8 O&#034;&#xff0c;\u5e38\u88ab\u8bef\u8bfb\u4e3a\u56e0\u679c O\u3002\u76f8\u5173\u6027 \u03a0 \u4e22\u5931\u4e86\u65b9\u5411\u4e0e\u65f6\u5e8f\u7ed3\u6784&#xff0c;\u636e\u6b64\u5e72\u9884\u5e38\u9002\u5f97\u5176\u53cd\u3002<\/li>\n<\/ul>\n<p>\u4e09\u4f8b\u7684\u5171\u540c\u6559\u8bad&#xff1a;\u6295\u5f71\u9009\u62e9\u9519\u8bef&#xff0c;\u4f1a\u5236\u9020\u51fa\u6839\u672c\u4e0d\u5b58\u5728\u7684 O\u3002\u8fd9\u6b63\u662f&#034;\u597d\u7684\u6295\u5f71&#034;\u6807\u51c6&#xff08;3.5 \u8282&#xff09;\u4e0e&#034;\u591a\u6295\u5f71\u4e09\u89d2\u9a8c\u8bc1&#034;&#xff08;7.2 \u8282&#xff09;\u5fc5\u987b\u88ab\u4e25\u683c\u6267\u884c\u7684\u539f\u56e0\u3002<\/p>\n<h3>3.7 \u91cf\u5b50\u6d4b\u91cf&#xff1a;\u5185\u7980\u7684\u4fe1\u606f\u635f\u5931<\/h3>\n<p>\u5728\u7ed3\u675f\u672c\u7ae0\u4e4b\u524d&#xff0c;\u5fc5\u987b\u8ba8\u8bba\u4e00\u4e2a\u5bf9\u6846\u67b6\u65e2\u652f\u6301\u53c8\u6311\u8845\u7684\u8303\u672c\u2014\u2014\u91cf\u5b50\u529b\u5b66\u3002\u5b83\u628a&#034;\u6295\u5f71&#034;\u4ece\u8ba4\u77e5\u9690\u55bb\u63a8\u5230\u4e86\u7269\u7406\u5b66\u7684\u6838\u5fc3\u673a\u5236&#xff1a;\u5728\u91cf\u5b50\u60c5\u5f62&#xff0c;\u6d4b\u91cf\u672c\u8eab\u5c31\u662f\u4e00\u6b21\u6295\u5f71\u3002<\/p>\n<p>\u91cf\u5b50\u6001 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u2223<\/p>\n<p>        \u03c8<\/p>\n<p>        \u27e9<\/p>\n<p>       |\\\\psi\\\\rangle<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c8<\/span><span class=\"mclose\">\u27e9<\/span><\/span><\/span><\/span><\/span> \u6d3b\u5728\u65e0\u9650\u7ef4\u7684 Hilbert \u7a7a\u95f4&#xff08;\u8fd9\u662f\u6700\u6781\u7aef\u7684 T&#xff1a;\u8fde&#034;\u4f4d\u7f6e&#034;&#034;\u52a8\u91cf&#034;\u90fd\u6ca1\u6709\u786e\u5b9a\u503c&#xff0c;\u53ea\u6709\u5e45\u503c\u4e0e\u76f8\u4f4d&#xff09;\u3002\u5f53\u6211\u4eec\u7528\u4e00\u4e2a\u53ef\u89c2\u6d4b\u91cf <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         A<\/p>\n<p>         ^<\/p>\n<p>       \\\\hat{A}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9468em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9468em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">A<\/span><\/span><span class=\"\" style=\"top: -3.2523em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1111em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u53bb\u6d4b\u91cf&#xff0c;\u6ce2\u51fd\u6570\u6309 Born \u89c4\u5219\u574d\u7f29\u5230 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         A<\/p>\n<p>         ^<\/p>\n<p>       \\\\hat{A}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.9468em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9468em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\">A<\/span><\/span><span class=\"\" style=\"top: -3.2523em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1111em\"><span class=\"mord\">^<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u7684\u67d0\u4e2a\u672c\u5f81\u6001\u4e0a&#xff0c;\u5f97\u5230\u4e00\u4e2a\u5177\u4f53\u8bfb\u6570\u3002\u8fd9\u4e2a\u8fc7\u7a0b&#xff0c;\u6570\u5b66\u4e0a\u5c31\u662f\u4e00\u4e2a\u6295\u5f71\u7b97\u5b50 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         P<\/p>\n<p>         a<\/p>\n<p>        &#061;<\/p>\n<p>        \u2223<\/p>\n<p>        a<\/p>\n<p>        \u27e9<\/p>\n<p>        \u27e8<\/p>\n<p>        a<\/p>\n<p>        \u2223<\/p>\n<p>       P_a &#061; |a\\\\rangle\\\\langle a|<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">P<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1514em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.1389em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mathnormal mtight\">a<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mclose\">\u27e9<\/span><span class=\"mopen\">\u27e8<\/span><span class=\"mord mathnormal\">a<\/span><span class=\"mord\">\u2223<\/span><\/span><\/span><\/span><\/span> \u4f5c\u7528\u5230 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u2223<\/p>\n<p>        \u03c8<\/p>\n<p>        \u27e9<\/p>\n<p>       |\\\\psi\\\\rangle<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">\u2223<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c8<\/span><span class=\"mclose\">\u27e9<\/span><\/span><\/span><\/span><\/span> \u4e0a&#xff0c;\u5e76\u4e22\u5931\u4e86\u6240\u6709\u4e0e <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        a<\/p>\n<p>       a<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\">a<\/span><\/span><\/span><\/span><\/span> \u6b63\u4ea4\u7684\u5206\u91cf\u3002\u6d4b\u91cf\u5373 \u03a0&#xff0c;\u6d4b\u91cf\u8bfb\u6570\u5373 O&#xff0c;\u800c\u88ab\u6d4b\u7cfb\u7edf <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         t<\/p>\n<p>         &gt;<\/p>\n<p>         0<\/p>\n<p>        t&gt;0<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6542em;vertical-align: -0.0391em\"><\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span> \u4e4b\u540e\u7684\u6f14\u5316\u6240\u670d\u4ece\u7684\u7edf\u8ba1\u89c4\u5f8b&#xff0c;\u5c31\u662f\u8be5 \u03a0 \u4e0b\u7684 O\u3002<\/p>\n<p>\u91cf\u5b50\u529b\u5b66\u7684\u7279\u6b8a\u4e4b\u5904&#xff0c;\u5728\u4e8e\u5b83\u5ba3\u544a&#xff1a;\u5373\u4f7f T \u672c\u8eab\u7531\u786e\u5b9a\u6027\u65b9\u7a0b&#xff08;\u859b\u5b9a\u8c14\u65b9\u7a0b&#xff09;\u6f14\u5316&#xff0c;\u6211\u4eec\u5bf9\u5b83\u7684\u6bcf\u4e00\u6b21&#034;\u7406\u89e3&#034;&#xff08;\u6d4b\u91cf&#xff09;\u90fd\u4e0d\u53ef\u907f\u514d\u5730\u3001\u4e14\u4ee5\u6982\u7387\u5f62\u5f0f\u5236\u9020\u4fe1\u606f\u635f\u5931\u3002 \u8fd9\u4e0e 2.2 \u8282&#034;\u63cf\u8ff0\u6027\u4e0d\u53ef\u5b8c\u5907&#034;\u547c\u5e94&#xff0c;\u4f46\u66f4\u5c16\u9510&#xff1a;\u5728\u91cf\u5b50\u5c42&#xff0c;\u4e0d\u53ef\u5b8c\u5907\u4e0d\u662f\u56e0\u4e3a&#034;\u6d4b\u4e0d\u51c6&#034;&#xff0c;\u800c\u662f\u56e0\u4e3a&#034;\u95ee\u4e00\u4e2a\u95ee\u9898&#xff08;\u9009\u4e00\u4e2a \u03a0&#xff09;\u5c31\u5fc5\u7136\u6392\u9664\u4e86\u5176\u4ed6\u95ee\u9898\u7684\u7b54\u6848&#034;\u3002\u6d77\u68ee\u5821\u4e0d\u786e\u5b9a\u6027\u539f\u7406&#xff0c;\u5728\u6b64\u53ef\u91cd\u8ff0\u4e3a&#xff1a;\u4f4d\u7f6e\u6295\u5f71\u4e0e\u52a8\u91cf\u6295\u5f71\u5f7c\u6b64\u975e\u5bf9\u6613&#xff0c;\u4f60\u9009\u4e86\u5176\u4e2d\u4e00\u4e2a \u03a0&#xff0c;\u53e6\u4e00\u4e2a \u03a0 \u4e0b\u7684 O \u5c31\u540c\u65f6\u53d8\u5f97\u6a21\u7cca\u3002<\/p>\n<p>\u8fd9\u7ed9\u672c\u6846\u67b6\u4e00\u4e2a\u6df1\u523b\u63d0\u9192&#xff1a;\u6295\u5f71\u7684\u4ee3\u4ef7&#xff0c;\u5728\u67d0\u4e9b\u7cfb\u7edf&#xff08;\u91cf\u5b50\u3001\u6df7\u6c8c&#xff09;\u4e2d\u662f\u5185\u7980\u7684\u3001\u6982\u7387\u6027\u7684\u3001\u4e0d\u53ef\u901a\u8fc7\u66f4\u7cbe\u5bc6\u4eea\u5668\u6d88\u9664\u7684\u3002 \u201c\u5b8c\u5168\u7406\u89e3\u201d&#xff08;2.4 \u8282\u6240\u62d2\u65a5\u7684\u6267\u5ff5&#xff09;&#xff0c;\u5728\u91cf\u5b50\u5c42\u8fde\u5b9a\u4e49\u90fd\u4e0d\u6210\u7acb\u3002\u91cf\u5b50\u529b\u5b66\u56e0\u6b64\u4e0d\u53ea\u662f\u672c\u6846\u67b6\u7684\u4e00\u4e2a\u5e94\u7528\u6848\u4f8b&#xff0c;\u800c\u662f\u5176\u6700\u7eaf\u7cb9\u7684\u5f62\u6001\u2014\u2014\u5b83\u8bc1\u660e&#xff0c;\u5373\u4fbf\u5728\u6700\u57fa\u7840\u7684\u7269\u7406\u5c42\u9762&#xff0c;\u7406\u89e3\u4e5f\u53ea\u80fd\u662f&#034;\u5728\u9009\u5b9a\u7684\u6295\u5f71\u4e2d\u8bfb\u53d6\u79e9\u5e8f&#034;&#xff0c;\u800c\u4e0d\u53ef\u80fd\u662f&#034;\u62b5\u8fbe T \u672c\u8eab&#034;\u3002\u8fd9\u4e00\u7ed3\u8bba&#xff0c;\u81ea\u7136\u5730\u628a\u6211\u4eec\u5f15\u5411\u672c\u7ae0\u7684\u5b6a\u751f\u95ee\u9898&#xff1a;\u65e2\u7136\u6295\u5f71\u4e0d\u53ef\u907f\u514d&#xff0c;\u90a3\u4e48\u6295\u5f71\u4e2d\u6240\u80fd\u7a33\u5b9a\u5448\u73b0\u7684&#034;\u79e9\u5e8f&#034;\u7a76\u7adf\u662f\u4ec0\u4e48&#xff1f;\u8fd9\u6b63\u662f\u4e0b\u4e00\u7ae0\u7684\u4e3b\u9898\u3002<\/p>\n<h2>\u7b2c 4 \u7ae0 \u53ef\u8ba1\u7b97\u79e9\u5e8f O&#xff1a;\u6df7\u6c8c\u80cc\u540e\u7684\u4e0d\u53d8\u7ed3\u6784<\/h2>\n<h3>4.1 O \u7684\u5b9a\u4e49\u4e0e\u5f62\u5f0f\u5316<\/h3>\n<p>\u5982\u679c\u8bf4 T \u662f\u6e90\u5934\u3001\u03a0 \u662f\u900f\u955c&#xff0c;\u90a3\u4e48 O \u5c31\u662f\u6211\u4eec\u5728\u900f\u955c\u4e2d\u5b9e\u9645\u770b\u89c1\u3001\u5e76\u80fd\u591f\u6293\u4f4f\u7684\u4e1c\u897f\u2014\u2014\u5b83\u662f&#034;\u79e9\u5e8f&#034;&#xff08;Order&#xff09;&#xff0c;\u662f\u5728\u67d0\u4e00\u4e2a\u6295\u5f71 \u03a0 \u4e4b\u4e0b&#xff0c;\u80fd\u591f\u88ab\u8ba1\u7b97\u3001\u88ab\u68c0\u9a8c\u3001\u88ab\u590d\u73b0\u7684\u89c4\u5f8b\u3002<\/p>\n<p>\u5fc5\u987b\u7acb\u523b\u6f84\u6e05 O \u7684\u672c\u4f53\u8bba\u8eab\u4efd&#xff1a;O \u4e0d\u662f\u7edd\u5bf9\u771f\u7406&#xff0c;\u5b83\u53ea\u662f\u5728\u67d0\u4e2a\u6295\u5f71\u4e0b\u7a33\u5b9a\u5b58\u5728\u7684\u79e9\u5e8f\u3002 \u4e00\u6761&#034;\u5b88\u6052\u5f8b&#034;\u4e4b\u6240\u4ee5\u88ab\u79f0\u4e3a\u89c4\u5f8b&#xff0c;\u4e0d\u662f\u56e0\u4e3a\u5b83\u63ed\u9732\u4e86 T \u7684\u7ec8\u6781\u672c\u8d28&#xff0c;\u800c\u662f\u56e0\u4e3a\u5728\u6240\u9009\u7684 \u03a0 \u4e0b&#xff0c;\u5b83\u5728\u53ef\u89c2\u6d4b\u7684\u7cbe\u5ea6\u4e0e\u65f6\u6bb5\u5185\u59cb\u7ec8\u6210\u7acb\u3002\u4e00\u65e6\u6211\u4eec\u5207\u6362\u6295\u5f71\u6216\u66f4\u6362\u5c3a\u5ea6&#xff0c;\u539f\u6765\u7684 O \u53ef\u80fd\u5931\u6548&#xff0c;\u800c\u65b0\u7684 O \u4f1a\u6d6e\u73b0\u3002\u56e0\u6b64&#xff0c;O \u662f\u76f8\u5bf9\u4e8e \u03a0 \u7684\u2014\u2014\u8c08\u8bba O \u800c\u4e0d\u6307\u660e\u5b83\u6240\u5728\u7684 \u03a0&#xff0c;\u5c31\u50cf\u8c08\u8bba&#034;\u9ad8\u5ea6&#034;\u800c\u4e0d\u6307\u660e\u53c2\u8003\u5e73\u9762\u4e00\u6837&#xff0c;\u662f\u6b8b\u7f3a\u7684\u3002<\/p>\n<p>\u4e3a\u4e86\u628a\u8fd9\u4e00\u76f8\u5bf9\u6027\u843d\u5230\u575a\u5b9e\u7684\u6570\u5b66\u5730\u57fa\u4e0a&#xff0c;\u6211\u4eec\u628a&#034;\u5728 \u03a0 \u4e0b\u6210\u7acb\u7684\u79e9\u5e8f O&#034;\u5f62\u5f0f\u5316\u4e3a\u6295\u5f71\u7a7a\u95f4 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Y<\/p>\n<p>       \\\\mathcal{Y}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7805em;vertical-align: -0.0972em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.0822em\">Y<\/span><\/span><\/span><\/span><\/span> \u4e0a\u7684\u4e00\u7ec4\u4e0d\u53d8\u7ed3\u6784\u3002\u6700\u4e25\u683c\u7684 O \u662f\u4e0d\u53d8\u6d4b\u5ea6&#xff1a;\u82e5 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03bd<\/p>\n<p>       \\\\nu<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0637em\">\u03bd<\/span><\/span><\/span><\/span><\/span> \u662f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        Y<\/p>\n<p>       \\\\mathcal{Y}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7805em;vertical-align: -0.0972em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.0822em\">Y<\/span><\/span><\/span><\/span><\/span> \u4e0a\u7684\u6982\u7387\u6d4b\u5ea6&#xff0c;\u4e14\u5bf9\u6295\u5f71\u540e\u7684\u6f14\u5316 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03d5<\/p>\n<p>         t<\/p>\n<p>        &#061;<\/p>\n<p>        \u03c0<\/p>\n<p>        \u2218<\/p>\n<p>         \u03a6<\/p>\n<p>         t<\/p>\n<p>        \u2218<\/p>\n<p>         \u03c0<\/p>\n<p>          \u2212<\/p>\n<p>          1<\/p>\n<p>       \\\\phi^{t}&#061;\\\\pi\\\\circ\\\\Phi^{t}\\\\circ\\\\pi^{-1}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.988em;vertical-align: -0.1944em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4445em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2218<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7936em\"><\/span><span class=\"mord\"><span class=\"mord\">\u03a6<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2218<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#xff08;\u5728 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03c0<\/p>\n<p>       \\\\pi<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><\/span><\/span><\/span><\/span> \u53ef\u9006\u7684\u5c40\u90e8\u610f\u4e49\u4e0b&#xff09;\u6ee1\u8db3 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03d5<\/p>\n<p>         #<\/p>\n<p>         t<\/p>\n<p>        \u03bd<\/p>\n<p>        &#061;<\/p>\n<p>        \u03bd<\/p>\n<p>       \\\\phi^{t}_{\\\\#}\\\\nu &#061; \\\\nu<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.2128em;vertical-align: -0.4192em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -2.4169em;margin-left: 0em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">#<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.4192em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0637em\">\u03bd<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0637em\">\u03bd<\/span><\/span><\/span><\/span><\/span>&#xff08;\u5373 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03bd<\/p>\n<p>       \\\\nu<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0637em\">\u03bd<\/span><\/span><\/span><\/span><\/span> \u5728\u6f14\u5316\u4e0b\u4fdd\u6301\u4e0d\u53d8&#xff09;&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03bd<\/p>\n<p>       \\\\nu<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0637em\">\u03bd<\/span><\/span><\/span><\/span><\/span> \u5c31\u662f\u4e00\u4e2a O\u3002\u5b88\u6052\u91cf\u662f\u8fd9\u4e00\u6982\u5ff5\u7684\u7279\u4f8b&#xff1a;\u82e5\u5b58\u5728\u4e00\u4e2a\u53ef\u6d4b\u51fd\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        f<\/p>\n<p>        :<\/p>\n<p>        Y<\/p>\n<p>        \u2192<\/p>\n<p>        R<\/p>\n<p>       f:\\\\mathcal{Y}\\\\to\\\\mathbb{R}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">:<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.7805em;vertical-align: -0.0972em\"><\/span><span class=\"mord mathcal\" style=\"margin-right: 0.0822em\">Y<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2192<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6889em\"><\/span><span class=\"mord mathbb\">R<\/span><\/span><\/span><\/span><\/span> \u4f7f\u5f97 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        f<\/p>\n<p>        (<\/p>\n<p>         \u03d5<\/p>\n<p>         t<\/p>\n<p>        (<\/p>\n<p>        y<\/p>\n<p>        )<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        f<\/p>\n<p>        (<\/p>\n<p>        y<\/p>\n<p>        )<\/p>\n<p>       f(\\\\phi^{t}(y))&#061;f(y)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0436em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">))<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u5bf9\u6240\u6709 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        t<\/p>\n<p>       t<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6151em\"><\/span><span class=\"mord mathnormal\">t<\/span><\/span><\/span><\/span><\/span> \u6210\u7acb&#xff0c;\u5219 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        f<\/p>\n<p>       f<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><\/span><\/span><\/span><\/span> \u7684\u7b49\u503c\u96c6\u5c31\u662f\u4e0d\u53d8\u96c6&#xff0c;\u800c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        f<\/p>\n<p>       f<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8889em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1076em\">f<\/span><\/span><\/span><\/span><\/span> \u672c\u8eab\u5373\u4e00\u4e2a\u5b88\u6052\u91cf\u3002\u5438\u5f15\u5b50\u5219\u662f\u53e6\u4e00\u7c7b O&#xff1a;\u5b83\u662f\u4f7f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03bd<\/p>\n<p>       \\\\nu<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.4306em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0637em\">\u03bd<\/span><\/span><\/span><\/span><\/span> \u5728\u957f\u65f6\u95f4\u6781\u9650\u4e0b\u96c6\u4e2d\u5176\u4e0a\u7684\u4e00\u7c7b\u4e0d\u53d8\u6d4b\u5ea6\u3002<\/p>\n<p>\u5f62\u5f0f\u5316\u544a\u8bc9\u6211\u4eec\u4e00\u53e5\u91cd\u8bdd&#xff1a;O \u4ece\u6765\u4e0d\u662f&#034;T \u7684\u6027\u8d28&#034;&#xff0c;\u800c\u662f&#034;<span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         (<\/p>\n<p>         \u03c0<\/p>\n<p>         ,<\/p>\n<p>          \u03d5<\/p>\n<p>          t<\/p>\n<p>         )<\/p>\n<p>        (\\\\pi,\\\\phi^{t})<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0436em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.7936em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mathnormal mtight\">t<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span> \u8fd9\u4e00\u5bf9\u7ec4\u5408\u4e0b\u7684\u4e0d\u53d8\u7ed3\u6784&#034;\u3002 \u4f46\u76f8\u5bf9\u6027\u4e0d\u7b49\u4e8e\u4efb\u610f\u6027\u3002\u4e00\u4e2a\u5408\u683c\u7684 O \u5fc5\u987b\u6ee1\u8db3\u4e09\u4e2a\u786c\u6027\u6761\u4ef6&#xff1a;\u53ef\u8ba1\u7b97&#xff08;\u80fd\u5728 \u03a0 \u4e0b\u88ab\u516c\u5f0f\u5316\u6216\u7b97\u6cd5\u5316&#xff09;\u3001\u53ef\u68c0\u9a8c&#xff08;\u80fd\u88ab\u72ec\u7acb\u89c2\u6d4b\u6216\u5b9e\u9a8c\u8bc1\u4f2a\u6216\u786e\u8bc1&#xff09;\u3001\u53ef\u590d\u73b0&#xff08;\u5728\u4e0d\u540c\u65f6\u95f4\u3001\u4e0d\u540c\u4e3b\u4f53\u3001\u4e0d\u540c\u5b9e\u73b0\u4e0b\u7ed9\u51fa\u4e00\u81f4\u7ed3\u679c&#xff09;\u3002\u8fd9\u4e09\u4e2a\u6761\u4ef6\u628a O \u4e0e&#034;\u76f4\u89c9&#034;\u201c\u8bd7\u610f\u7684\u7c7b\u6bd4\u201d\u201c\u672a\u88ab\u8bc1\u5b9e\u7684\u731c\u60f3&#034;\u533a\u5206\u5f00\u6765&#xff0c;\u4e5f\u628a\u672c\u6846\u67b6\u4e0e\u7eaf\u7cb9\u7684\u76f8\u5bf9\u4e3b\u4e49\u533a\u5206\u5f00\u6765\u3002O \u662f\u76f8\u5bf9\u7684&#xff0c;\u4f46\u4e0d\u662f\u968f\u5fc3\u6240\u6b32\u7684&#xff1b;\u5b83\u662f\u5d4c\u5165\u5728\u6295\u5f71\u4e2d\u7684\u3001\u5177\u6709\u7ecf\u9a8c\u7ea6\u675f\u7684\u79e9\u5e8f\u3002\u672c\u7ae0\u6b64\u540e\u5c06\u56f4\u7ed5\u8fd9\u4e00\u5f62\u5f0f\u5316&#xff0c;\u9010\u5c42\u5c55\u793a O \u5728\u590d\u6742\u7cfb\u7edf\u4e2d\u5982\u4f55&#034;\u53d8\u5f62\u201d\u3001\u5982\u4f55&#034;\u5206\u5c42&#034;\u3001\u53c8\u5982\u4f55\u6210\u4e3a\u6df7\u6c8c\u80cc\u540e\u771f\u6b63\u7684\u79e9\u5e8f\u5185\u6838\u3002<\/p>\n<h3>4.2 \u4e8c\u4f53\u4e0e\u4e09\u4f53&#xff1a;O \u7684\u5f62\u5f0f\u8f6c\u6362<\/h3>\n<p>\u7528 N \u4f53\u95ee\u9898\u6700\u80fd\u8bf4\u660e O \u5982\u4f55\u968f\u6295\u5f71\u4e0e\u95ee\u9898\u5c42\u6b21\u800c&#034;\u53d8\u5f62&#034;\u3002\u5728 N&#061;2 \u7684\u60c5\u5f62\u4e0b&#xff0c;\u725b\u987f\u6295\u5f71\u6781\u5176\u6210\u529f&#xff1a;\u5b58\u5728\u5c01\u95ed\u5f62\u5f0f\u7684\u692d\u5706\u8f68\u9053&#xff0c;\u5b58\u5728\u4e25\u683c\u7684\u80fd\u91cf\u5b88\u6052\u4e0e\u89d2\u52a8\u91cf\u5b88\u6052&#xff0c;\u5b58\u5728\u5f00\u666e\u52d2\u7b2c\u4e09\u5b9a\u5f8b\u628a\u5468\u671f\u4e0e\u534a\u957f\u8f74\u8054\u7cfb\u8d77\u6765\u3002\u6b64\u65f6 O \u662f&#034;\u8f68\u9053\u5c42\u9762&#034;\u7684\u2014\u2014\u4f60\u4e0d\u4ec5\u80fd\u8bf4\u51fa\u4e24\u9897\u5929\u4f53&#034;\u7ed5\u8f6c&#034;&#xff0c;\u8fd8\u80fd\u7cbe\u786e\u5199\u51fa\u5b83\u4eec\u6bcf\u4e00\u523b\u7684\u4f4d\u7f6e\u3002O \u5728\u8fd9\u91cc\u662f\u5b8c\u6574\u3001\u663e\u5f0f\u3001\u53ef\u9010\u9879\u9a8c\u8bc1\u7684\u3002<\/p>\n<p>\u4f46\u5728 N&#061;3 \u7684\u60c5\u5f62\u4e0b&#xff0c;\u7cbe\u786e\u8f68\u9053\u5728\u539f\u5219\u4e0a\u4e0d\u53ef\u6c42\u5f97&#xff08;\u4e0d\u5b58\u5728\u901a\u7528\u7684\u521d\u7b49\u51fd\u6570\u901a\u89e3&#xff09;&#xff0c;\u957f\u671f\u8f68\u9053\u5bf9\u521d\u59cb\u6761\u4ef6\u6307\u6570\u654f\u611f\u3002\u8fd9\u662f\u5426\u610f\u5473\u7740\u4e09\u4f53\u7cfb\u7edf\u4e2d&#034;\u6ca1\u6709 O&#034;&#xff1f;\u7edd\u975e\u5982\u6b64\u3002\u5e9e\u52a0\u83b1\u4e0e\u540e\u4e16\u52a8\u529b\u5b66\u5bb6\u53d1\u73b0&#xff0c;\u4e09\u4f53\u7cfb\u7edf\u5728\u76f8\u7a7a\u95f4\u4e2d\u4ecd\u7136\u8574\u542b\u7740\u6781\u5176\u4e30\u5bcc\u3001\u53ef\u8ba1\u7b97\u7684 O&#xff0c;\u53ea\u662f\u5b83\u4eec\u6362\u4e86\u5f62\u5f0f\u3001\u6362\u4e86\u5c42\u6b21&#xff1a;<\/p>\n<ul>\n<li>\u5468\u671f\u8f68\u9053\u7684\u5b58\u5728\u6027&#xff1a;\u5c3d\u7ba1\u4e00\u822c\u8f68\u8ff9\u6df7\u4e71&#xff0c;\u4f46\u76f8\u7a7a\u95f4\u4e2d\u4ecd&#034;\u9576\u5d4c&#034;\u7740\u65e0\u7a77\u591a\u6761\u5468\u671f\u8f68\u9053&#xff08;\u5e9e\u52a0\u83b1\u6700\u5148\u8bba\u8bc1\u5176\u5b58\u5728&#xff09;&#xff0c;\u5b83\u4eec\u662f\u7cfb\u7edf\u7ed3\u6784\u6027\u7684&#034;\u9aa8\u67b6&#034;&#xff1b;<\/li>\n<li>\u7a33\u5b9a\u6027\u8fb9\u754c&#xff1a;\u53ef\u4ee5\u8ba1\u7b97\u54ea\u4e9b\u521d\u59cb\u6784\u578b\u5bfc\u81f4\u675f\u7f1a\u8fd0\u52a8\u3001\u54ea\u4e9b\u5bfc\u81f4\u6563\u5c04\u9003\u9038&#xff0c;\u5373\u6240\u8c13 Hill \u7a33\u5b9a\u533a\u57df&#xff1b;<\/li>\n<li>Lyapunov \u6307\u6570&#xff1a;\u53ef\u4ee5\u91cf\u5316\u8f68\u8ff9\u5bf9\u521d\u59cb\u6761\u4ef6\u7684\u654f\u611f\u7a0b\u5ea6&#xff0c;\u7ed9\u51fa\u4e00\u4e2a&#034;\u53ef\u9884\u6d4b\u65f6\u95f4\u5c3a\u5ea6&#034;&#xff08;Lyapunov \u65f6\u95f4&#xff09;&#xff1b;<\/li>\n<li>\u7edf\u8ba1\u5206\u5e03&#xff1a;\u5728\u957f\u65f6\u95f4\u6781\u9650\u4e0b&#xff0c;\u5927\u91cf\u8f68\u8ff9\u5728\u76f8\u7a7a\u95f4\u4e2d\u670d\u4ece\u53ef\u63cf\u8ff0\u7684\u7edf\u8ba1\u5206\u5e03&#xff08;\u904d\u5386\u6027\u5047\u8bbe\u4e0b\u7684\u5fae\u6b63\u5219\u5206\u5e03&#xff09;\u3002<\/li>\n<\/ul>\n<p>\u8fd9\u63ed\u793a\u4e86\u4e00\u6761\u8d2f\u7a7f\u5168\u6587\u7684\u6df1\u523b\u89c4\u5f8b&#xff1a;\u5f53\u67d0\u4e00\u5c42\u6b21\u4e0a\u7684 O \u6d88\u5931&#xff08;\u5982\u4e09\u4f53\u7684\u7cbe\u786e\u8f68\u9053&#xff09;&#xff0c;\u5e76\u4e0d\u7b49\u4e8e\u79e9\u5e8f\u672c\u8eab\u6d88\u5931&#xff0c;\u800c\u662f\u63d0\u793a\u6211\u4eec\u2014\u2014O \u5728\u66f4\u9ad8\u6216\u66f4\u4f4e\u7684\u5c42\u6b21\u4e0a&#xff0c;\u4ee5\u53e6\u4e00\u79cd\u5f62\u5f0f\u5b58\u5728\u3002 \u4ece&#034;\u8f68\u9053 O&#034;\u9000\u5230&#034;\u7ed3\u6784 O&#034;\u4e0e&#034;\u7edf\u8ba1 O&#034;&#xff0c;\u6b63\u662f\u79e9\u5e8f\u5728\u6295\u5f71\u53d8\u6362\u4e0b\u7684\u5f62\u5f0f\u8f6c\u6362\u3002\u8fd9\u4e00\u8ba4\u8bc6&#xff0c;\u662f\u628a&#034;\u6df7\u6c8c&#034;\u4ece\u6050\u60e7\u5bf9\u8c61\u8f6c\u5316\u4e3a\u7814\u7a76\u5bf9\u8c61\u7684\u5173\u952e\u3002<\/p>\n<h3>4.3 \u4ece\u5355\u6446\u5230\u53cc\u6446&#xff1a;O \u968f\u590d\u6742\u5ea6\u7684\u53d8\u5f62<\/h3>\n<p>\u62bd\u8c61\u6846\u67b6\u9700\u8981\u843d\u5230\u5177\u4f53\u7684\u8ba1\u7b97\u5bf9\u8c61\u4e0a\u624d\u53ef\u88ab\u68c0\u9a8c\u3002\u672c\u8282\u7528\u4e00\u4e2a\u4eba\u4eba\u53ef\u590d\u73b0\u7684\u8303\u4f8b\u2014\u2014\u5355\u6446\u4e0e\u53cc\u6446\u2014\u2014\u6f14\u793a O \u5982\u4f55\u968f\u7cfb\u7edf\u590d\u6742\u5ea6\u7684\u4e0a\u5347\u800c&#034;\u53d8\u5f62&#034;&#xff0c;\u4ece\u800c\u628a\u524d\u6587\u7684\u8bba\u70b9\u53d8\u6210\u53ef\u8ba1\u7b97\u7684\u5b9e\u4f8b\u3002<\/p>\n<p>\u5355\u6446&#xff1a;\u8f68\u9053\u5c42\u9762\u7684\u5b8c\u6574 O\u3002 \u5355\u6446\u7684\u8fd0\u52a8\u65b9\u7a0b\u4e3a <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         \u03b8<\/p>\n<p>         \u00a8<\/p>\n<p>        &#043;<\/p>\n<p>        (<\/p>\n<p>        g<\/p>\n<p>        \/<\/p>\n<p>        L<\/p>\n<p>        )<\/p>\n<p>        sin<\/p>\n<p>        \u2061<\/p>\n<p>        \u03b8<\/p>\n<p>        &#061;<\/p>\n<p>        0<\/p>\n<p>       \\\\ddot{\\\\theta}&#043;(g\/L)\\\\sin\\\\theta&#061;0<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.0146em;vertical-align: -0.0833em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9313em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.1667em\"><span class=\"mord\">\u00a8<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mord\">\/<\/span><span class=\"mord mathnormal\">L<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">sin<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">0<\/span><\/span><\/span><\/span><\/span>\u3002\u5728\u5c0f\u5e45\u8fd1\u4f3c\u4e0b\u5b83\u9000\u5316\u4e3a\u7b80\u8c10\u632f\u52a8&#xff0c;\u5b58\u5728\u7cbe\u786e\u7684\u89e3\u6790\u89e3 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03b8<\/p>\n<p>        (<\/p>\n<p>        t<\/p>\n<p>        )<\/p>\n<p>        &#061;<\/p>\n<p>        A<\/p>\n<p>        cos<\/p>\n<p>        \u2061<\/p>\n<p>        (<\/p>\n<p>        \u03c9<\/p>\n<p>        t<\/p>\n<p>        &#043;<\/p>\n<p>        \u03d5<\/p>\n<p>        )<\/p>\n<p>       \\\\theta(t)&#061;A\\\\cos(\\\\omega t&#043;\\\\phi)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mclose\">)<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">A<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mop\">cos<\/span><span class=\"mopen\">(<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c9<\/span><span class=\"mord mathnormal\">t<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">\u03d5<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5468\u671f <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        T<\/p>\n<p>        &#061;<\/p>\n<p>        2<\/p>\n<p>        \u03c0<\/p>\n<p>          L<\/p>\n<p>          \/<\/p>\n<p>          g<\/p>\n<p>       T&#061;2\\\\pi\\\\sqrt{L\/g}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.24em;vertical-align: -0.305em\"><\/span><span class=\"mord\">2<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">\u03c0<\/span><span class=\"mord sqrt\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.935em\"><span class=\"svg-align\" style=\"top: -3.2em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"mord\" style=\"padding-left: 1em\"><span class=\"mord mathnormal\">L<\/span><span class=\"mord\">\/<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><\/span><\/span><span class=\"\" style=\"top: -2.895em\"><span class=\"pstrut\" style=\"height: 3.2em\"><\/span><span class=\"hide-tail\" style=\"min-width: 1.02em;height: 1.28em\"><\/p>\n<p>           <\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.305em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u4e0e\u632f\u5e45\u65e0\u5173&#xff08;\u7b49\u65f6\u6027&#xff09;\u3002\u6b64\u65f6 O \u662f\u5b8c\u6574\u3001\u663e\u5f0f\u3001\u8f68\u9053\u5c42\u9762\u7684&#xff1a;\u4f60\u4e0d\u4ec5\u80fd\u8bf4&#034;\u6446\u4f1a\u6765\u56de\u6446\u52a8&#034;&#xff0c;\u8fd8\u80fd\u7cbe\u786e\u5199\u51fa\u6bcf\u4e00\u77ac\u7684\u89d2\u4f4d\u7f6e\u3002\u5b88\u6052\u91cf&#xff08;\u673a\u68b0\u80fd <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        E<\/p>\n<p>        &#061;<\/p>\n<p>          1<\/p>\n<p>          2<\/p>\n<p>        m<\/p>\n<p>         L<\/p>\n<p>         2<\/p>\n<p>          \u03b8<\/p>\n<p>          \u02d9<\/p>\n<p>         2<\/p>\n<p>        &#043;<\/p>\n<p>        m<\/p>\n<p>        g<\/p>\n<p>        L<\/p>\n<p>        (<\/p>\n<p>        1<\/p>\n<p>        \u2212<\/p>\n<p>        cos<\/p>\n<p>        \u2061<\/p>\n<p>        \u03b8<\/p>\n<p>        )<\/p>\n<p>       E&#061;\\\\tfrac12 mL^{2}\\\\dot{\\\\theta}^{2}&#043;mgL(1-\\\\cos\\\\theta)<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0576em\">E<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.2763em;vertical-align: -0.345em\"><\/span><span class=\"mord\"><span class=\"mopen nulldelimiter\"><\/span><span class=\"mfrac\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8451em\"><span class=\"\" style=\"top: -2.655em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><span class=\"\" style=\"top: -3.23em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"frac-line\" style=\"border-bottom-width: 0.04em\"><\/span><\/span><span class=\"\" style=\"top: -3.394em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">1<\/span><\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.345em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mord\"><span class=\"mord mathnormal\">L<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mord\"><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.9313em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><\/span><span class=\"\" style=\"top: -3.2634em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.0556em\"><span class=\"mord\">\u02d9<\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">2<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">&#043;<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord mathnormal\">m<\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">g<\/span><span class=\"mord mathnormal\">L<\/span><span class=\"mopen\">(<\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mop\">cos<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">\u03b8<\/span><span class=\"mclose\">)<\/span><\/span><\/span><\/span><\/span>&#xff09;\u9489\u6b7b\u4e86\u6f14\u5316&#xff0c;\u5bf9\u5e94\u4e00\u4e2a\u4e00\u7ef4\u7684\u4e0d\u53d8\u6d41\u5f62\u3002\u5355\u6446&#xff0c;\u662f&#034;\u4f4e\u7ef4\u53ef\u79ef\u7cfb\u7edf\u4e2d O \u5b8c\u6574&#034;\u7684\u8303\u672c\u3002<\/p>\n<p>\u53cc\u6446&#xff1a;\u6df7\u6c8c\u767b\u573a\u3002 \u628a\u4e24\u4e2a\u6446\u4e32\u8054&#xff0c;\u5f97\u5230\u53cc\u6446\u7cfb\u7edf\u3002\u5176\u8fd0\u52a8\u65b9\u7a0b\u867d\u4ecd\u662f\u786e\u5b9a\u7684&#xff08;\u62c9\u683c\u6717\u65e5\u65b9\u7a0b\u7ed9\u51fa\u56db\u4e2a\u4e00\u9636\u65b9\u7a0b&#xff09;&#xff0c;\u5374\u5df2\u7ecf\u5728\u6570\u503c\u4e0a\u8868\u73b0\u51fa\u5178\u578b\u7684\u6df7\u6c8c&#xff1a;\u521d\u59cb\u89d2\u5ea6\u76f8\u5dee <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         10<\/p>\n<p>          \u2212<\/p>\n<p>          3<\/p>\n<p>       10^{-3}<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8141em\"><\/span><span class=\"mord\">1<\/span><span class=\"mord\"><span class=\"mord\">0<\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.8141em\"><span class=\"\" style=\"top: -3.063em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\"><span class=\"mord mtight\">\u2212<\/span><span class=\"mord mtight\">3<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u5f27\u5ea6\u7684\u4e24\u6761\u8f68\u8ff9&#xff0c;\u5728\u6570\u79d2\u4e4b\u5185\u4fbf\u5f7b\u5e95\u5206\u79bb\u3002\u53cc\u6446\u6ca1\u6709\u7b80\u5355\u7684\u5c01\u95ed\u8f68\u9053\u89e3&#xff0c;\u957f\u671f\u884c\u4e3a\u4e0d\u53ef\u9884\u6d4b\u2014\u2014\u5b83\u6b63\u662f&#034;\u4e09\u4f53\u7cbe\u795e&#034;\u5728\u684c\u9762\u4e0a\u7684\u5fae\u578b\u4f53\u73b0\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f&#xff0c;\u53cc\u6446\u7684\u6df7\u6c8c\u5e76\u975e\u56e0\u4e3a\u65b9\u7a0b&#034;\u9519\u4e86&#034;\u6216&#034;\u968f\u673a&#034;&#xff0c;\u800c\u662f\u56e0\u4e3a\u786e\u5b9a\u6027\u7684\u975e\u7ebf\u6027\u8026\u5408\u672c\u8eab\u4ea7\u751f\u4e86\u5bf9\u521d\u59cb\u6761\u4ef6\u7684\u6307\u6570\u654f\u611f&#xff0c;\u4e0e 2.2 \u8282&#034;\u4e0d\u53ef\u9884\u6d4b\u6027\u5185\u751f\u4e8e T \u7684\u52a8\u529b\u5b66\u7ed3\u6784&#034;\u5b8c\u5168\u543b\u5408\u3002<\/p>\n<p>\u5728\u53cc\u6446\u4e2d\u91cd\u65b0\u5b9a\u4f4d O\u3002 \u53cc\u6446&#034;\u6ca1\u6709\u8f68\u9053\u5c42\u9762\u7684 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space&#xff09;\u601d\u60f3\u7684\u6210\u719f\u8fd0\u7528&#xff08;\u5176\u96cf\u5f62\u53ef\u8ffd\u6eaf\u81f3\u73bb\u5c14\u5179\u66fc&#xff0c;\u540e\u88ab\u5409\u5e03\u65af\u5f62\u5f0f\u5316\u4e3a\u7edf\u8ba1\u529b\u5b66\u7684\u57fa\u7840&#xff09;\u3002<\/p>\n<p>\u5728\u76f8\u7a7a\u95f4\u4e2d&#xff0c;\u7cfb\u7edf\u7684\u6bcf\u4e00\u4e2a\u77ac\u65f6\u72b6\u6001\u5bf9\u5e94\u4e00\u4e2a\u70b9&#xff0c;\u6f14\u5316\u5bf9\u5e94\u4e00\u6761\u8f68\u8ff9\u3002\u5e9e\u52a0\u83b1\u4e0d\u518d\u8ffd\u95ee&#034;\u8fd9\u6761\u8f68\u8ff9\u5177\u4f53\u53bb\u54ea&#034;&#xff0c;\u800c\u662f\u8ffd\u95ee&#xff1a;\u8fd9\u4e9b\u8f68\u8ff9\u6574\u4f53\u4e0a\u6784\u6210\u4ec0\u4e48\u7ed3\u6784&#xff1f;\u4ed6\u7531\u6b64\u5f00\u521b\u4e86\u5fae\u5206\u65b9\u7a0b\u7684\u5b9a\u6027\u7406\u8bba&#xff08;qualitative theory&#xff09;\u2014\u2014\u4e0d\u89e3\u65b9\u7a0b&#xff0c;\u800c\u7814\u7a76\u65b9\u7a0b\u89e3\u7684\u62d3\u6251\u4e0e\u51e0\u4f55\u6027\u8d28\u3002<\/p>\n<p>\u5728\u8fd9\u4e00\u89c6\u89d2\u4e0b&#xff0c;\u5e9e\u52a0\u83b1\u53d1\u73b0\u4e86\u65e5\u540e\u6df7\u6c8c\u7406\u8bba\u7684\u6807\u5fd7\u6027\u7ed3\u6784&#xff1a;\u540c\u5bbf\u8f68\u9053&#xff08;homoclinic orbit&#xff09;\u4e0e\u5f02\u5bbf\u8f68\u9053&#xff08;heteroclinic orbit&#xff09;\u3002\u7b80\u5355\u6765\u8bf4&#xff0c;\u67d0\u4e9b\u7279\u6b8a\u8f68\u8ff9\u4f1a\u4ece\u4e0d\u7a33\u5b9a\u5e73\u8861\u70b9\u9644\u8fd1\u51fa\u53d1&#xff0c;\u7ed5\u884c\u4e00\u5708\u540e\u53c8\u65e0\u9650\u903c\u8fd1\u8be5\u5e73\u8861\u70b9\u2014\u2014\u5b83&#034;\u79bb\u5f00\u5bb6\u53c8\u56de\u5bb6&#034;&#xff0c;\u5374\u6c38\u4e0d\u771f\u6b63\u62b5\u8fbe\u3002\u5f53\u591a\u6761\u8fd9\u6837\u7684\u8f68\u9053\u76f8\u4e92\u4ea4\u9519&#xff0c;\u76f8\u7a7a\u95f4\u5c31\u88ab&#034;\u7f20\u7ed3&#034;&#xff08;tangle&#xff09;\u6210\u6781\u5176\u590d\u6742\u7684\u6a2a\u622a\u540c\u5bbf\u7ed3\u6784\u3002\u6a2a\u622a\u540c\u5bbf\u70b9\u7684\u5b58\u5728&#xff0c;\u76f4\u63a5\u8574\u542b\u4e86\u8f68\u8ff9\u5bf9\u521d\u59cb\u6761\u4ef6\u7684\u654f\u611f\u4f9d\u8d56\u2014\u2014\u4e5f\u5c31\u662f\u6df7\u6c8c\u3002<\/p>\n<p>\u8fd9\u4e00\u53d1\u73b0\u7684\u65b9\u6cd5\u8bba\u610f\u4e49\u5728\u4e8e&#xff1a;\u5e9e\u52a0\u83b1\u7528&#034;\u51e0\u4f55\u7ed3\u6784&#034;\u66ff\u4ee3\u4e86&#034;\u89e3\u6790\u516c\u5f0f&#034;&#xff0c;\u4f5c\u4e3a\u7406\u89e3\u590d\u6742\u7cfb\u7edf\u7684\u65b0 O\u3002 \u4ed6\u8bc1\u660e\u4e86&#xff0c;\u5373\u4fbf\u9010\u70b9\u8f68\u9053\u4e0d\u53ef\u6c42&#xff0c;\u7cfb\u7edf\u6574\u4f53\u4ecd\u53ef\u88ab\u51e0\u4f55\u5730\u7406\u89e3\u3002\u8fd9\u6b63\u662f T-\u03a0-O \u6846\u67b6\u4e2d&#034;\u6362\u6295\u5f71\u3001\u6362\u5c42\u6b21\u627e O&#034;\u7684\u6700\u65e9\u3001\u6700\u5149\u8f89\u7684\u5b9e\u8df5&#xff0c;\u4e5f\u628a\u7b2c 4 \u7ae0&#034;\u6362\u5c42\u6b21\u627e O&#034;\u7684\u62bd\u8c61\u5904\u65b9&#xff0c;\u951a\u5b9a\u5728\u4e86\u4e00\u4e2a\u771f\u5b9e\u7684\u5386\u53f2\u65f6\u523b\u3002<\/p>\n<h3>5.3 \u4ece&#034;\u8f68\u9053 O&#034;\u5230&#034;\u7ed3\u6784 O&#034;&#xff1a;\u95ee\u9898\u5c42\u6b21\u7684\u6539\u53d8<\/h3>\n<p>\u628a\u672c\u7ae0\u4e0e\u7b2c 4 \u7ae0\u8054\u7cfb\u8d77\u6765&#xff0c;\u53ef\u4ee5\u770b\u5230\u5e9e\u52a0\u83b1\u5b8c\u6210\u7684\u662f\u4e00\u6b21\u7814\u7a76\u5bf9\u8c61\u5c42\u9762\u7684\u8303\u5f0f\u8dc3\u8fc1&#xff1a;<\/p>\n<ul>\n<li>\u65e7\u8303\u5f0f\u95ee&#xff1a;\u201c\u7ed9\u5b9a\u521d\u59cb\u6761\u4ef6&#xff0c;\u7cfb\u7edf\u4e0b\u4e00\u523b\u5728\u54ea&#xff1f;\u201d&#xff08;\u8f68\u9053\u5c42\u9762\u7684 O&#xff09;<\/li>\n<li>\u65b0\u8303\u5f0f\u95ee&#xff1a;\u201c\u7cfb\u7edf\u7684\u53ef\u80fd\u72b6\u6001\u7a7a\u95f4&#xff0c;\u6574\u4f53\u4e0a\u957f\u4ec0\u4e48\u6837&#xff1f;\u54ea\u4e9b\u533a\u57df\u7a33\u5b9a\u3001\u54ea\u4e9b\u8106\u5f31\u3001\u54ea\u4e9b\u5f7c\u6b64\u8fde\u901a&#xff1f;\u201d&#xff08;\u7ed3\u6784\u5c42\u9762\u7684 O&#xff09;<\/li>\n<\/ul>\n<p>\u8fd9\u4e00\u8dc3\u8fc1\u7684\u8981\u5bb3&#xff0c;\u662f\u91cd\u65b0\u5b9a\u4e49\u4e86\u4ec0\u4e48\u53eb&#034;\u7406\u89e3\u4e86\u7cfb\u7edf&#034;\u3002\u5728\u4f20\u7edf\u89c2\u5ff5\u91cc&#xff0c;\u7406\u89e3 &#061; \u80fd\u7b97\u51fa\u6bcf\u4e00\u6761\u8f68\u9053&#xff1b;\u5e9e\u52a0\u83b1\u5c55\u793a\u51fa&#xff0c;\u7406\u89e3 &#061; \u80fd\u523b\u753b\u72b6\u6001\u7a7a\u95f4\u7684\u62d3\u6251\u4e0e\u52a8\u529b\u5b66\u7ed3\u6784\u3002\u5f53\u67d0\u4e00\u5c42\u6b21\u7684 O&#xff08;\u9010\u70b9\u8f68\u9053&#xff09;\u5931\u6548\u65f6&#xff0c;\u63d0\u5347\u62bd\u8c61\u5c42\u6b21&#xff0c;\u5728\u65b0\u7684\u5c42\u6b21\u4e0a\u5bfb\u627e\u4ecd\u7136\u6210\u7acb\u7684 O&#xff08;\u76f8\u7a7a\u95f4\u7ed3\u6784&#xff09;&#xff0c;\u53cd\u800c\u83b7\u5f97\u4e86\u66f4\u672c\u8d28\u7684\u7406\u89e3\u3002<\/p>\n<p>\u8fd9\u4e00\u601d\u60f3\u5728\u8ba1\u7b97\u673a\u79d1\u5b66\u3001\u63a7\u5236\u8bba\u3001\u4e43\u81f3\u73b0\u4ee3\u673a\u5668\u5b66\u4e60\u4e2d\u53cd\u590d\u56de\u54cd\u3002\u4f8b\u5982&#xff0c;\u7814\u7a76\u4e00\u4e2a\u9ad8\u7ef4\u4f18\u5316\u95ee\u9898\u7684\u635f\u5931\u666f\u89c2&#xff08;loss landscape&#xff09;\u65f6&#xff0c;\u6211\u4eec\u5f88\u5c11\u53bb\u8ffd\u8e2a&#034;\u8fd9\u6761\u5177\u4f53\u8f68\u8ff9\u600e\u4e48\u8d70\u5230\u6781\u5c0f\u503c&#034;&#xff0c;\u800c\u662f\u53bb\u523b\u753b landscape \u7684\u6574\u4f53\u51e0\u4f55&#xff08;\u6709\u591a\u5c11\u978d\u70b9\u3001\u6781\u5c0f\u503c\u6709\u591a\u5e73\u5766\u3001\u662f\u5426\u5b58\u5728\u7a84\u8c37&#xff09;\u2014\u2014\u8fd9\u672c\u8d28\u4e0a\u5c31\u662f\u5728\u76f8\u7a7a\u95f4\/\u53c2\u6570\u7a7a\u95f4\u4e2d\u5bfb\u627e&#034;\u7ed3\u6784\u5c42\u9762\u7684 O&#034;\u3002\u5e9e\u52a0\u83b1\u5728\u4e00\u767e\u4e09\u5341\u5e74\u524d\u79cd\u4e0b\u7684\u79cd\u5b50&#xff0c;\u4eca\u5929\u4ecd\u5728\u5f00\u82b1\u3002\u5177\u4f53\u5230\u64cd\u4f5c\u5c42\u9762&#xff0c;\u5e9e\u52a0\u83b1\u7684\u9057\u4ea7\u51dd\u7ed3\u6210\u4e00\u5957&#034;\u6539\u53d8\u95ee\u9898\u5c42\u6b21&#034;\u7684\u624b\u518c&#xff1a;\u4e0d\u770b\u5355\u4e2a\u8f68\u8ff9\u770b\u5438\u5f15\u5b50&#xff1b;\u4e0d\u770b\u7cbe\u786e\u503c\u770b\u7edf\u8ba1\u5206\u5e03&#xff1b;\u4e0d\u770b\u77ed\u671f\u6f14\u5316\u770b\u957f\u671f\u7a33\u5b9a\u6027\u4e0e\u5206\u5c94&#xff08;bifurcation&#xff09;&#xff1b;\u4e0d\u770b\u6bcf\u4e2a\u7c92\u5b50\u770b\u5e8f\u53c2\u91cf&#xff08;order parameter&#xff09;\u4e0e\u6807\u5ea6\u5f8b\u3002\u6240\u6709\u8fd9\u4e9b&#034;\u770b X \u800c\u4e0d\u770b Y&#034;\u7684\u53e5\u5f0f&#xff0c;\u672c\u8d28\u4e0a\u90fd\u662f\u540c\u4e00\u53e5\u8bdd\u7684\u53d8\u4f53&#xff1a;\u5f53\u4f4e\u5c42\u6b21\u7684 O \u627e\u4e0d\u5230&#xff0c;\u5c31\u5347\u7ef4\u6216\u964d\u7ef4&#xff0c;\u5230\u7ed3\u6784\u5c42\u9762\u53bb\u91cd\u65b0\u5b9a\u4f4d O\u3002 \u8fd9\u4e5f\u4e3a\u7b2c 7 \u7ae0\u7684\u65b9\u6cd5\u8bba\u63d0\u4f9b\u4e86\u6700\u65e9\u7684\u5386\u53f2\u539f\u578b\u3002<\/p>\n<h3>5.4 \u6df7\u6c8c\u7684\u73b0\u4ee3\u8c31\u7cfb<\/h3>\n<p>\u5e9e\u52a0\u83b1\u7684\u5929\u624d\u6d1e\u89c1&#xff0c;\u5e76\u6ca1\u6709\u7acb\u5373\u88ab\u6574\u4e2a\u79d1\u5b66\u754c\u5438\u6536\u3002\u672c\u8282\u8865\u5168\u5386\u53f2\u94fe\u6761&#xff0c;\u5c55\u793a&#034;\u7ed3\u6784\u5c42\u9762\u7684 O&#034;\u5982\u4f55\u5728\u8fd1\u767e\u5e74\u95f4\u88ab\u4e00\u5c42\u5c42\u63d0\u70bc&#xff0c;\u4f7f\u6df7\u6c8c\u4ece&#034;\u6570\u5b66\u602a\u8c61&#034;\u53d8\u4e3a&#034;\u79d1\u5b66\u8303\u5f0f&#034;\u3002<\/p>\n<ul>\n<li>\u5e9e\u52a0\u83b1&#xff08;1890 \u524d\u540e&#xff09;&#xff1a;\u53d1\u73b0\u540c\u5bbf\u7f20\u7ed3&#xff0c;\u5f00\u542f\u5b9a\u6027\u7406\u8bba&#xff08;5.2 \u8282&#xff09;\u3002<\/li>\n<li>\u4f2f\u514b\u970d\u592b&#xff08;G. D. Birkhoff, 1927&#xff09;&#xff1a;\u5efa\u7acb\u5168\u5c40\u52a8\u529b\u7cfb\u7edf\u7406\u8bba&#xff0c;\u8bc1\u660e\u5468\u671f\u8f68\u9053\u7684\u5b58\u5728\u5b9a\u7406&#xff0c;\u628a\u5e9e\u52a0\u83b1\u7684\u51e0\u4f55\u76f4\u89c9\u7cfb\u7edf\u5316\u4e3a\u4e25\u683c\u6570\u5b66\u3002<\/li>\n<li>KAM \u7406\u8bba&#xff08;\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b 1954&#xff0c;\u963f\u8bfa\u5fb7\u3001\u83ab\u6cfd 1960s&#xff09;&#xff1a;\u8bc1\u660e\u5728\u5f31\u6270\u52a8\u4e0b&#xff0c;\u53ef\u79ef\u7cfb\u7edf\u7684\u7edd\u5927\u90e8\u5206\u4e0d\u53d8\u73af\u9762\u4ecd\u5b58\u6d3b&#xff0c;\u4f46\u5176\u4f59\u533a\u57df\u9677\u5165\u6df7\u6c8c\u2014\u2014\u7ed9\u51fa\u4e86&#034;\u89c4\u5219\u4e0e\u6df7\u6c8c\u5982\u4f55\u5171\u5b58&#034;\u7684 O\u3002<\/li>\n<li>\u6d1b\u4f26\u5179&#xff08;1963&#xff09;&#xff1a;\u7528\u622a\u65ad\u5bf9\u6d41\u6a21\u578b\u6570\u503c\u53d1\u73b0\u5947\u602a\u5438\u5f15\u5b50&#xff0c;\u628a\u6df7\u6c8c\u5e26\u5165\u53ef\u8ba1\u7b97\u65f6\u4ee3&#xff08;4.4 \u8282&#xff09;\u3002<\/li>\n<li>\u674e\u5929\u5ca9\u2013\u7ea6\u514b&#xff08;1975&#xff09;&#xff1a;\u4ee5 \u201cPeriod Three Implies Chaos\u201d \u7ed9\u51fa\u6df7\u6c8c\u7684\u4e25\u683c\u5b9a\u4e49&#xff0c;\u6df7\u6c8c\u6b63\u5f0f\u8fdb\u5165\u6570\u5b66\u8bed\u8a00\u3002<\/li>\n<li>\u8d39\u6839\u9c8d\u59c6&#xff08;1978&#xff09;&#xff1a;\u53d1\u73b0\u500d\u5468\u671f\u5206\u5c94\u7684\u666e\u9002\u5e38\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         \u03b4<\/p>\n<p>         \u2248<\/p>\n<p>         4.669<\/p>\n<p>        \\\\delta\\\\approx 4.669<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0379em\">\u03b4<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">4.669<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u63ed\u793a\u4e0d\u540c\u7cfb\u7edf\u6df7\u6c8c\u9053\u8def\u7684\u666e\u9002 O&#xff08;4.5 \u8282\u6807\u5ea6\u5f8b\u7684\u5178\u8303&#xff09;\u3002<\/li>\n<\/ul>\n<p>\u8fd9\u6761\u8c31\u7cfb\u672c\u8eab&#xff0c;\u5c31\u662f T-\u03a0-O \u5faa\u73af\u7684\u6d3b\u6837\u672c&#xff1a;\u6bcf\u4e00\u4ee3\u4eba\u90fd\u7528\u65b0\u7684 \u03a0&#xff08;\u66f4\u7cbe\u7ec6\u7684\u6570\u503c\u3001\u66f4\u62bd\u8c61\u7684\u5b9a\u4e49\u3001\u66f4\u7edf\u4e00\u7684\u6807\u5ea6&#xff09;&#xff0c;\u4ece\u540c\u4e00\u7c7b T&#xff08;\u786e\u5b9a\u6027\u975e\u7ebf\u6027\u7cfb\u7edf&#xff09;\u4e2d&#xff0c;\u63d0\u70bc\u51fa\u66f4\u6df1\u523b\u7684 O\u3002\u5b83\u96c4\u8fa9\u5730\u8bc1\u660e&#xff1a;\u6df7\u6c8c\u4e0d\u662f&#034;\u65e0\u79e9\u5e8f&#034;&#xff0c;\u800c\u662f&#034;\u79e9\u5e8f\u88ab\u53d1\u73b0\u5f97\u683c\u5916\u6162\u3001\u4e14\u9700\u8981\u4e0d\u65ad\u6362\u6295\u5f71&#034;\u3002\u5386\u53f2\u7531\u6b64\u6210\u4e3a\u6846\u67b6\u6700\u6709\u529b\u7684\u8bc1\u6210\u3002<\/p>\n<h3>5.5 \u9a7e\u9a6d\u6df7\u6c8c&#xff1a;OGY \u63a7\u5236<\/h3>\n<p>5.2\u20135.3 \u8282\u8bf4\u5e9e\u52a0\u83b1\u6559\u6211\u4eec&#034;\u770b\u7ed3\u6784\u800c\u975e\u8f68\u9053&#034;\u3002\u4e00\u4e2a\u60ca\u4eba\u7684\u540e\u7eed\u662f&#xff1a;\u7406\u89e3\u7ed3\u6784\u5c42\u9762\u7684 O&#xff0c;\u7adf\u80fd\u8ba9\u6211\u4eec&#034;\u9a7e\u9a6d&#034;\u6df7\u6c8c\u3002 \u8fd9\u6b63\u662f Ott\u3001Grebogi \u4e0e Yorke \u5728 1990 \u5e74\u63d0\u51fa\u7684 OGY 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0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.8623em;vertical-align: -0.1944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><span class=\"mpunct\">,<\/span><span class=\"mspace\">\u00a0<\/span><span class=\"mspace\" style=\"margin-right: 0.1667em\"><\/span><span class=\"mord accent\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.6679em\"><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0359em\">y<\/span><\/span><span class=\"\" style=\"top: -3em\"><span class=\"pstrut\" style=\"height: 3em\"><\/span><span class=\"accent-body\" style=\"left: -0.0833em\"><span class=\"mord\">\u02d9<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.1944em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&#061;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6667em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">\u2212<\/span><span class=\"mord mathnormal\">x<\/span><\/span><\/span><\/span><\/span> \u8fd9\u7c7b\u65b9\u7a0b\u3002\u8fd9\u662f\u628a&#034;\u6570\u636e\u4e2d\u6d6e\u73b0\u7684 O&#034;\u7ffb\u8bd1\u56de\u4eba\u7c7b\u53ef\u8bfb\u65b9\u7a0b\u7684\u5173\u952e\u4e00\u6b65&#xff0c;\u4e5f\u662f AI \u4ece&#034;\u9ed1\u7bb1\u6295\u5f71&#034;\u8d70\u5411&#034;\u767d\u7bb1\u89c4\u5f8b&#034;\u7684\u6865\u6881\u3002<\/p>\n<p>\u56fe\u795e\u7ecf\u7f51\u7edc&#xff08;GNN&#xff09;\u4e0e\u591a\u4f53\u76f8\u4e92\u4f5c\u7528\u3002 \u6750\u6599\u3001\u86cb\u767d\u8d28\u6298\u53e0\u3001\u793e\u4f1a\u7f51\u7edc\u7b49\u7cfb\u7edf&#xff0c;\u672c\u8d28\u4e0a\u662f\u4e00\u7ec4\u76f8\u4e92\u4f5c\u7528\u7684&#034;\u8282\u70b9&#034;\u3002GNN \u901a\u8fc7\u6d88\u606f\u4f20\u9012&#xff08;message passing&#xff09;\u81ea\u52a8\u5b66\u4e60\u8282\u70b9\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u7ed3\u6784&#xff0c;\u628a\u5b83\u6295\u5f71\u4e3a\u8282\u70b9\u5d4c\u5165&#xff0c;\u4ece\u800c\u9884\u6d4b\u6750\u6599\u6027\u8d28\u3001\u86cb\u767d\u8d28\u6784\u8c61\u6216\u8206\u8bba\u4f20\u64ad\u3002\u8fd9\u91cc&#xff0c;GNN \u5b66\u5230\u7684&#034;\u76f8\u4e92\u4f5c\u7528\u56fe&#034;&#xff0c;\u6b63\u662f\u5bf9\u591a\u4f53\u7cfb\u7edf T \u7684\u4e00\u79cd\u7ed3\u6784\u5316\u6295\u5f71\u3002<\/p>\n<p>\u8fd9\u56db\u4e2a\u4f8b\u5b50\u5171\u540c\u8bf4\u660e&#xff1a;AI \u65e2\u80fd\u5b66\u4e60\u6295\u5f71&#xff08;\u628a\u9ad8\u7ef4 T \u538b\u5230\u4f4e\u7ef4\u7279\u5f81&#xff09;&#xff0c;\u4e5f\u80fd\u5728\u8be5\u6295\u5f71\u4e2d\u53d1\u73b0\u3001\u751a\u81f3\u5199\u51fa O&#xff08;\u5b88\u6052\u91cf\u3001\u63a7\u5236\u65b9\u7a0b\u3001\u76f8\u4e92\u4f5c\u7528\u7ed3\u6784&#xff09;\u3002 \u8fd9\u6b63\u662f T-\u03a0-O \u5faa\u73af\u88ab AI \u52a0\u901f\u540e\u7684\u65b0\u5f62\u6001&#xff0c;\u4e5f\u628a\u7b2c 3 \u7ae0&#034;\u6295\u5f71&#034;\u4e0e\u7b2c 4 \u7ae0&#034;\u79e9\u5e8f&#034;\u7684\u62bd\u8c61\u8ba8\u8bba&#xff0c;\u843d\u5230\u4e86\u5f53\u4ee3\u65b9\u6cd5\u8bba\u7684\u6700\u524d\u6cbf\u3002<\/p>\n<h3>6.3 \u5f52\u7eb3\u504f\u7f6e\u4e0e\u57fa\u7840\u6a21\u578b<\/h3>\n<p>6.1\u20136.2 \u8282\u8bf4 AI \u81ea\u52a8\u5b66\u4e60\u6295\u5f71\u3002\u672c\u8282\u5fc5\u987b\u8865\u5145\u4e00\u4e2a\u9650\u5b9a&#xff1a;AI \u7684\u6295\u5f71\u65b9\u5411\u5e76\u975e\u5b8c\u5168\u81ea\u7531&#xff0c;\u800c\u662f\u88ab&#034;\u5f52\u7eb3\u504f\u7f6e&#xff08;inductive bias&#xff09;&#034;\u9884\u5148\u5851\u5f62\u2014\u2014\u8fd9\u51b3\u5b9a\u4e86\u5b83\u64c5\u957f\u53d1\u73b0\u54ea\u7c7b O\u3002<\/p>\n<ul>\n<li>\u5377\u79ef\u7f51\u7edc&#xff08;CNN&#xff09; \u7684\u504f\u7f6e\u662f\u5e73\u79fb\u7b49\u53d8&#xff1a;\u5b83\u5929\u7136\u9002\u5408\u628a&#034;\u5c40\u90e8\u6a21\u5f0f\u91cd\u590d\u51fa\u73b0&#034;\u7684 T&#xff08;\u5982\u56fe\u50cf\u3001\u89c4\u5219\u683c\u5b50&#xff09;\u6295\u5f71\u4e3a\u5c42\u6b21\u7279\u5f81\u3002<\/li>\n<li>\u56fe\u7f51\u7edc&#xff08;GNN&#xff09; \u7684\u504f\u7f6e\u662f\u7f6e\u6362\u7b49\u53d8&#xff1a;\u5b83\u5929\u7136\u9002\u5408\u628a&#034;\u76f8\u4e92\u4f5c\u7528\u7f51\u7edc&#034;\u7c7b T \u6295\u5f71\u4e3a\u8282\u70b9\u5d4c\u5165&#xff08;6.2 \u8282&#xff09;\u3002<\/li>\n<li>Transformer \u7684\u504f\u7f6e\u662f\u6ce8\u610f\u529b\u4e0e\u5e8f\u5217\u4f4d\u7f6e&#xff1a;\u5b83\u5929\u7136\u9002\u5408\u628a&#034;\u7b26\u53f7\/ token \u5e8f\u5217&#034;\u7c7b T \u6295\u5f71\u4e3a\u4e0a\u4e0b\u6587\u8868\u5f81\u3002<\/li>\n<\/ul>\n<p>\u8fd1\u5e74\u5174\u8d77\u7684\u57fa\u7840\u6a21\u578b&#xff08;Foundation Models&#xff0c;\u542b\u5927\u8bed\u8a00\u6a21\u578b&#xff09;&#xff0c;\u53ef\u88ab\u89c6\u4e3a\u4e00\u79cd&#034;\u901a\u7528\u6295\u5f71\u5668&#034;&#xff1a;\u5b83\u628a\u6781\u5176\u5e7f\u6cdb\u7684\u7b26\u53f7\/\u8bed\u8a00\/\u591a\u6a21\u6001 T&#xff0c;\u7edf\u4e00\u6295\u5f71\u4e3a&#034;\u4e0b\u4e00\u4e2a token \u7684\u5206\u5e03&#034;\u3002\u5176\u53d1\u73b0\u7684 O&#xff08;\u8bed\u8a00\u7ed3\u6784\u3001\u4e16\u754c\u77e5\u8bc6\u7684\u67d0\u79cd\u538b\u7f29&#xff09;\u5177\u6709\u8de8\u4efb\u52a1\u53ef\u8fc1\u79fb\u6027&#xff0c;\u56e0\u800c\u9707\u64bc\u3002\u4f46\u672c\u8282\u5fc5\u987b\u91cd\u7533\u4e0b\u4e00\u8282\u7684\u544a\u8beb&#xff1a;\u57fa\u7840\u6a21\u578b\u4ecd\u662f \u03a0&#xff0c;\u4e14\u5176\u5f52\u7eb3\u504f\u7f6e&#xff08;\u4e0b\u4e00\u4e2a token \u9884\u6d4b&#xff09;\u5e76\u4e0d\u4fdd\u8bc1\u5b83\u53d1\u73b0\u7684 O \u5177\u6709\u56e0\u679c\u6216\u7269\u7406\u610f\u4e49&#xff1b;\u5b83\u6781\u6613\u5728\u5206\u5e03\u5916\u4ea7\u751f\u6d41\u7545\u800c\u9519\u8bef\u7684 O&#xff08;\u5e7b\u89c9&#xff09;\u3002\u628a\u57fa\u7840\u6a21\u578b\u5f53\u4f5c T&#xff0c;\u662f\u5f53\u4e0b\u6700\u76db\u884c\u7684\u6295\u5f71\u8bef\u7528\u3002<\/p>\n<h3>6.4 AI \u53ea\u662f\u65b0\u7684 \u03a0&#xff1a;\u9ed1\u7bb1\u9884\u6d4b\u4e0d\u7b49\u4e8e\u7406\u89e3<\/h3>\n<p>\u7136\u800c&#xff0c;\u5fc5\u987b\u5bf9 AI \u7684\u72c2\u70ed\u4fdd\u6301\u6e05\u9192\u3002\u672c\u6846\u67b6\u7684\u4e00\u4e2a\u786c\u7ea6\u675f\u662f&#xff1a;\u4efb\u4f55\u6a21\u578b&#xff0c;\u65e0\u8bba\u591a\u5f3a\u5927&#xff0c;\u90fd\u53ea\u662f \u03a0&#xff0c;\u800c\u4e0d\u662f T\u3002 AI \u6a21\u578b\u5c24\u5176\u5982\u6b64&#xff0c;\u539f\u56e0\u6709\u4e09&#xff1a;<\/p>\n<p>\u5176\u4e00&#xff0c;AI \u6295\u5f71\u540c\u6837\u662f\u4fe1\u606f\u6709\u635f\u7684\u3002 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\u7684\u4fe1\u5fc3\u5c31\u663e\u8457\u589e\u5f3a&#xff1b;\u5f53\u5b83\u4eec\u5f7c\u6b64\u77db\u76fe&#xff0c;\u77db\u76fe\u672c\u8eab\u6070\u6070\u66b4\u9732\u4e86\u67d0\u4e2a\u6295\u5f71\u7684\u76f2\u533a&#xff0c;\u6307\u5f15\u6211\u4eec\u6539\u8fdb\u3002\u591a\u6295\u5f71\u4e5f\u662f\u4e00\u79cd\u9c81\u68d2\u6027\u68c0\u9a8c&#xff1a;\u4e00\u4e2a\u53ea\u5728\u5355\u4e00\u6a21\u578b\u91cc\u6210\u7acb\u7684&#034;\u89c4\u5f8b&#034;&#xff0c;\u5f88\u53ef\u80fd\u53ea\u662f\u8be5 \u03a0 \u7684 artifacts&#xff1b;\u4e00\u4e2a\u5728\u65b9\u7a0b\u3001\u6a21\u62df\u3001\u6570\u636e\u4e09\u91cd\u6295\u5f71\u4e0b\u90fd\u5b58\u6d3b\u7684 O&#xff0c;\u624d\u66f4\u53ef\u80fd\u662f T \u4e2d\u771f\u5b9e\u5b58\u5728\u7684\u79e9\u5e8f\u3002<\/p>\n<h3>7.3 \u5bfb\u627e\u4e0d\u53d8\u91cf\u4e0e\u68c0\u9a8c\u8fb9\u754c<\/h3>\n<p>\u7b2c\u4e09\u6b65&#xff1a;\u5bfb\u627e\u4e0d\u53d8\u91cf\u3002 \u5728\u786e\u7acb\u6295\u5f71\u4e4b\u540e&#xff0c;\u7406\u89e3\u5de5\u4f5c\u7684\u6838\u5fc3\u4efb\u52a1\u662f\u5bfb\u627e\u4e0d\u53d8\u91cf\u2014\u2014\u90a3\u4e9b\u5728\u6295\u5f71\u53d8\u6362\u3001\u53c2\u6570\u6270\u52a8\u3001\u4e43\u81f3\u5c3a\u5ea6\u53d8\u5316\u4e0b\u4ecd\u7136\u5b58\u6d3b\u7684 O&#xff08;4.5 \u8282\u8be6\u5217\u5176\u5bb6\u65cf&#xff09;\u3002\u5177\u4f53\u53ef\u4ece\u4ee5\u4e0b\u6e05\u5355\u5165\u624b&#xff1a;\u5bf9\u79f0\u6027\u4e0e\u5b88\u6052\u91cf&#xff08;\u7cfb\u7edf\u662f\u5426\u6ee1\u8db3\u65f6\u95f4\/\u7a7a\u95f4\/\u5c3a\u5ea6\u7684\u67d0\u79cd\u5bf9\u79f0&#xff1f;\u5bf9\u5e94\u7684\u5b88\u6052\u91cf\u662f\u4ec0\u4e48&#xff1f;&#xff09;&#xff1b;\u5438\u5f15\u5b50\u4e0e\u6781\u9650\u96c6&#xff08;\u957f\u671f\u884c\u4e3a\u88ab\u5438\u5411\u4ec0\u4e48\u7ed3\u6784&#xff1f;\u5176\u7ef4\u6570\u4e0e\u62d3\u6251\u5982\u4f55&#xff1f;&#xff09;&#xff1b;\u4e34\u754c\u73b0\u8c61\u4e0e\u666e\u9002\u7c7b&#xff08;\u662f\u5426\u5b58\u5728\u76f8\u53d8\u70b9&#xff1f;\u4e34\u754c\u6307\u6570\u5c5e\u4e8e\u54ea\u4e00\u7c7b\u666e\u9002\u6027&#xff1f;&#xff09;&#xff1b;\u6807\u5ea6\u5f8b&#xff08;\u8de8\u8d8a\u591a\u4e2a\u6570\u91cf\u7ea7&#xff0c;\u662f\u5426\u5b58\u5728\u5e42\u5f8b\u6216\u5bf9\u6570\u5173\u7cfb&#xff1f;&#xff09;&#xff1b;\u62d3\u6251\u4e0d\u53d8\u91cf&#xff08;\u76f8\u7a7a\u95f4\u4e2d\u662f\u5426\u5b58\u5728\u4e0d\u80fd\u88ab\u8fde\u7eed\u5f62\u53d8\u6d88\u9664\u7684&#034;\u6d1e&#034;\u6216&#034;\u7f20\u7ed3&#034;&#xff1f;&#xff09;\u3002\u5bfb\u627e\u4e0d\u53d8\u91cf&#xff0c;\u672c\u8d28\u4e0a\u662f\u5728\u6df7\u6c8c\u4e2d\u6253\u635e\u79e9\u5e8f\u3002\u5b83\u4eec\u4e4b\u6240\u4ee5\u73cd\u8d35&#xff0c;\u662f\u56e0\u4e3a\u5b83\u4eec\u6700\u6709\u53ef\u80fd\u8de8\u6295\u5f71\u3001\u8de8\u5c3a\u5ea6\u5730\u6210\u7acb&#xff0c;\u662f T-\u03a0-O \u6846\u67b6\u4e2d O \u7684\u6700\u7a33\u56fa\u5185\u6838\u3002\u4e00\u4e2a\u7814\u7a76\u82e5\u6b62\u6b65\u4e8e&#034;\u62df\u5408\u4e86\u4e00\u6761\u66f2\u7ebf&#034;\u800c\u672a\u80fd\u6307\u5411\u4efb\u4f55\u4e0d\u53d8\u91cf&#xff0c;\u5176\u7ed3\u8bba\u7684\u5bff\u547d\u5f80\u5f80\u5f88\u77ed\u3002<\/p>\n<p>\u7b2c\u56db\u6b65&#xff1a;\u68c0\u9a8c\u53ef\u9884\u6d4b\u6027\u8fb9\u754c\u3002 \u7406\u89e3\u590d\u6742\u7cfb\u7edf&#xff0c;\u5fc5\u987b\u540c\u65f6\u7406\u89e3\u5b83\u7684\u4e0d\u53ef\u9884\u6d4b\u6027\u3002\u672c\u6846\u67b6\u575a\u51b3\u53cd\u5bf9&#034;\u8981\u4e48\u5168\u77e5\u3001\u8981\u4e48\u65e0\u77e5&#034;\u7684\u4e8c\u5206\u6cd5&#xff0c;\u4e3b\u5f20\u8bda\u5b9e\u5730\u6807\u5b9a\u53ef\u9884\u6d4b\u6027\u8fb9\u754c&#xff1a;\u2460 Lyapunov \u65f6\u95f4\u2014\u2014\u5bf9\u4e8e\u6df7\u6c8c\u7cfb\u7edf&#xff0c;\u8ba1\u7b97\u6700\u5927 Lyapunov \u6307\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        \u03bb<\/p>\n<p>       \\\\lambda<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\">\u03bb<\/span><\/span><\/span><\/span><\/span>&#xff0c;\u5176\u5012\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        1<\/p>\n<p>        \/<\/p>\n<p>        \u03bb<\/p>\n<p>       1\/\\\\lambda<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1\/<\/span><span class=\"mord mathnormal\">\u03bb<\/span><\/span><\/span><\/span><\/span> \u5927\u81f4\u7ed9\u51fa\u4e86&#034;\u53ef\u9884\u6d4b\u65f6\u95f4\u5c3a\u5ea6&#034;&#xff0c;\u8d85\u8fc7\u8fd9\u4e2a\u65f6\u95f4&#xff0c;\u521d\u59cb\u8bef\u5dee\u5df2\u88ab\u653e\u5927\u5230\u6df9\u6ca1\u4fe1\u53f7&#xff1b;\u2461 \u8bef\u5dee\u589e\u957f\u66f2\u7ebf\u2014\u2014\u76f4\u63a5\u6a21\u62df&#034;\u521d\u59cb\u6761\u4ef6\u52a0\u5fae\u5c0f\u6270\u52a8&#034;\u540e\u7684\u8f68\u8ff9\u5206\u79bb&#xff0c;\u6bd4\u4efb\u4f55\u54f2\u5b66\u8ba8\u8bba\u90fd\u66f4\u76f4\u89c2\u5730\u5212\u5b9a\u8fb9\u754c&#xff1b;\u2462 \u6a21\u578b\u4e0d\u786e\u5b9a\u6027\u91cf\u5316\u2014\u2014\u5bf9 AI \u4e0e\u7edf\u8ba1\u6a21\u578b&#xff0c;\u8f93\u51fa\u9884\u6d4b\u65f6\u5e94\u540c\u65f6\u8f93\u51fa\u7f6e\u4fe1\u533a\u95f4\u6216\u96c6\u5408\u9884\u62a5&#xff08;ensemble forecast&#xff09;&#xff0c;\u660e\u786e&#034;\u6211\u4e0d\u77e5\u9053&#034;\u7684\u533a\u57df\u3002\u68c0\u9a8c\u8fb9\u754c\u7684\u610f\u4e49&#xff0c;\u4e0d\u53ea\u662f\u8c26\u900a&#xff0c;\u66f4\u662f\u8d1f\u8d23\u4efb\u3002\u6c14\u5019\u79d1\u5b66\u4e4b\u6240\u4ee5\u7528&#034;\u96c6\u5408\u9884\u62a5&#034;\u800c\u975e\u5355\u4e00\u786e\u5b9a\u9884\u6d4b&#xff0c;\u6b63\u662f\u56e0\u4e3a\u627f\u8ba4\u4e86 T \u7684\u4e0d\u53ef\u5b8c\u5907\u6027&#xff08;2.2 \u8282&#xff09;\u3002\u4e00\u4e2a\u6210\u719f\u7684\u7406\u89e3&#xff0c;\u5fc5\u7136\u5305\u542b\u4e00\u4efd\u6e05\u6670\u7684&#034;\u80fd\u529b\u8bf4\u660e\u4e66&#034;&#xff1a;\u54ea\u4e9b\u53ef\u9884\u6d4b\u3001\u9884\u6d4b\u591a\u8fdc\u3001\u54ea\u4e9b\u539f\u5219\u4e0a\u4e0d\u53ef\u9884\u6d4b\u3002<\/p>\n<h3>7.4 \u8fed\u4ee3\u903c\u8fd1\u4e0e\u8de8\u5c3a\u5ea6\u8fde\u63a5<\/h3>\n<p>\u7b2c\u4e94\u6b65&#xff1a;\u8fed\u4ee3\u903c\u8fd1\u3002 \u7406\u89e3\u662f\u4e00\u4e2a\u5faa\u73af&#xff0c;\u4e0d\u662f\u4e00\u6b21\u5b8c\u6210\u3002\u6211\u4eec\u628a\u5b83\u5199\u4e3a\u4e00\u4e2a\u660e\u786e\u7684\u8fed\u4ee3\u5f0f&#xff1a;<\/p>\n<p><span class=\"katex--display\"><span class=\"katex-display\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         T<br \/>\n        \u2005\u200a<\/p>\n<p>          \u2192<\/p>\n<p>           \u6295\u5f71<\/p>\n<p>        \u2005\u200a<\/p>\n<p>         \u03a0<br \/>\n        \u2005\u200a<\/p>\n<p>          \u2192<\/p>\n<p>           \u63d0\u53d6<\/p>\n<p>        \u2005\u200a<\/p>\n<p>         O<br \/>\n        \u2005\u200a<\/p>\n<p>          \u2192<\/p>\n<p>           \u5b9e\u9a8c\/\u89c2\u6d4b\u9a8c\u8bc1<\/p>\n<p>        \u2005\u200a<\/p>\n<p>          {<\/p>\n<p>               \u901a\u8fc7<\/p>\n<p>               \u21d2<\/p>\n<p>               \u4fee\u6b63\/\u7ec6\u5316\u00a0<\/p>\n<p>               \u03a0<\/p>\n<p>               \u5931\u8d25<\/p>\n<p>               \u21d2<\/p>\n<p>               \u66f4\u6362\u00a0<\/p>\n<p>               \u03a0<\/p>\n<p>        \u2005\u200a<\/p>\n<p>         \u27f6<br \/>\n        \u2005\u200a<\/p>\n<p>         T<\/p>\n<p>        T \\\\;\\\\xrightarrow{\\\\text{\u6295\u5f71}}\\\\; \\\\Pi \\\\;\\\\xrightarrow{\\\\text{\u63d0\u53d6}}\\\\; O \\\\;\\\\xrightarrow{\\\\text{\u5b9e\u9a8c\/\u89c2\u6d4b\u9a8c\u8bc1}}\\\\; \\\\begin{cases} \\\\text{\u901a\u8fc7} \\\\Rightarrow \\\\text{\u4fee\u6b63\/\u7ec6\u5316 }\\\\Pi \\\\\\\\ \\\\text{\u5931\u8d25} \\\\Rightarrow \\\\text{\u66f4\u6362 }\\\\Pi \\\\end{cases} \\\\;\\\\longrightarrow\\\\; T<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 1.1113em;vertical-align: -0.011em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel x-arrow\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1003em\"><span class=\"\" style=\"top: -3.322em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight x-arrow-pad\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord cjk_fallback mtight\">\u6295\u5f71<\/span><\/span><\/span><\/span><\/span><span class=\"svg-align\" style=\"top: -2.689em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"hide-tail\" style=\"height: 0.522em;min-width: 1.469em\"><\/p>\n<p>            <\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.011em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.1113em;vertical-align: -0.011em\"><\/span><span class=\"mord\">\u03a0<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel x-arrow\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.1003em\"><span class=\"\" style=\"top: -3.322em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight x-arrow-pad\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord cjk_fallback mtight\">\u63d0\u53d6<\/span><\/span><\/span><\/span><\/span><span class=\"svg-align\" style=\"top: -2.689em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"hide-tail\" style=\"height: 0.522em;min-width: 1.469em\"><\/p>\n<p>            <\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.011em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1.158em;vertical-align: -0.011em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0278em\">O<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel x-arrow\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.147em\"><span class=\"\" style=\"top: -3.322em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight x-arrow-pad\"><span class=\"mord mtight\"><span class=\"mord text mtight\"><span class=\"mord cjk_fallback mtight\">\u5b9e\u9a8c<\/span><span class=\"mord mtight\">\/<\/span><span class=\"mord cjk_fallback mtight\">\u89c2\u6d4b\u9a8c\u8bc1<\/span><\/span><\/span><\/span><\/span><span class=\"svg-align\" style=\"top: -2.689em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"hide-tail\" style=\"height: 0.522em;min-width: 1.469em\"><\/p>\n<p>            <\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.011em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 3em;vertical-align: -1.25em\"><\/span><span class=\"minner\"><span class=\"mopen delimcenter\" style=\"top: 0em\"><span class=\"delimsizing size4\">{<\/span><\/span><span class=\"mord\"><span class=\"mtable\"><span class=\"col-align-l\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.69em\"><span class=\"\" style=\"top: -3.69em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u901a\u8fc7<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u4fee\u6b63<\/span><span class=\"mord\">\/<\/span><span class=\"mord cjk_fallback\">\u7ec6\u5316<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord\">\u03a0<\/span><\/span><\/span><span class=\"\" style=\"top: -2.25em\"><span class=\"pstrut\" style=\"height: 3.008em\"><\/span><span class=\"mord\"><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u5931\u8d25<\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u21d2<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mord text\"><span class=\"mord cjk_fallback\">\u66f4\u6362<\/span><span class=\"mord\">\u00a0<\/span><\/span><span class=\"mord\">\u03a0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 1.19em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mclose nulldelimiter\"><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u27f6<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6833em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.1389em\">T<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p>\u6bcf\u4e00\u6b21\u5faa\u73af&#xff0c;validated \u7684 O \u53cd\u8fc7\u6765 refined \u6211\u4eec\u5bf9 T \u7684\u8ba4\u8bc6&#xff0c;\u4e5f\u63d0\u793a\u6211\u4eec\u4e0b\u4e00\u4e2a\u66f4\u597d\u7684 \u03a0 \u8be5\u5f80\u54ea\u4e2a\u65b9\u5411\u6784\u9020\u3002\u79d1\u5b66\u7684\u8fdb\u6b65&#xff0c;\u4ece\u65e5\u5fc3\u8bf4\u5230\u91cf\u5b50\u573a\u8bba&#xff0c;\u672c\u8d28\u4e0a\u90fd\u662f\u8fd9\u53f0\u5faa\u73af\u673a\u5668\u4e00\u6b21\u6b21\u8f6c\u52a8\u7684\u7ed3\u679c\u3002\u9700\u8981\u7279\u522b\u5f3a\u8c03\u5faa\u73af\u4e2d\u7684\u9a8c\u8bc1\u73af\u8282\u2014\u2014\u5b83\u628a\u672c\u6846\u67b6\u4e0e&#034;\u968f\u4fbf\u7f16\u4e2a\u6a21\u578b&#034;\u533a\u5206\u5f00\u6765\u3002\u4e00\u4e2a O \u82e5\u4e0d\u80fd\u5728\u72ec\u7acb\u6570\u636e\u6216\u5b9e\u9a8c\u4e2d\u590d\u73b0&#xff08;4.1 \u8282\u4e09\u6761\u4ef6&#xff09;&#xff0c;\u5c31\u5fc5\u987b\u88ab\u964d\u7ea7\u6216\u629b\u5f03&#xff0c;\u800c\u4e0d\u662f\u88ab\u8fa9\u62a4\u3002T-\u03a0-O \u5faa\u73af\u4e4b\u6240\u4ee5\u80fd\u4ea7\u751f\u53ef\u9760\u77e5\u8bc6&#xff0c;\u9760\u7684\u4e0d\u662f\u67d0\u6b21\u7075\u5149\u4e00\u73b0&#xff0c;\u800c\u662f\u9a8c\u8bc1\u73af\u8282\u7684\u4e0d\u65ad\u5426\u51b3\u4e0e\u7b5b\u9009\u3002<\/p>\n<p>\u7b2c\u516d\u6b65&#xff1a;\u8de8\u5c3a\u5ea6\u8fde\u63a5\u3002 \u516d\u6b65\u4e4b\u4e2d&#xff0c;\u6700\u96be\u4e5f\u6700\u5173\u952e\u7684\u662f\u8de8\u5c3a\u5ea6\u8fde\u63a5&#xff1a;\u5fae\u89c2\u673a\u5236\u5982\u4f55\u6d8c\u73b0\u51fa\u5b8f\u89c2\u79e9\u5e8f&#xff1f;\u8fd9\u662f\u590d\u6742\u7cfb\u7edf\u7814\u7a76\u7684&#034;\u5723\u676f&#034;&#xff0c;\u4e5f\u662f\u6846\u67b6\u6700\u5403\u52b2\u7684\u5730\u65b9\u3002\u8de8\u5c3a\u5ea6\u4e4b\u6240\u4ee5\u96be&#xff0c;\u662f\u56e0\u4e3a\u5b83\u8981\u6c42\u6211\u4eec\u5728\u4e24\u4e2a&#xff08;\u751a\u81f3\u591a\u4e2a&#xff09;\u5c42\u6b21\u7684\u6295\u5f71\u4e4b\u95f4&#xff0c;\u5efa\u7acb\u53ef\u63a8\u5bfc\u7684\u6865\u6881\u3002\u7edf\u8ba1\u7269\u7406\u7528\u7c97\u7c92\u5316&#xff08;coarse-graining&#xff09;\u4e0e\u91cd\u6574\u5316\u7fa4&#xff08;renormalization group&#xff09;\u628a\u5fae\u89c2\u54c8\u5bc6\u987f\u91cf\u6620\u5c04\u5230\u5b8f\u89c2\u76f8\u53d8\u2014\u2014\u8fd9\u662f\u6700\u6210\u529f\u7684\u8de8\u5c3a\u5ea6\u8303\u5f0f&#xff1b;\u591a\u5c3a\u5ea6\u5efa\u6a21&#xff08;multiscale modeling&#xff09;\u5728\u6750\u6599\u4e0e\u751f\u7269\u4e2d\u5c1d\u8bd5\u628a\u539f\u5b50\u5c3a\u5ea6\u4e0e\u8fde\u7eed\u4ecb\u8d28\u5c3a\u5ea6\u8026\u5408&#xff1b;\u673a\u5668\u5b66\u4e60\u4e2d\u7684&#034;\u795e\u7ecf\u79d1\u5b66\u7684\u5c3a\u5ea6\u8de8\u8d8a&#034;\u5219\u8bd5\u56fe\u4ece\u7a81\u89e6\u53ef\u5851\u6027\u63a8\u5bfc\u5230\u8ba4\u77e5\u529f\u80fd\u3002\u4f46\u6211\u4eec\u5fc5\u987b\u8bda\u5b9e&#xff1a;\u8de8\u5c3a\u5ea6\u8fde\u63a5\u81f3\u4eca\u8fdc\u672a\u89e3\u51b3\u3002\u8bb8\u591a\u53f7\u79f0&#034;\u6d8c\u73b0&#034;\u7684\u73b0\u8c61&#xff0c;\u5176\u5b9e\u53ea\u662f\u6211\u4eec\u5728\u67d0\u4e00\u5c3a\u5ea6\u4e0a\u5c1a\u672a\u627e\u5230\u5408\u9002\u7684 \u03a0 \u800c\u5df2&#xff08;\u89c1 8.4 \u8282\u5bf9&#034;\u6d8c\u73b0&#034;\u6982\u5ff5\u7684\u6279\u5224&#xff09;\u3002\u672c\u6846\u67b6\u5bf9\u8de8\u5c3a\u5ea6\u7684\u6001\u5ea6\u662f&#xff1a;\u628a\u5b83\u4f5c\u4e3a\u6301\u4e45\u7684\u7814\u7a76\u7eb2\u9886&#xff0c;\u800c\u975e\u5df2\u7ecf\u5151\u73b0\u7684\u627f\u8bfa\u2014\u2014\u6bcf\u4e00\u6b21\u6211\u4eec\u5728\u5fae\u89c2 \u03a0 \u4e0e\u5b8f\u89c2 \u03a0 \u4e4b\u95f4\u6210\u529f\u67b6\u6865&#xff0c;\u90fd\u662f\u5bf9 T \u7406\u89e3\u7684\u5b9e\u8d28\u6027\u63a8\u8fdb\u3002<\/p>\n<h3>7.5 \u8d2f\u7a7f\u6848\u4f8b&#xff1a;\u4f20\u67d3\u75c5\u6269\u6563<\/h3>\n<p>\u628a\u516d\u6b65\u6cd5\u5e94\u7528\u4e8e\u4e00\u4e2a\u5177\u4f53\u53c8\u719f\u6089\u7684\u95ee\u9898\u2014\u2014\u4f20\u67d3\u75c5\u6269\u6563&#xff0c;\u53ef\u4ee5\u6700\u76f4\u89c2\u5730\u6f14\u793a\u65b9\u6cd5\u8bba\u5982\u4f55\u843d\u5730\u3002<\/p>\n<p>\u7b2c\u4e00\u6b65 \u754c\u5b9a\u5c42\u6b21\u3002 \u6211\u4eec\u8981\u7684\u662f&#034;\u9884\u6d4b\u5cf0\u503c\u4e0e\u5236\u5b9a\u5e72\u9884&#034;&#xff0c;\u800c\u975e&#034;\u6a21\u62df\u6bcf\u4e2a\u4e2a\u4f53\u7684\u6bcf\u4e00\u6b21\u6027\u63a5\u89e6&#034;\u3002\u76ee\u6807\u51b3\u5b9a\u4e86\u5408\u9002\u7684\u6295\u5f71\u5c3a\u5ea6\u3002<\/p>\n<p>\u7b2c\u4e8c\u6b65 \u5efa\u7acb\u591a\u6295\u5f71\u3002 \u2460 compartmental \u6a21\u578b&#xff08;SIR\/SEIR \u65b9\u7a0b&#xff09;&#xff1a;\u628a\u4eba\u7fa4\u6295\u5f71\u4e3a S\/E\/I\/R \u51e0\u4e2a\u603b\u91cf&#xff0c;\u7ed9\u51fa\u57fa\u672c\u518d\u751f\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         R<\/p>\n<p>         0<\/p>\n<p>       R_0<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0077em\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0077em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span> \u8fd9\u4e00 O&#xff1b;\u2461 \u7f51\u7edc\u6a21\u578b&#xff1a;\u628a\u63a5\u89e6\u7ed3\u6784\u6295\u5f71\u4e3a\u56fe&#xff0c;\u6355\u83b7&#034;\u8d85\u7ea7\u4f20\u64ad\u8005&#034;\u7ed3\u6784 O&#xff1b;\u2462 \u57fa\u4e8e\u4e3b\u4f53\u7684\u4eff\u771f&#xff08;ABM&#xff09;&#xff1a;\u4fdd\u7559\u4e2a\u4f53\u5f02\u8d28\u6027&#xff1b;\u2463 \u673a\u5668\u5b66\u4e60&#xff1a;\u4ece\u624b\u673a\u4fe1\u4ee4\u6570\u636e\u4e2d\u6295\u5f71\u51fa mobility \u7279\u5f81\u4ee5\u6821\u51c6\u3002<\/p>\n<p>\u7b2c\u4e09\u6b65 \u5bfb\u627e\u4e0d\u53d8\u91cf\u3002 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>         R<\/p>\n<p>         0<\/p>\n<p>        &gt;<\/p>\n<p>        1<\/p>\n<p>       R_0&gt;1<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.8333em;vertical-align: -0.15em\"><\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0077em\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0077em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">&gt;<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">1<\/span><\/span><\/span><\/span><\/span> \u65f6\u7206\u53d1\u3001\u7f51\u7edc\u5ea6\u5206\u5e03\u7684\u5e42\u5f8b&#xff08;4.5 \u8282&#xff09;\u3001\u4ee5\u53ca&#034;\u4e34\u754c\u63a5\u79cd\u9608\u503c <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\"><\/p>\n<p>        1<\/p>\n<p>        \u2212<\/p>\n<p>        1<\/p>\n<p>        \/<\/p>\n<p>         R<\/p>\n<p>         0<\/p>\n<p>       1-1\/R_0<\/p>\n<p>    <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.7278em;vertical-align: -0.0833em\"><\/span><span class=\"mord\">1<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><span class=\"mbin\">\u2212<\/span><span class=\"mspace\" style=\"margin-right: 0.2222em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 1em;vertical-align: -0.25em\"><\/span><span class=\"mord\">1\/<\/span><span class=\"mord\"><span class=\"mord mathnormal\" style=\"margin-right: 0.0077em\">R<\/span><span class=\"msupsub\"><span class=\"vlist-t vlist-t2\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.3011em\"><span class=\"\" style=\"top: -2.55em;margin-left: -0.0077em;margin-right: 0.05em\"><span class=\"pstrut\" style=\"height: 2.7em\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mord mtight\">0<\/span><\/span><\/span><\/span><span class=\"vlist-s\">\u200b<\/span><\/span><span class=\"vlist-r\"><span class=\"vlist\" style=\"height: 0.15em\"><span class=\"\"><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>&#034;\u2014\u2014\u8fd9\u4e9b\u662f\u8de8\u6295\u5f71\u5b58\u6d3b\u7684 O\u3002<\/p>\n<p>\u7b2c\u56db\u6b65 \u68c0\u9a8c\u8fb9\u754c\u3002 \u65e9\u671f\u6570\u636e\u7a00\u758f\u65f6&#xff0c;\u4e0d\u786e\u5b9a\u6027\u6781\u5927&#xff1b;\u53d8\u5f02\u682a\u51fa\u73b0\u5373\u5206\u5e03\u6f02\u79fb&#xff08;6.4 \u8282&#xff09;&#xff0c;\u9700\u4ee5\u96c6\u5408\u9884\u62a5\u6807\u5b9a&#034;\u6211\u4eec\u77e5\u9053\u4ec0\u4e48\u3001\u4e0d\u77e5\u9053\u4ec0\u4e48&#034;\u3002<\/p>\n<p>\u7b2c\u4e94\u6b65 \u8fed\u4ee3\u3002 \u7528\u771f\u5b9e\u75c5\u4f8b\u6570\u636e\u56de\u68c0\u5404 \u03a0 \u7684 O&#xff0c;\u6dd8\u6c70\u5931\u771f\u6295\u5f71&#xff0c;\u4fee\u6b63\u53c2\u6570\u3002<\/p>\n<p>\u7b2c\u516d\u6b65 \u8de8\u5c3a\u5ea6\u3002 \u4e2a\u4f53\u63a5\u89e6&#xff08;\u5fae\u89c2 \u03a0&#xff09;\u5982\u4f55\u6d8c\u73b0\u51fa\u7fa4\u4f53\u66f2\u7ebf&#xff08;\u5b8f\u89c2 O&#xff09;&#xff0c;\u6b63\u662f SIR \u7c97\u7c92\u5316\u7684\u6210\u529f\u6848\u4f8b&#xff0c;\u4e5f\u662f\u8de8\u5c3a\u5ea6\u8fde\u63a5\u6700\u4eb2\u5207\u7684\u8303\u672c\u3002<\/p>\n<p>\u6b64\u6848\u4f8b\u8bf4\u660e&#xff1a;T-\u03a0-O \u4e0d\u662f\u7a7a\u4e2d\u697c\u9601&#xff0c;\u800c\u662f\u4e00\u5957\u80fd\u771f\u6b63&#034;\u7528\u8d77\u6765&#034;\u7684\u7814\u7a76\u89c4\u7a0b\u3002\u5b83\u4e5f\u793a\u8303\u4e86\u516d\u6b65\u5982\u4f55\u9996\u5c3e\u76f8\u63a5\u5f62\u6210\u5faa\u73af&#xff0c;\u800c\u975e\u4e00\u6b21\u6027\u6d41\u7a0b\u3002<\/p>\n<h3>7.6 \u6df1\u5ea6\u4e0d\u786e\u5b9a\u6027\u4e0b\u7684\u7a33\u5065\u51b3\u7b56<\/h3>\n<p>7.3 \u8282\u5f3a\u8c03\u6807\u5b9a\u53ef\u9884\u6d4b\u6027\u8fb9\u754c\u3002\u672c\u8282\u628a\u8fd9\u4e00\u7cbe\u795e\u63a8\u5230\u653f\u7b56\u5c42\u9762&#xff1a;\u5f53 T \u7684\u4e0d\u53ef\u5b8c\u5907\u6027\u610f\u5473\u7740&#034;\u957f\u671f\u7cbe\u786e\u9884\u6d4b\u4e0d\u53ef\u5f97&#034;\u65f6&#xff0c;\u6211\u4eec\u8be5\u5982\u4f55\u51b3\u7b56&#xff1f;<\/p>\n<p>\u4f20\u7edf\u51b3\u7b56\u4f9d\u8d56&#034;\u9884\u6d4b\u2013\u4f18\u5316&#034;&#xff1a;\u5148\u9884\u6d4b\u672a\u6765\u72b6\u6001&#xff0c;\u518d\u9009\u6700\u4f18\u884c\u52a8\u3002\u4f46\u5728\u6df1\u5ea6\u4e0d\u786e\u5b9a\u6027&#xff08;deep uncertainty&#xff0c;\u5373\u8fde\u672a\u6765\u53ef\u80fd\u72b6\u6001\u7684\u6982\u7387\u5206\u5e03\u90fd\u65e0\u6cd5\u53ef\u9760\u7ed9\u51fa&#xff09;\u4e0b&#xff0c;\u8fd9\u6761\u8def\u5931\u6548\u2014\u2014\u91d1\u878d\u5371\u673a\u3001\u6c14\u5019\u4e34\u754c\u70b9\u3001\u5927\u6d41\u884c\u7686\u5c5e\u6b64\u5217\u3002\u66ff\u4ee3\u8303\u5f0f\u662f\u7a33\u5065\u51b3\u7b56&#xff08;Robust Decision Making&#xff09;\u4e0e\u97e7\u6027&#xff08;resilience&#xff09;&#xff1a;<\/p>\n<ul>\n<li>\u7a33\u5065\u51b3\u7b56 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\/>\n<h2>\u53c2\u8003\u6587\u732e<\/h2>\n<p>\u8bf4\u660e&#xff1a;\u4ee5\u4e0b\u4e3a\u672c\u6587\u8bba\u53ca\u7684\u4ee3\u8868\u6027\u6587\u732e&#xff0c;\u5747\u4e3a\u76f8\u5173\u9886\u57df\u7684\u7ecf\u5178\u6216\u5960\u57fa\u6027\u5de5\u4f5c&#xff1b;\u6b63\u5f0f\u6295\u7a3f\u524d\u8bf7\u6309\u76ee\u6807\u4f1a\u8bae\/\u671f\u520a\u7684\u5f15\u7528\u683c\u5f0f&#xff08;\u5982 APA\u3001IEEE\u3001GB\/T 7714&#xff09;\u6838\u6821\u51fa\u7248\u4fe1\u606f\u3001\u5377\u671f\u9875\u7801\u4e0e DOI\u3002\u90e8\u5206\u65e9\u671f\u6587\u732e\u7684\u7cbe\u786e\u5377\u671f\u4ee5\u539f\u59cb\u51fa\u7248\u7269\u4e3a\u51c6\u3002<\/p>\n<li>Poincar\u00e9, H. (1892\u20131899). Les M\u00e9thodes Nouvelles de la M\u00e9canique C\u00e9leste (\u4e09\u5377). Gauthier-Villars, Paris.&#xff08;\u5e9e\u52a0\u83b1\u5960\u5b9a\u73b0\u4ee3\u52a8\u529b\u7cfb\u7edf\u4e0e\u6df7\u6c8c\u7406\u8bba\u7684\u5960\u57fa\u4e4b\u4f5c&#xff0c;\u542b\u540c\u5bbf\u8f68\u9053\u7ed3\u6784\u7684\u53d1\u73b0\u3002&#xff09;<\/li>\n<li>Lyapunov, A. M. (1892). The General Problem of the Stability of Motion&#xff08;\u535a\u58eb\u8bba\u6587&#xff0c;\u4fc4\u8bed\u539f\u7248&#xff1b;\u540e\u88ab\u8bd1\u4e3a\u591a\u56fd\u6587\u5b57&#xff09;\u3002&#xff08;Lyapunov \u7a33\u5b9a\u6027\u7406\u8bba\u4e0e\u6307\u6570\u5224\u636e\u7684\u6e90\u5934\u3002&#xff09;<\/li>\n<li>Birkhoff, G. D. (1927). Dynamical Systems. American Mathematical Society.&#xff08;\u5168\u5c40\u52a8\u529b\u7cfb\u7edf\u7406\u8bba&#xff0c;\u5468\u671f\u8f68\u9053\u5b58\u5728\u5b9a\u7406\u3002&#xff09;<\/li>\n<li>Kolmogorov, A. N. (1954). On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian. Doklady Akademii Nauk SSSR.&#xff08;KAM \u7406\u8bba\u7684\u8d77\u70b9\u3002&#xff09;<\/li>\n<li>Arnold, V. I. &amp; Moser, J. (1960s). \u5173\u4e8e\u8fd1\u53ef\u79ef\u54c8\u5bc6\u987f\u7cfb\u7edf\u7684\u4e0d\u53d8\u73af\u9762&#xff08;KAM \u5b9a\u7406&#xff09;\u3002&#xff08;\u89c4\u5219\u4e0e\u6df7\u6c8c\u5171\u5b58\u7684\u4e25\u683c\u6570\u5b66\u3002&#xff09;<\/li>\n<li>Lorenz, E. N. (1963). Deterministic Nonperiodic Flow. Journal of the Atmospheric Sciences, 20(2), 130\u2013141.&#xff08;\u8774\u8776\u6548\u5e94\u4e0e\u5947\u602a\u5438\u5f15\u5b50\u7684\u5f00\u521b\u6027\u6570\u503c\u5b9e\u9a8c\u3002&#xff09;<\/li>\n<li>Li, T.-Y. &amp; Yorke, J. A. (1975). Period Three Implies Chaos. Journal of Mathematical Analysis and Applications, 16(2), 181\u2013192.&#xff08;\u6df7\u6c8c\u7684\u4e25\u683c\u6570\u5b66\u5b9a\u4e49\u3002&#xff09;<\/li>\n<li>Feigenbaum, M. J. (1978). Quantitative universality for a class of nonlinear transformations. Journal of Statistical Physics, 19, 25\u201352.&#xff08;\u500d\u5468\u671f\u5206\u5c94\u666e\u9002\u5e38\u6570 <span class=\"katex--inline\"><span class=\"katex\"><span class=\"katex-mathml\">\n<p>         \u03b4<\/p>\n<p>         \u2248<\/p>\n<p>         4.669<\/p>\n<p>        \\\\delta\\\\approx 4.669<\/p>\n<p>     <\/span><span class=\"katex-html\"><span class=\"base\"><span class=\"strut\" style=\"height: 0.6944em\"><\/span><span class=\"mord mathnormal\" style=\"margin-right: 0.0379em\">\u03b4<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><span class=\"mrel\">\u2248<\/span><span class=\"mspace\" style=\"margin-right: 0.2778em\"><\/span><\/span><span class=\"base\"><span class=\"strut\" style=\"height: 0.6444em\"><\/span><span class=\"mord\">4.669<\/span><\/span><\/span><\/span><\/span>\u3002&#xff09;<\/li>\n<li>Ott, E., Grebogi, C. &amp; Yorke, J. A. (1990). Controlling chaos. Physical Review Letters, 64(11), 1196\u20131199.&#xff08;OGY \u6df7\u6c8c\u63a7\u5236\u6cd5\u3002&#xff09;<\/li>\n<li>Noether, E. (1918). Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu G\u00f6ttingen, 235\u2013257.&#xff08;\u8bfa\u7279\u5b9a\u7406&#xff1a;\u5bf9\u79f0\u6027\u4e0e\u5b88\u6052\u91cf\u7684\u5bf9\u5e94\u3002&#xff09;<\/li>\n<li>Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27, 379\u2013423 &amp; 623\u2013656.&#xff08;\u4fe1\u606f\u8bba\u5960\u57fa&#xff0c;\u4f5c\u4e3a&#034;\u8bed\u4e49\u6295\u5f71&#034;\u7684\u8303\u4f8b\u3002&#xff09;<\/li>\n<li>Turing, A. M. (1936). On Computable Numbers, with an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42, 230\u2013265.&#xff08;\u505c\u673a\u95ee\u9898\u4e0e\u53ef\u8ba1\u7b97\u6027\u8fb9\u754c\u3002&#xff09;<\/li>\n<li>G\u00f6del, K. (1931). \u00dcber formal unentscheidbare S\u00e4tze der Principia Mathematica und verwandter Systeme I. Monatshefte f\u00fcr Mathematik und Physik, 38, 173\u2013198.&#xff08;\u4e0d\u5b8c\u5907\u5b9a\u7406\u3002&#xff09;<\/li>\n<li>Wolfram, S. (2002). A New Kind of Science. Wolfram Media.&#xff08;\u5143\u80de\u81ea\u52a8\u673a\u4e0e\u8ba1\u7b97\u4e0d\u53ef\u7ea6\u6027\u3002&#xff09;<\/li>\n<li>Chaitin, G. J. (1966\/1975). \u7b97\u6cd5\u4fe1\u606f\u8bba\u76f8\u5173\u8bba\u6587&#xff08;\u5982 \u201cOn the length of programs for computing finite binary sequences\u201d&#xff09;\u3002&#xff08;\u4e0d\u53ef\u538b\u7f29\u6027\u4e0e\u590d\u6742\u5ea6\u4e0d\u53ef\u8ba1\u7b97\u3002&#xff09;<\/li>\n<li>Prigogine, I. (1969, \u53ca\u540e\u7eed). \u8017\u6563\u7ed3\u6784\u7406\u8bba\u76f8\u5173\u8bba\u6587\u3002&#xff08;\u8fdc\u79bb\u5e73\u8861\u6001\u7cfb\u7edf\u4e2d\u7684\u6709\u5e8f\u7ed3\u6784\u3002&#xff09;<\/li>\n<li>Bak, P., Tang, C., &amp; Wiesenfeld, K. (1987). Self-organized criticality. Physical Review Letters, 59(4), 381\u2013384.&#xff08;\u81ea\u7ec4\u7ec7\u4e34\u754c\u6027\u4e0e\u6807\u5ea6\u5f8b\u3002&#xff09;<\/li>\n<li>Anderson, P. W. (1972). More Is Different. Science, 177(4047), 393\u2013396.&#xff08;\u6d8c\u73b0\u4e0e\u8de8\u5c3a\u5ea6\u5c42\u6b21\u7684\u7ecf\u5178\u8bba\u8ff0\u3002&#xff09;<\/li>\n<li>Holland, J. H. (1992). Adaptation in Natural and Artificial Systems (rev. ed.). MIT Press.&#xff08;\u590d\u6742\u9002\u5e94\u7cfb\u7edf\u4e0e\u6d8c\u73b0\u3002&#xff09;<\/li>\n<li>Crutchfield, J. P. (1994). The Calculi of Emergence. Physica D, 75, 11\u201354.&#xff08;\u8ba1\u7b97\u529b\u5b66\u4e0e\u6d8c\u73b0\u7684\u5b9a\u91cf\u5316\u3002&#xff09;<\/li>\n<li>Brunton, S. L., Proctor, J. L., &amp; Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932\u20133937.&#xff08;SINDy \u7b26\u53f7\u56de\u5f52\u65b9\u6cd5&#xff0c;\u4ece\u6570\u636e\u8fa8\u8bc6\u63a7\u5236\u65b9\u7a0b\u3002&#xff09;<\/li>\n<li>Raissi, M., Perdikaris, P., &amp; Karniadakis, G. E. (2019). Physics-informed neural networks. Journal of Computational Physics, 378, 686\u2013707.&#xff08;PINN \u628a\u7269\u7406\u65b9\u7a0b\u4f5c\u4e3a\u5148\u9a8c\u5d4c\u5165\u7f51\u7edc\u3002&#xff09;<\/li>\n<li>Greydanus, S., Dzamba, M., &amp; Yosinski, J. (2019). Hamiltonian Neural Networks. Advances in Neural Information Processing Systems (NeurIPS), 32.&#xff08;HNN \u628a\u80fd\u91cf\u5b88\u6052\u4f5c\u4e3a\u7f51\u7edc\u786c\u7ea6\u675f\u3002&#xff09;<\/li>\n<li>Feynman, R. P., Leighton, R. B., &amp; Sands, M. (1963\u20131965). The Feynman Lectures on Physics. Addison-Wesley.&#xff08;\u7ecf\u5178\u529b\u5b66\u3001\u5bf9\u79f0\u4e0e\u5b88\u6052\u7684\u6e05\u6670\u9610\u8ff0\u3002&#xff09;<\/li>\n<li>Strogatz, S. H. (2018). Nonlinear Dynamics and Chaos (2nd ed.). CRC Press.&#xff08;\u6df7\u6c8c\u3001\u5206\u5c94\u4e0e\u76f8\u7a7a\u95f4\u51e0\u4f55\u7684\u6807\u51c6\u6559\u6750\u3002&#xff09;<\/li>\n<li>Gleick, J. (1987). Chaos: Making a New Science. Viking.&#xff08;\u6df7\u6c8c\u7406\u8bba\u7684\u79d1\u5b66\u53f2\u666e\u53ca&#xff0c;\u5e9e\u52a0\u83b1\u8f6c\u5411\u7684\u751f\u52a8\u53d9\u8ff0\u3002&#xff09;<\/li>\n<li>Popper, K. R. (1959\/1963). The Logic of Scientific Discovery&#xff08;\u53ca Conjectures and Refutations&#xff09;. Routledge.&#xff08;\u8bc1\u4f2a\u4e3b\u4e49\u4e0e\u53ef\u9519\u8bba\u3002&#xff09;<\/li>\n<li>Kuhn, T. S. (1962). The Structure of Scientific Revolutions. University of Chicago Press.&#xff08;\u8303\u5f0f\u4e0e\u4e0d\u53ef\u901a\u7ea6\u6027\u3002&#xff09;<\/li>\n<li>Kant, I. (1781\/1787). Kritik der reinen Vernunft&#xff08;\u7eaf\u7cb9\u7406\u6027\u6279\u5224&#xff09;. &#xff08;\u7269\u81ea\u4f53\u4e0e\u73b0\u8c61\u4e4b\u8fa8\u3002&#xff09;<\/li>\n<hr \/>\n<h2>\u9644\u5f55 A \u672f\u8bed\u4e0e\u7b26\u53f7\u8868<\/h2>\n<table>\n<tr>\u7b26\u53f7 \/ \u672f\u8bed\u542b\u4e49<\/tr>\n<tbody>\n<tr>\n<td>T<\/td>\n<td>the real Thing&#xff1a;\u4e0d\u53ef\u88ab\u5b8c\u5168\u5bb9\u7eb3\u7684\u771f\u5b9e\u7cfb\u7edf<\/td>\n<\/tr>\n<tr>\n<td>\u03a0<\/td>\n<td>Projection&#xff1a;\u5bf9 T \u7684\u6570\u5b66\/\u8ba4\u77e5\u6295\u5f71<\/td>\n<\/tr>\n<tr>\n<td>O<\/td>\n<td>Order&#xff1a;\u5728 \u03a0 \u4e0b\u7a33\u5b9a\u6210\u7acb\u7684\u53ef\u8ba1\u7b97\u79e9\u5e8f<\/td>\n<\/tr>\n<tr>\n<td>\u76f8\u7a7a\u95f4<\/td>\n<td>\u4ee5\u7cfb\u7edf\u5168\u90e8\u53d8\u91cf\u4e3a\u5750\u6807\u7684\u62bd\u8c61\u72b6\u6001\u7a7a\u95f4<\/td>\n<\/tr>\n<tr>\n<td>\u540c\u5bbf\u8f68\u9053<\/td>\n<td>\u4ece\u4e0d\u7a33\u5b9a\u5e73\u8861\u70b9\u51fa\u53d1\u53c8\u65e0\u9650\u903c\u8fd1\u5b83\u7684\u8f68\u8ff9<\/td>\n<\/tr>\n<tr>\n<td>Lyapunov \u6307\u6570<\/td>\n<td>\u91cf\u5316\u8f68\u8ff9\u5bf9\u521d\u59cb\u6761\u4ef6\u654f\u611f\u7a0b\u5ea6\u7684\u6807\u91cf<\/td>\n<\/tr>\n<tr>\n<td>\u5947\u602a\u5438\u5f15\u5b50<\/td>\n<td>\u6df7\u6c8c\u7cfb\u7edf\u957f\u671f\u884c\u4e3a\u88ab\u5438\u5411\u7684\u5206\u5f62\u7ed3\u6784<\/td>\n<\/tr>\n<tr>\n<td>\u4e0d\u53d8\u91cf<\/td>\n<td>\u8de8\u6295\u5f71\/\u5c3a\u5ea6\u4ecd\u5b58\u6d3b\u7684 O&#xff08;\u5b88\u6052\u91cf\u3001\u6807\u5ea6\u5f8b\u7b49&#xff09;<\/td>\n<\/tr>\n<tr>\n<td>\u6295\u5f71\u591a\u5143\u4e3b\u4e49<\/td>\n<td>\u65e0\u552f\u4e00\u6b63\u786e\u6a21\u578b&#xff0c;\u53ea\u6709\u9002\u5408\u95ee\u9898\u7684\u6295\u5f71<\/td>\n<\/tr>\n<tr>\n<td>PINN \/ HNN \/ SINDy \/ GNN<\/td>\n<td>\u56db\u7c7b\u5c06 AI \u7528\u4f5c \u03a0 \u4e0e O \u53d1\u73b0\u5668\u7684\u65b9\u6cd5<\/td>\n<\/tr>\n<tr>\n<td>\u4e09\u89d2\u9a8c\u8bc1<\/td>\n<td>\u591a\u4e2a\u6295\u5f71\u5f7c\u6b64\u5370\u8bc1\u4ee5\u63d0\u5347 O \u7684\u53ef\u4fe1\u5ea6<\/td>\n<\/tr>\n<tr>\n<td>\u8ba1\u7b97\u4e0d\u53ef\u7ea6\u6027<\/td>\n<td>\u67d0\u4e9b\u7cfb\u7edf\u7684\u957f\u671f\u884c\u4e3a\u5728\u539f\u5219\u4e0a\u4e0d\u53ef\u88ab\u52a0\u901f\u9884\u6d4b<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>\u6458\u8981<br \/>\n\u4eba\u7c7b\u7406\u89e3\u590d\u6742\u4e16\u754c\u7684\u65b9\u5f0f&#xff0c;\u957f\u671f\u88ab\u4e00\u4e2a\u9690\u542b\u7684\u8bc9\u6c42\u652f\u914d&#xff1a;\u5bfb\u627e\u4e00\u6761\u80fd\u591f\u7cbe\u786e\u9884\u8a00\u672a\u6765\u7684\\&#8221;\u7ec8\u6781\u516c\u5f0f\\&#8221;\u3002\u7136\u800c&#xff0c;\u81ea\u5341\u4e5d\u4e16\u7eaa\u672b\u5e9e\u52a0\u83b1&#xff08;Henri Poincar\u00e9&#xff09;\u7814\u7a76\u4e09\u4f53\u95ee\u9898\u8d77&#xff0c;\u6570\u5b66\u4e0e\u7269\u7406\u5b66\u5c31\u4e00\u6b21\u6b21\u649e\u4e0a\u540c\u4e00\u5835\u5899\u2014\u2014\u5bf9\u4e8e\u8db3\u591f\u590d\u6742\u7684\u7cfb\u7edf&#xff0c;\u7cbe\u786e\u5730\u590d\u73b0\u5176\u5168\u90e8\u7ec6\u8282\u5728\u539f\u5219\u4e0a\u65e2\u4e0d\u53ef\u884c\u4e5f\u4e0d\u53ef\u6c42\u3002\u672c\u6587\u63d0\u51fa\u5e76\u7cfb\u7edf\u8bba\u8bc1\u4e00\u4e2a\u7edf\u4e00\u7684\u7406\u89e3\u6846\u67b6 T-\u03a0-O&#xff1a;\u5176\u4e2d T \u8868\u793a\u4e0d\u53ef\u88ab\u5b8c\u5168\u5bb9\u7eb3\u7684\u771f\u5b9e\u7cfb\u7edf&amp;#xf<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[9789,50],"topic":[],"class_list":["post-97183","post","type-post","status-publish","format-standard","hentry","category-server","tag-9789","tag-50"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.3 - 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