{"id":54097,"date":"2024-12-07T19:15:28","date_gmt":"2024-12-07T11:15:28","guid":{"rendered":"https:\/\/swarma.org\/?p=54097"},"modified":"2024-12-07T19:15:28","modified_gmt":"2024-12-07T11:15:28","slug":"%e5%a6%82%e4%bd%95%e7%94%a8koopman%e5%88%86%e6%9e%90%e8%a7%a3%e7%a0%81%e9%9d%9e%e7%ba%bf%e6%80%a7%e5%8a%a8%e5%8a%9b%e5%ad%a6-%e6%96%b0%e8%af%be%e4%b8%8a%e7%ba%bf","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=54097","title":{"rendered":"\u5982\u4f55\u7528Koopman\u5206\u6790\u89e3\u7801\u975e\u7ebf\u6027\u52a8\u529b\u5b66 | \u65b0\u8bfe\u4e0a\u7ebf"},"content":{"rendered":"
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\u8fd9\u4e9b\u73b0\u8c61\u80cc\u540e\u90fd\u6307\u5411\u4e86\u540c\u4e00\u4e2a\u6838\u5fc3\u6311\u6218\uff1a\u5982\u4f55\u5728\u590d\u6742\u6027\u4e2d\u5bfb\u627e\u79e9\u5e8f\uff0c\u5728\u975e\u7ebf\u6027\u4e2d\u53d1\u73b0\u89c4\u5f8b\uff1f<\/strong><\/span><\/p>\n


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\u5176\u4e2dKoopman\u7b97\u7b26\u662f\u6700\u91cd\u8981\u7684\u65b9\u6cd5\u4e4b\u4e00\uff0c\u8be5\u7406\u8bba\u6700\u521d\u7531\u8457\u540d\u7684\u6570\u5b66\u5bb6Bernard Koopman\u57281931\u5e74\u63d0\u51fa\uff0c\u76ee\u524d\u5df2\u7ecf\u88ab\u53d1\u5c55\u6210\u4e3a\u4e00\u79cd\u65b0\u9896\u7684\u7528\u4e8e\u7814\u7a76\u975e\u7ebf\u6027\u52a8\u529b\u7cfb\u7edf\u7684\u6709\u529b\u7406\u8bba\uff0c\u53ef\u4ee5\u7528\u65e0\u7a77\u7ef4\u7ebf\u6027\u7b97\u5b50\u8868\u5f81\u975e\u7ebf\u6027\u7cfb\u7edf\u7684\u6f14\u5316\u3002\u5728\u521a\u63d0\u51fa\u65f6\uff0c\u7531\u4e8e\u5f53\u65f6\u7814\u7a76\u95ee\u9898\u548c\u8ba1\u7b97\u80fd\u529b\u7684\u9650\u5236\uff0c\u8fd9\u4e2a\u7406\u8bba\u5e76\u672a\u88ab\u5927\u89c4\u6a21\u4f7f\u7528\u3002\u968f\u7740\u6570\u636e\u7684\u79ef\u7d2f\u3001\u8ba1\u7b97\u673a\u7b97\u529b\u7684\u63d0\u5347\uff0c\u7279\u522b\u662f\u6700\u8fd1\u5341\u51e0\u5e74\u6765\uff0c\u76f8\u5173\u7684\u7814\u7a76\u5982\u6eda\u96ea\u7403\u822c\u589e\u957f\uff0c\u5728\u6d41\u4f53\u529b\u5b66\u3001\u7535\u7f51\u8fd0\u884c\u3001\u667a\u80fd\u5efa\u7b51\u3001\u8f6f\u673a\u5668\u4eba\u3001\u795e\u7ecf\u79d1\u5b66\u3001\u6c14\u5019\u7b49\u591a\u4e2a\u9886\u57df\u5f97\u5230\u5e7f\u6cdb\u5e94\u7528\u3002<\/span><\/p>\n


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Koopman\u7b97\u7b26\u4ee5\u5176\u5c06\u975e\u7ebf\u6027\u95ee\u9898\u8f6c\u5316\u4e3a\u7ebf\u6027\u95ee\u9898\u7684\u80fd\u529b\uff0c\u4e3a\u590d\u6742\u7cfb\u7edf\u63d0\u4f9b\u4e86\u4e00\u79cd\u76f4\u89c2\u800c\u5f3a\u5927\u7684\u5206\u6790\u5de5\u5177\uff0c\u4e0d\u4ec5\u80fd\u591f\u89e3\u91ca\u7269\u7406\u548c\u5de5\u7a0b\u73b0\u8c61\uff0c\u8fd8\u4e3a\u6570\u636e\u9a71\u52a8\u7684\u79d1\u5b66\u7814\u7a76\u548c\u673a\u5668\u5b66\u4e60\u63d0\u4f9b\u4e86\u575a\u5b9e\u7684\u7406\u8bba\u57fa\u7840\u3002<\/span><\/p>\n


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\u8be5\u9886\u57df\u7684\u91cd\u8981\u5b66\u8005Igor Mezi\u0107\u57282013\u5e74\u7684\u6587\u7ae0\u4e2d\uff0c\u7efc\u8ff0\u4e86Koopman\u6a21\u6001\u5728\u6d41\u4f53\u529b\u5b66\u4e2d\u7684\u7406\u8bba\u4e0e\u5e94\u7528\uff0c\u8be5\u7406\u8bba\u7edf\u4e00\u4e86\u6d41\u4f53\u529b\u5b66\u4e2d\u8bb8\u591a\u4e0d\u540c\u7684\u6982\u5ff5\uff0c\u5e76\u63d0\u4f9b\u4e86\u4e25\u683c\u7684\u80cc\u666f\uff1b\u5317\u90ae\u5170\u5cb3\u6052\u6559\u6388\u57282022\u5e74\u7684\u6587\u7ae0\u4e2d\uff0c\u7ed3\u5408Koopman\u5206\u6790\uff0c\u8bbe\u8ba1\u4e86\u795e\u7ecf\u7f51\u7edc\u63d0\u53d6\u6700\u7b80\u6a21\u5f0f\u6765\u5b9e\u73b0\u52a8\u529b\u7cfb\u7edf\u7684\u91cd\u6784\u548c\u6f14\u5316\uff1b\u6e05\u534e\u7530\u6d0b\u56e2\u961f\u57282024\u5e74\u7684\u6587\u7ae0\u4e2d\uff0c\u63d0\u51fa\u4e86\u4e00\u79cd\u540d\u4e3aKoopman\u795e\u7ecf\u7b97\u5b50\u7684\u6a21\u578b\uff0c\u8be5\u6a21\u578b\u5b9e\u73b0\u4e86\u504f\u5fae\u5206\u65b9\u7a0b\u6216\u771f\u5b9e\u52a8\u529b\u7cfb\u7edf\u957f\u65f6\u95f4\u6f14\u5316\u7684\u80fd\u529b\u3002<\/span><\/p>\n


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\u5173\u4e8eKoopman\u7b97\u7b26\uff0c\u8fd8\u6709\u5f88\u591a\u91cd\u8981\u7684\u7814\u7a76\u6210\u679c\u548c\u601d\u60f3\u503c\u5f97\u6211\u4eec\u5b66\u4e60\uff0c\u540c\u65f6\u4e5f\u6709\u5f88\u591a\u95ee\u9898\u503c\u5f97\u6211\u4eec\u53bb\u89e3\u51b3\u3002\u4e3a\u6b64\uff0c\u96c6\u667a\u5b66\u56ed\u9080\u8bf7\u4e86\u5170\u5cb3\u6052\u6559\u6388\u5f00\u8bbe\u76f8\u5173\u8bfe\u7a0b\uff0c\u4eceKoopman\u7b97\u7b26\u7684\u5b9a\u4e49\u51fa\u53d1\uff0c\u8bb2\u8ff0\u5176\u5b9a\u4e49\u3001\u57fa\u672c\u6027\u8d28\u548c\u8c31\u7279\u5f81\u7684\u6570\u503c\u8ba1\u7b97\u65b9\u6cd5\uff0c\u4ee5\u53ca\u5728\u51e0\u4e2a\u5178\u578b\u975e\u7ebf\u6027\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528\u3002\u9080\u8bf7\u5173\u6ce8\u8fd9\u4e00\u9886\u57df\u53d1\u5c55\u7684\u540c\u5b66\u52a0\u5165\uff0c\u5171\u540c\u63a2\u8ba8\u3002<\/span><\/p>\n


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    \n
  1. \n

    Koopman, Bernard O. "Hamiltonian systems and transformation in Hilbert space." Proceedings of the National Academy of Sciences 17.5 (1931): 315-318.<\/span><\/p>\n<\/li>\n

  2. \n

    Mezi\u0107, Igor. "Analysis of fluid flows via spectral properties of the Koopman operator." Annual review of fluid mechanics 45.1 (2013): 357-378.<\/span><\/p>\n<\/li>\n

  3. \n

    Li, Chufan, and Yueheng Lan. "Koopman analysis of nonlinear systems with a neural network representation." Communications in Theoretical Physics 74.9 (2022): 095604.<\/span><\/p>\n<\/li>\n

  4. \n

    C. Zhang, H. Li and Y. Lan. \u201cPhase space partition with Koopman analysis\u201d. Chaos 32 (2022): 063132.<\/span><\/p>\n<\/li>\n

  5. \n

    S. Wang and Y. Lan. \u201cProbing the phase space of coupled oscillators with Koopman analysis\u201d. Phys. Rev. E 104 (2021): 034211.<\/span><\/p>\n<\/li>\n

  6. \n

    J. Hu and Y. Lan. \u201cKoopman analysis in oscillator synchronization\u201d. Phys. Rev. E 102 (2020): 062216.<\/span><\/p>\n<\/li>\n

  7. \n

    Xiong, Wei, et al. "Koopman neural operator as a mesh-free solver of non-linear partial differential equations." Journal of Computational Physics<\/em> (2024): 113194.<\/span><\/p>\n<\/li>\n<\/ol>\n


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      \u8fd9\u4e9b\u90fd\u662f\u590d\u6742\u7cfb\u7edf\u7814\u7a76\u7684\u57fa\u672c\u95ee\u9898\uff0c\u9700\u8981\u6709\u4e00\u5957\u7cfb\u7edf\u7684\u5206\u6790\u65b9\u6cd5\u3002<\/span><\/p>\n


      <\/span><\/p>\n

      Koopman\u7b97\u7b26\u7406\u8bba\u76f4\u63a5\u4ece\u53ef\u89c2\u6d4b\u91cf\u51fa\u53d1\uff0c\u8003\u5bdf\u5b9a\u4e49\u5728\u72b6\u6001\u7a7a\u95f4\u4e2d\u76f8\u5173\u51fd\u6570\u7684\u6f14\u5316\uff0c\u4ece\u800c\u5c06\u4e00\u4e2a\u975e\u7ebf\u6027\u95ee\u9898\u8f6c\u5316\u4e3a\u65e0\u7a77\u7ef4\u51fd\u6570\u7a7a\u95f4\u4e2d\u7684\u7ebf\u6027\u95ee\u9898\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u53ea\u9700\u8981\u9009\u62e9\u5408\u9002\u7684\u51fd\u6570\u7a7a\u95f4\uff0c\u5c31\u53ef\u4ee5\u7528\u4e00\u4e2a\u9ad8\u7ef4\u7684\u7ebf\u6027\u95ee\u9898\u6765\u8fd1\u4f3c\u539f\u6709\u7684\u975e\u7ebf\u6027\u95ee\u9898\uff0c\u5e76\u4fdd\u6301\u76f8\u5173\u7684\u4f4e\u9891\u65f6\u7a7a\u7279\u5f81\u4e0d\u53d8\u3002\u76ee\u524d\u8fd9\u4e00\u7406\u8bba\u88ab\u7528\u4e8e\u5404\u79cd\u6a21\u578b\u548c\u6570\u636e\u7684\u5206\u6790\uff0c\u4ea7\u751f\u4e86\u4e00\u6279\u6709\u91cd\u8981\u610f\u4e49\u7684\u7ed3\u679c\uff0c\u6f5c\u5728\u5e94\u7528\u5305\u62ec\u590d\u6742\u7cfb\u7edf\u65f6\u95f4\u5e8f\u5217\u5206\u6790\u3001\u91cd\u8981\u52a8\u529b\u5b66\u6a21\u5f0f\u7684\u63d0\u53d6\u3001\u52a8\u529b\u5b66\u6a21\u578b\u7684\u91cd\u6784\u3001\u4ee5\u53ca\u673a\u5668\u5b66\u4e60\u7684\u65f6\u7a7a\u6570\u636e\u9884\u5904\u7406\u7b49\u3002<\/span><\/p>\n


      <\/span><\/p>\n

      \u5176\u4e2d\u8fd8\u5b58\u5728\u5f88\u591a\u503c\u5f97\u7814\u7a76\u7684\u95ee\u9898\uff0c\u5305\u62ec\u5982\u4f55\u9009\u62e9\u5408\u9002\u57fa\u51fd\u6570\u8fdb\u884c\u6700\u6709\u6548\u5206\u6790\uff1f\u5982\u4f55\u4eceKoopman\u8c31\u5206\u89e3\u4e2d\u786e\u5b9a\u91cd\u8981\u7684\u672c\u5f81\u503c\u548c\u672c\u5f81\u5411\u91cf\uff1f\u5bf9\u4e8e\u6709\u9650\u65f6\u95f4\u957f\u5ea6\u548c\u6709\u9650\u7a7a\u95f4\u8303\u56f4\u7684\u6570\u636e\uff0c\u63d0\u53d6\u7684\u7279\u5f81\u53ef\u4ee5\u53cd\u6620\u7cfb\u7edf\u7684\u4ec0\u4e48\u6027\u8d28\uff1f<\/span><\/p>\n


      <\/span><\/p>\n

      \u5728\u672c\u7cfb\u5217\u8bfe\u7a0b\u4e2d\u5c06\u4eceKoopman\u7b97\u7b26\u7684\u5b9a\u4e49\u51fa\u53d1\uff0c\u8bb2\u8ff0\u5176\u5b9a\u4e49\u3001\u57fa\u672c\u6027\u8d28\u548c\u8c31\u7279\u5f81\u7684\u6570\u503c\u8ba1\u7b97\u65b9\u6cd5\uff0c\u4ee5\u53ca\u5728\u51e0\u4e2a\u5178\u578b\u975e\u7ebf\u6027\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528\u3002<\/span><\/p>\n


      <\/h3>\n

      \u8bfe\u7a0b\u5927\u7eb2<\/span><\/strong><\/span><\/h3>\n\n\n\n\n<\/colgroup>\n\n\n\n\n\n\n\n\n
      \n

      \u8282\u5e8f<\/span><\/p>\n<\/td>\n

      		\n

      \u8bfe\u7a0b\u5927\u7eb2<\/span><\/p>\n<\/td>\n

      		\n

      \u7b80\u4ecb<\/span><\/p>\n<\/td>\n<\/tr>\n

      \n

      1<\/span><\/p>\n<\/td>\n

      		\n

      \u590d\u6742\u7cfb\u7edf\u548c\u975e\u7ebf\u6027\u52a8\u529b\u5b66<\/strong><\/span><\/p>\n

        \n
      1. \n

        \u590d\u6742\u7cfb\u7edf\u53ca\u5176\u7279\u5f81<\/span><\/p>\n<\/li>\n

      2. \n

        \u52a8\u529b\u5b66\u7cfb\u7edf\u4e0e\u76f8\u7a7a\u95f4<\/span><\/p>\n<\/li>\n

      3. \n

        \u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7814\u7a76\u7684\u673a\u4f1a\u4e0e\u6311\u6218<\/span><\/p>\n<\/li>\n

      4. \n

        \u5404\u7c7b\u5206\u5c94\u73b0\u8c61\u53ca\u5176\u63cf\u8ff0<\/span><\/p>\n<\/li>\n<\/ol>\n<\/td>\n

      		\n

      \u590d\u6742\u7cfb\u7edf\u90fd\u662f\u9ad8\u7ef4\u975e\u7ebf\u6027\u4f53\u7cfb\uff0c\u5177\u6709\u5c42\u7ea7\u7ed3\u6784\u548c\u6d8c\u73b0\u52a8\u529b\u5b66\uff0c\u80fd\u591f\u5b66\u4e60\u548c\u9002\u5e94\u73af\u5883\u53d8\u5316\u3002\u7814\u7a76\u590d\u6742\u7cfb\u7edf\uff0c\u9996\u5148\u8981\u719f\u6089\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7684\u7406\u8bba\u53ca\u5176\u7814\u7a76\u7684\u5e38\u89c1\u5de5\u5177\u3002\u8fd9\u91cc\u4ecb\u7ecd\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7684\u51e0\u4f55\u63cf\u8ff0\u65b9\u6cd5\u548c\u53c2\u6570\u53d8\u5316\u4e0b\u5404\u7c7b\u52a8\u529b\u5b66\u7684\u4ea7\u751f-\u5206\u5c94\u73b0\u8c61\u3002\u540e\u9762\u53ef\u4ee5\u770b\u5230\uff0c\u5728Koopman\u7b97\u7b26\u7684\u5e2e\u52a9\u4e0b\uff0c\u8fd9\u4e9b\u5de5\u5177\u53ef\u4ee5\u5e94\u7528\u5230\u9ad8\u7ef4\u4f53\u7cfb\u7684\u5206\u6790\u4e2d\u3002<\/span><\/p>\n<\/td>\n<\/tr>\n

      \n

      2<\/span><\/p>\n<\/td>\n

      		\n

      \u5c40\u90e8\u548c\u5168\u5c40\u7ebf\u6027\u5316<\/strong><\/span><\/p>\n

        \n
      1. \n

        \u975e\u7ebf\u6027\u7cfb\u7edf\u7684\u7ebf\u6027\u5316<\/span><\/p>\n<\/li>\n

      2. \n

        \u5173\u4e8e\u7ebf\u6027\u5316\u7684\u95ee\u9898\u4e0e\u601d\u8003<\/span><\/p>\n<\/li>\n

      3. \n

        \u4e24\u4e2a\u6848\u4f8b<\/span><\/p>\n<\/li>\n<\/ol>\n<\/td>\n

      		\n

      \u7ebf\u6027\u65b9\u7a0b\u7531\u4e8e\u5176\u89e3\u5177\u6709\u53ef\u53e0\u52a0\u6027\uff0c\u53ef\u4ee5\u901a\u8fc7\u6c42\u89e3\u672c\u5f81\u95ee\u9898\u83b7\u5f97\u5b8c\u7f8e\u89e3\u51b3\uff0c\u6240\u4ee5\uff0c\u975e\u7ebf\u6027\u95ee\u9898\u5982\u679c\u80fd\u591f\u8f6c\u5316\u4e3a\u7ebf\u6027\u95ee\u9898\uff0c\u5373\u80fd\u591f\u8fdb\u884c\u7ebf\u6027\u5316\uff0c\u95ee\u9898\u5c31\u5927\u5927\u7b80\u5316\u4e86\u3002\u53ef\u60dc\u7684\u662f\uff0c\u5927\u591a\u6570\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u95ee\u9898\u4ec5\u4ec5\u80fd\u591f\u8fdb\u884c\u5c40\u90e8\u7ebf\u6027\u5316\uff0c\u5168\u5c40\u7684\u7ebf\u6027\u5316\u975e\u5e38\u56f0\u96be\u6216\u8005\u4e0d\u53ef\u80fd\u3002\u4f46\u67d0\u4e9b\u65f6\u5019\u6211\u4eec\u53ef\u4ee5\u5728\u8f83\u5927\u8303\u56f4\u5185\u6216\u8005\u5728\u7279\u5b9a\u6db5\u4e49\u4e0b\uff08Koopman\u5206\u6790\uff09\u8fdb\u884c\u7ebf\u6027\u5316\uff0c\u4ece\u800c\u7aa5\u5f97\u89e3\u7a7a\u95f4\u4e4b\u7ed3\u6784\u3002<\/span><\/p>\n<\/td>\n<\/tr>\n

      \n

      3<\/span><\/p>\n<\/td>\n

      		\n

      \u52a8\u529b\u5b66\u4e0e\u7edf\u8ba1<\/strong><\/span><\/p>\n

        \n
      1. \n

        \u7edf\u8ba1\u7269\u7406\u539f\u7406<\/span><\/p>\n<\/li>\n

      2. \n

        \u6700\u5927\u71b5\u539f\u7406\u548c\u80fd\u91cf\u666f\u89c2<\/span><\/p>\n<\/li>\n<\/ol>\n<\/td>\n

      		\n

      \u5173\u4e8e\u9ad8\u81ea\u7531\u5ea6\u975e\u7ebf\u6027\u4f53\u7cfb\u7684\u5206\u6790\uff0c\u7269\u7406\u5b66\u5bb6\u65e9\u5c31\u52aa\u529b\u8fdb\u884c\u4e86\uff0c\u4e5f\u53d1\u5c55\u51fa\u4e00\u4e9b\u6709\u529b\u5de5\u5177\uff0c\u8fd9\u5c31\u662f\u7edf\u8ba1\u7269\u7406\uff0c\u5305\u62ec\u5e73\u8861\u548c\u975e\u5e73\u8861\u7edf\u8ba1\uff0c\u7740\u773c\u70b9\u4ece\u5355\u4e2a\u7684\u72b6\u6001\u8d70\u5411\u72b6\u6001\u96c6\u5408\u3002\u8fd9\u91cc\u6211\u4eec\u5f3a\u8c03\u7edf\u8ba1\u7269\u7406\u548c\u52a8\u529b\u5b66\u7684\u8054\u7cfb\uff0c\u770b\u770b\u4ece\u52a8\u529b\u5b66\u5230\u7ecf\u5178\u7edf\u8ba1\u7269\u7406\u9700\u8981\u7684\u7b80\u5316\u5047\u8bbe\uff0c\u4ecb\u7ecd\u4e00\u4e0b\u52a8\u529b\u5b66\u7684\u6d4b\u5ea6\u7406\u8bba\u601d\u60f3\u3002\u8ba8\u8bba\u7edf\u8ba1\u7269\u7406\u6846\u67b6\u662f\u5426\u80fd\u591f\u6db5\u76d6\u975e\u54c8\u5bc6\u987f\u4f53\u7cfb\uff0c\u5173\u6ce8\u6700\u5927\u71b5\u539f\u7406\u548c\u80fd\u91cf\u666f\u89c2\u3002<\/span><\/p>\n<\/td>\n<\/tr>\n

      \n

      4<\/span><\/p>\n<\/td>\n

      		\n

      Koopman\u7b97\u7b26\u548c\u52a8\u529b\u5b66\u6a21\u5f0f<\/strong><\/span><\/p>\n

        \n
      1. \n

        Koopman\u7b97\u7b26<\/span><\/p>\n<\/li>\n

          \n
        1. \n

          \u566a\u58f0\u548cSRB\u6d4b\u5ea6<\/span><\/p>\n<\/li>\n

        2. \n

          \u7279\u5f81\u503c\u4e0e\u7279\u5f81\u6a21\u5f0f<\/span><\/p>\n<\/li>\n

        3. \n

          \u91cd\u8981\u7684\u6848\u4f8b<\/span><\/p>\n<\/li>\n<\/ol>\n

        4. \n

          \u52a8\u529b\u5b66\u6a21\u5f0f\u5206\u89e3<\/span><\/p>\n<\/li>\n<\/ol>\n<\/td>\n

      		\n

      Koopman\u7b97\u7b26\u7528\u53e6\u4e00\u79cd\u65b9\u5f0f\u5c06\u7edf\u8ba1\u7269\u7406\u548c\u52a8\u529b\u5b66\u601d\u60f3\u7ed3\u5408\uff0c\u5c06\u6211\u4eec\u7684\u6ce8\u610f\u529b\u4ece\u5355\u6761\u8f68\u9053\u8f6c\u6362\u5230\u72b6\u6001\u7a7a\u95f4\u7684\u51fd\u6570\u53d8\u5316\uff0c\u8fd9\u6837\u975e\u7ebf\u6027\u95ee\u9898\u53d8\u6210\u4e86\u51fd\u6570\u7a7a\u95f4\u4e2d\u7684\u7ebf\u6027\u95ee\u9898\u3002\u6211\u4eec\u53ef\u4ee5\u501f\u52a9\u672c\u5f81\u503c\u548c\u672c\u5f81\u51fd\u6570\u6765\u7814\u7a76\u4f53\u7cfb\u7684\u6027\u8d28\uff0c\u5b9a\u4e49\u52a8\u529b\u5b66\u6a21\u5f0f\uff0c\u5b9e\u73b0\u590d\u6742\u95ee\u9898\u7684\u5206\u89e3\uff0c\u5728\u67d0\u79cd\u610f\u4e49\u4e0a\u5b9e\u73b0\u4e86\u7cfb\u7edf\u7684\u7ebf\u6027\u5316\u3002Koopman\u7b97\u7b26\u5b9e\u9645\u4e0a\u5305\u542b\u4e86\u4f53\u7cfb\u7684\u6240\u6709\u52a8\u529b\u5b66\u4fe1\u606f\uff0c\u4f46\u53ef\u4ee5\u7528\u4e0d\u540c\u7684\u89c2\u6d4b\u91cf\u5b9e\u73b0\u7c97\u7c92\u5316\uff0c\u5341\u5206\u5951\u5408\u590d\u6742\u4f53\u7cfb\u7684\u5c42\u6b21\u5316\u7279\u5f81\u3002<\/span><\/p>\n<\/td>\n<\/tr>\n

      \n

      5<\/span><\/p>\n<\/td>\n

      		\n

      Koopman\u7b97\u7b26\u5728\u7b26\u53f7\u52a8\u529b\u5b66\u7684\u5e94\u7528<\/strong><\/span><\/p>\n

        \n
      1. \n

        Logistic\u56fe\u7684\u7b26\u53f7\u5212\u5206<\/span><\/p>\n<\/li>\n

      2. \n

        H\u00e9non \u56fe\u4e2d\u7684\u5206\u533a<\/span><\/p>\n<\/li>\n<\/ol>\n<\/td>\n

      		\n

      \u4e00\u4e2a\u91cd\u8981\u7684\u5e94\u7528\u5c31\u662f\u5728\u7b26\u53f7\u52a8\u529b\u5b66\u4e2d\u7684\u5e94\u7528\u3002\u7b26\u53f7\u52a8\u529b\u5b66\u662f\u975e\u7ebf\u6027\u8f68\u9053\u7684\u4e00\u79cd\u4e25\u683c\u62d3\u6251\u63cf\u8ff0\uff0c\u7528\u4e0d\u540c\u7b26\u53f7\u63cf\u8ff0\u52a8\u529b\u5b66\u7684\u4e0d\u540c\u90e8\u5206\uff0c\u8f68\u9053\u53ef\u4ee5\u7528\u7b26\u53f7\u5e8f\u5217\u6765\u8868\u793a\uff0c\u56e0\u800c\u4e5f\u662f\u8054\u7cfb\u4fe1\u606f\u7406\u8bba\u7684\u91cd\u8981\u5de5\u5177\uff0c\u5728\u667a\u80fd\u65f6\u4ee3\u5177\u6709\u7279\u6b8a\u7684\u610f\u4e49\u3002\u5176\u4e2d\u6700\u5927\u7684\u56f0\u96be\u662f\u52a8\u529b\u5b66\u4e0d\u540c\u90e8\u5206\u600e\u6837\u5212\u5206\uff0cKoopman\u7b97\u7b26\u65b9\u6cd5\u63d0\u4f9b\u4e86\u4e00\u4e2a\u5168\u65b0\u7684\u505a\u6cd5\u3002<\/span><\/p>\n<\/td>\n<\/tr>\n

      \n

      6<\/span><\/p>\n<\/td>\n

      		\n

      Koopman\u7b97\u7b26\u7684\u66f4\u591a\u5e94\u7528<\/strong><\/span><\/p>\n

        \n
      1. \n

        \u7f51\u7edc\u4e0a\u7684Kuramoto\u6a21\u578b<\/span><\/p>\n<\/li>\n

      2. \n

        \u54c8\u5bc6\u987f\u52a8\u529b\u5b66\u4e2d\u7684\u5e94\u7528<\/span><\/p>\n<\/li>\n

      3. \n

        \u6c14\u5019\u95ee\u9898\u7814\u7a76<\/span><\/p>\n<\/li>\n

      4. \n

        \u7cfb\u5217\u8bfe\u7a0b\u603b\u7ed3<\/span><\/p>\n<\/li>\n<\/ol>\n<\/td>\n

      		\n

      Koopman\u7b97\u7b26\u6709\u5f88\u591a\u7684\u5176\u5b83\u5e94\u7528\uff0c\u7279\u522b\u662f\u5728\u5206\u6790\u590d\u6742\u7cfb\u7edf\u52a8\u529b\u5b66\u65b9\u9762\u3002\u8fd9\u91cc\u6211\u4eec\u8fd8\u4f1a\u5217\u4e3e\u51e0\u4e2a\uff1a1. \u7f51\u7edc\u4e0aKuramoto\u632f\u5b50\u6a21\u578b\u540c\u6b65\u76f8\u53d8\u70b9\u7684\u786e\u5b9a\uff08\u590d\u6742\u7cfb\u7edf\u4e2d\u76f8\u53d8\u7684\u8bc6\u522b\uff09\uff1b2. \u9ad8\u7ef4\u76f8\u7a7a\u95f4\u4e2d\u5b9a\u6027\u6027\u8d28\u4e0d\u540c\u8f68\u9053\u7684\u641c\u5bfb\uff08\u975e\u7ebf\u6027\u8fd0\u52a8\u6a21\u5f0f\u7684\u786e\u8ba4\uff09\uff1b3. \u6c14\u5019\u5b66\u7684\u7814\u7a76\u7b49\u7b49\u3002<\/span><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n


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

      \u8bfe\u7a0b\u4e3b\u8bb2\u4eba<\/span><\/strong><\/span><\/h3>\n
      <\/section>\n

      \u5170\u5cb3\u6052<\/strong>\uff0c\u5317\u4eac\u90ae\u7535\u5927\u5b66\u7406\u5b66\u9662\u6559\u6388\uff0c\u535a\u58eb\u5b66\u4f4d\u5728\u4f50\u6cbb\u4e9a\u7406\u5de5\u5b66\u9662\uff08Georgia Institute of Technology\uff09\u83b7\u5f97\u3002\u5148\u540e\u5728\u56fd\u5185\u5916\u591a\u4e2a\u8457\u540d\u5927\u5b66\u5b66\u4e60\u548c\u5de5\u4f5c\u8fc7\uff0c\u6709\u4e30\u5bcc\u7684\u5b66\u79d1\u4ea4\u53c9\u7814\u7a76\u7ecf\u5386\u3002\u4e3b\u8981\u4ece\u4e8b\u975e\u7ebf\u6027\u79d1\u5b66\u3001\u7edf\u8ba1\u7269\u7406\u3001\u751f\u7269\u7269\u7406\u3001\u590d\u6742\u4fe1\u606f\u548c\u667a\u80fd\u7cfb\u7edf\u7b49\u65b9\u9762\u7684\u7814\u7a76\u5de5\u4f5c\uff0c\u6ce8\u91cd\u57fa\u672c\u7406\u8bba\u65b9\u6cd5\u7684\u53d1\u5c55\u548c\u4e0e\u5b9e\u9a8c\u7d27\u5bc6\u7ed3\u5408\u7684\u5e94\u7528\u3002\u73b0\u4e3a\u5317\u4eac\u90ae\u7535\u5927\u5b66\u201c\u6570\u5b66\u4e0e\u4fe1\u606f\u7f51\u7edc\u201d\u6559\u80b2\u90e8\u91cd\u70b9\u5b9e\u9a8c\u5ba4\u526f\u4e3b\u4efb\uff0c\u591a\u6b21\u88ab\u9080\u8bf7\u5728\u56fd\u5185\u5916\u5b66\u672f\u4f1a\u8bae\u4e0a\u62a5\u544a\u81ea\u5df1\u7684\u5de5\u4f5c\uff0c\u540c\u65f6\u62c5\u4efb\u671f\u520a\u201c\u7406\u8bba\u7269\u7406\u901a\u4fe1\u201d\uff08Communications in Theoretical Physics\uff09\u548c\u201c\u73b0\u4ee3\u6570\u5b66\u7269\u7406\u201d\uff08Modern Mathematical Physics\uff09\u7684\u7f16\u59d4\uff0c\u4e5f\u662f\u591a\u4e2a\u56fd\u9645\u8457\u540d\u6742\u5fd7\u7684\u5ba1\u7a3f\u4eba\u3002\u53d1\u8868\u5b66\u672f\u8bba\u6587100\u4f59\u7bc7\uff0c\u5305\u62ec\u56fd\u9645\u9876\u7ea7\u6742\u5fd7PRL\uff0c PNAS\uff0c Nature\u5b50\u520a\u8bba\u6587\u591a\u7bc7\u3002<\/span><\/p>\n


      <\/span><\/p>\n


      <\/span><\/h2>\n

      \n
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      <\/section>\n

      <\/section>\n
      \n
      <\/svg><\/section>\n<\/section>\n<\/section>\n
      \n

      \u8bfe\u7a0b\u4fe1\u606f<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

      \n

      <\/section>\n

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


      <\/p>\n

      \u8bfe\u7a0b\u9002\u7528\u5bf9\u8c61<\/span><\/strong><\/span><\/h3>\n
        \n
      1. \n
        \u7406\u5de5\u79d1\u7814\u7a76\u751f\u6216\u9ad8\u5e74\u7ea7\u672c\u79d1\u751f<\/strong>\uff1a<\/span><\/section>\n<\/li>\n
          \n
        1. \n
          \u5bf9\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u3001\u6570\u5b66\u5efa\u6a21\u3001\u6570\u5b66\u7269\u7406\u6216\u7edf\u8ba1\u529b\u5b66\u611f\u5174\u8da3\u3002<\/span><\/section>\n<\/li>\n
        2. \n
          \u5177\u5907\u4e00\u5b9a\u7684\u5fae\u5206\u65b9\u7a0b\u3001\u7ebf\u6027\u4ee3\u6570\u53ca\u8ba1\u7b97\u57fa\u7840\u3002<\/span><\/section>\n<\/li>\n<\/ol>\n
        3. \n
          \u5bf9\u7406\u8bba\u4e0e\u5b9e\u8df5\u7ed3\u5408\u611f\u5174\u8da3\u7684\u8de8\u5b66\u79d1\u7814\u7a76\u8005<\/strong>\uff1a<\/span><\/section>\n<\/li>\n
            \n
          1. \n
            \u5e0c\u671b\u901a\u8fc7Koopman\u7b97\u5b50\u63a2\u7d22\u975e\u7ebf\u6027\u7cfb\u7edf\u5728\u81ea\u5df1\u7684\u7814\u7a76\u65b9\u5411\u4e2d\u7684\u5e94\u7528\u3002<\/span><\/section>\n<\/li>\n
          2. \n
            \u4ece\u4e8b\u5404\u7c7b\u590d\u6742\u7cfb\u7edf\u7814\u7a76\u3001\u5bfb\u627e\u6709\u529b\u5206\u6790\u5de5\u5177\u3002<\/span><\/section>\n<\/li>\n<\/ol>\n
          3. \n
            \u5177\u6709\u63a2\u7a76\u7cbe\u795e\u7684\u5b66\u751f<\/strong>\uff1a<\/span><\/section>\n<\/li>\n
              \n
            1. \n

              \u4e50\u4e8e\u53c2\u4e0e\u8ba8\u8bba\u3001\u5047\u8bbe\u63a8\u5bfc\u548c\u95ee\u9898\u53cd\u601d\u7684\u5b66\u751f\u3002<\/span><\/p>\n<\/li>\n<\/ol>\n<\/ol>\n


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

              \u8bfe\u7a0b\u8bc1\u4e66<\/strong><\/span><\/h3>\n

              \u8981\u60f3\u89e3\u5f00\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7684\u5965\u79d8\u5e76\u4e0d\u7b80\u5355\uff0c\u4f46\u524d\u8fdb\u7684\u6bcf\u4e00\u6b65\uff0c\u90fd\u503c\u5f97\u6211\u4eec\u6b23\u559c\u3002\u672c\u7cfb\u5217\u8bfe\u7a0b\uff0c\u6211\u4eec\u4f1a\u8fdb\u884c\u4e25\u683c\u7684\u8bfe\u5802\u7ba1\u7406\uff0c\u9f13\u52b1\u5404\u4f4d\u540c\u5b66\u79ef\u6781\u601d\u8003\u3001\u8ba8\u8bba\uff0c\u5e0c\u671b\u80fd\u591f\u901a\u8fc7\u672c\u8bfe\u7a0b\u8ba9\u540c\u5b66\u4eec\u80fd\u5bf9Koopman\u7b97\u7b26\u7406\u8bba\u6709\u6df1\u5165\u7684\u7814\u7a76\uff0c\u5e76\u80fd\u8fdb\u884c\u76f8\u5e94\u7684\u7406\u8bba\u7814\u7a76\u548c\u5e94\u7528\u5b9e\u8df5\u3002\u5bf9\u4e8e\u6ee1\u8db3\u4ee5\u4e0b\u6761\u4ef6\u7684\u540c\u5b66\uff0c\u4f1a\u53d1\u653e\u5b9e\u4f53\u8bc1\u4e66\uff0c\u5c06\u9009\u51fa3\u540d\u4f18\u79c0\u7684\u540c\u5b66\u6bcf\u4eba\u8d60\u90011\u4ef6\u96c6\u667a\u5b9a\u5236T\u6064\u3002\u8ba9\u6211\u4eec\u5171\u540c\u5f00\u542f\u4e00\u6b21\u82cf\u683c\u62c9\u5e95\u5f0f\u7684\u8bfe\u7a0b\u5427\u3002<\/span><\/p>\n


              <\/strong><\/span><\/p>\n

              \u8bfe\u7a0b\u8bc1\u4e66\u53d1\u653e\u6807\u51c6\uff1a<\/strong><\/span><\/section>\n
                \n
              1. \n
                \u62a5\u540d\u65f6\u95f4\uff1a2024\u5e7412\u670821\u65e5\u524d\u62a5\u540d\u7684\u6210\u5458\uff1b<\/span><\/section>\n<\/li>\n
              2. \n
                \u53c2\u4e0e\u8bfe\u7a0b\u76f4\u64ad\uff1a\u4e0d\u4f4e\u4e8e80%\uff0c\u6839\u636e\u817e\u8baf\u4f1a\u8bae\u7684\u5728\u7ebf\u65f6\u95f4\u8fdb\u884c\u7edf\u8ba1\uff1b<\/span><\/section>\n<\/li>\n
              3. \n
                \u52a0\u5206\u9879\uff1a\u8bfe\u7a0b\u76f4\u64ad\u548c\u8bfe\u7a0b\u5fae\u4fe1\u7fa4\u5185\u79ef\u6781\u63d0\u95ee\uff1b<\/span><\/section>\n<\/li>\n
              4. \n

                \u52a0\u5206\u9879\uff1a\u5b8c\u6210\u8bfe\u7a0b\u8bbe\u7f6e\u7684\u4efb\u52a1\uff1b<\/span><\/p>\n<\/li>\n<\/ol>\n


                <\/span><\/h2>\n


                <\/span><\/p>\n

                \n
                \n
                \n
                \n
                \n

                <\/section>\n

                <\/section>\n
                \n
                <\/svg><\/section>\n<\/section>\n<\/section>\n
                \n

                \u62a5\u540d\u987b\u77e5<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

                \n

                <\/section>\n

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


                <\/p>\n

                  \n
                1. \n

                  \u8bfe\u7a0b\u5f62\u5f0f\uff1a\u817e\u8baf\u4f1a\u8bae\u76f4\u64ad\uff0c\u96c6\u667a\u5b66\u56ed\u7f51\u7ad9\u5f55\u64ad\u3002\u672c\u7cfb\u5217\u8bfe\u7a0b\u4e0d\u5b89\u6392\u514d\u8d39\u76f4\u64ad\u3002<\/span><\/p>\n<\/li>\n

                2. \n

                  \u8bfe\u7a0b\u5468\u671f\uff1a2024\u5e7412\u670821\u65e5-2025\u5e741\u670825\u65e5\uff0c\u6bcf\u5468\u516d\u4e0b\u53482\u70b9-4\u70b9\u8fdb\u884c\u3002<\/span><\/p>\n<\/li>\n

                3. \n

                  \u8bfe\u7a0b\u5b9a\u4ef7\uff1a\u539f\u4ef7599\uff0c\u65e9\u9e1f\u4ef7449\uff0c\u65e9\u9e1f\u4f18\u60e0\u622a\u6b62\u52302024\u5e7412\u670821\u65e5\u4e2d\u534812\u70b9\u3002<\/span><\/p>\n<\/li>\n<\/ol>\n

                  <\/p>\n

                  \u626b\u7801\u4ed8\u8d39\u62a5\u540d\u8bfe\u7a0b<\/span><\/section>\n
                  \u8bfe\u7a0b\u94fe\u63a5\uff1ahttps:\/\/campus.swarma.org\/course\/5419?from=wechat<\/span><\/section>\n


                  <\/span><\/p>\n

                  \u4ed8\u8d39\u6d41\u7a0b\uff1a<\/span><\/section>\n
                    \n
                  1. \n
                    \u626b\u7801\u4ed8\u8d39\uff1b<\/span><\/section>\n<\/li>\n
                  2. \n
                    \u8bfe\u7a0b\u9875\u9762\u6dfb\u52a0\u5b66\u5458\u767b\u8bb0\u8868\uff0c\u6dfb\u52a0\u52a9\u6559\u5fae\u4fe1\u5165\u7fa4\uff1b<\/span><\/section>\n<\/li>\n
                  3. \n

                    \u8bfe\u7a0b\u53ef\u5f00\u53d1\u7968\u3002<\/span><\/p>\n<\/li>\n<\/ol>\n


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

                    \u8bfe\u7a0b\u5956\u5b66\u91d1\u673a\u5236<\/span><\/strong><\/span><\/h3>\n
                      \n
                    1. \n
                      \u9014\u5f84\u4e00\uff1a\u53d1\u5e03\u9ad8\u8d28\u91cf\u8bfe\u7a0b\u7b14\u8bb0<\/span><\/strong><\/span><\/section>\n<\/li>\n<\/ol>\n
                      \u5728\u96c6\u667a\u6591\u56fe\u7f51\u7ad9\uff08pattern.swarma.org\uff09\u5b8c\u6210\u672c\u8bfe\u7a0b\u4f53\u7cfb\u4e0b\u67d0\u4e2a\u65b9\u5411\u7684\u603b\u7ed3\u6587\u7ae0\u6216\u5b66\u4e60\u8def\u5f84\u3002\u7ecf\u96c6\u667a\u5b66\u56ed\u52a9\u6559\u56e2\u961f\u8bc4\u5b9a\u8ba4\u53ef\u540e\uff0c\u53ef\u4f5c\u4e3a\u4e00\u6761\u8d21\u732e\u3002\u4e00\u6761\u8d21\u732e\u5956\u52b1200\u5143\u5956\u5b66\u91d1\uff0c\u8d28\u91cf\u4f18\u5f02\u7684\u5185\u5bb9\uff0c\u4f1a\u6709\u6d6e\u52a8\u5956\u52b1\u3002<\/span><\/section>\n
                      \u53ef\u53c2\u8003\uff1a<\/span><\/section>\n