{"id":63961,"date":"2025-08-29T17:25:10","date_gmt":"2025-08-29T09:25:10","guid":{"rendered":"https:\/\/swarma.org\/?p=63961"},"modified":"2025-08-29T17:25:10","modified_gmt":"2025-08-29T09:25:10","slug":"%e8%87%aa%e7%84%b6%e9%80%9a%e8%ae%af-%e5%a4%8d%e6%9d%82%e7%bd%91%e7%bb%9c%e4%b8%8a%e7%9a%84%e9%80%9a%e7%94%a8%e7%a5%9e%e7%bb%8f%e7%ac%a6%e5%8f%b7%e5%9b%9e%e5%bd%92%e7%ae%97%e6%b3%95","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=63961","title":{"rendered":"\u81ea\u7136\u2022\u901a\u8baf: \u590d\u6742\u7f51\u7edc\u4e0a\u7684\u901a\u7528\u795e\u7ecf\u7b26\u53f7\u56de\u5f52\u7b97\u6cd5"},"content":{"rendered":"
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\u5173\u952e\u8bcd\uff1a\u590d\u6742\u7f51\u7edc\u52a8\u529b\u5b66\uff08complex network dynamics\uff09\u3001\u7b26\u53f7\u56de\u5f52\uff08symbolic regression\uff09\u3001\u795e\u7ecf\u7f51\u7edc\uff08neural networks\uff09\u3001\u7269\u7406\u5148\u9a8c\uff08physical priors\uff09\u3001\u53ef\u89e3\u91ca\u6a21\u578b\uff08interpretable models\uff09\u3001\u6df7\u6c8c\u7cfb\u7edf\uff08chaotic systems\uff09\u3002<\/span><\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u8bba\u6587\u9898\u76ee\uff1aLearning Interpretable Network Dynamics via Universal Neural Symbolic Regression<\/span><\/span><\/p>\n

\u6591\u56fe\u94fe\u63a5\uff1ahttps:\/\/pattern.swarma.org\/paper\/4f63d7b2-aaa9-11ef-ab2f-0242ac17000f<\/span><\/span><\/p>\n

\u53d1\u8868\u671f\u520a\uff1a<\/span>Nature Communication<\/span><\/span><\/p>\n<\/blockquote>\n

\u590d\u6742\u7cfb\u7edf\u65e0\u5904\u4e0d\u5728\uff0c\u4ece\u751f\u7269\u7f51\u7edc\u5230\u793e\u4f1a\u52a8\u529b\u5b66\uff0c\u5176\u72b6\u6001\u53d8\u5316\u5f80\u5f80\u7531\u975e\u7ebf\u6027\u5fae\u5206\u65b9\u7a0b\u4e3b\u5bfc\uff0c\u5374\u5e38\u5e38\u672a\u77e5\uff0c\u8fd9\u9650\u5236\u4e86\u6211\u4eec\u5bf9\u73b0\u8c61\u5982\u6d41\u884c\u75c5\u7206\u53d1\u6216\u751f\u6001\u5d29\u6e83\u7684\u7406\u89e3\u3002\u968f\u7740\u6570\u636e\u83b7\u53d6\u7684\u4fbf\u5229\uff0c\u6570\u636e\u9a71\u52a8\u65b9\u6cd5\u80fd\u591f\u5e2e\u52a9\u4e13\u5bb6\u4ece\u89c2\u6d4b\u4e2d\u63d0\u70bc\u6a21\u5f0f\u3002\u7136\u800c\uff0c\u5bf9\u4e8e\u9ad8\u7ef4\u7f51\u7edc\u52a8\u529b\u5b66\uff0c\u73b0\u6709\u7684\u7b26\u53f7\u56de\u5f52\u6280\u672f\u9762\u4e34\u7ef4\u5ea6\u707e\u96be\u548c\u8ba1\u7b97\u6548\u7387\u4f4e\u4e0b\u7684\u6311\u6218\u3002<\/span><\/span>\u672c\u7814\u7a76\u521b\u65b0\u6027\u63d0\u51faLLC\uff08Learning Law of Changes\uff09\uff0c\u878d\u5408\u6df1\u5ea6\u5b66\u4e60\u4e0e\u9884\u8bad\u7ec3\u7b26\u53f7\u56de\u5f52\uff0c\u4ec5\u9700\u5c11\u91cf\u5148\u9a8c\u77e5\u8bc6\uff0c\u5c31\u80fd\u9ad8\u6548\u89e3\u6790\u590d\u6742\u7f51\u7edc\u7684\u6f14\u5316\u89c4\u5f8b<\/span><\/span><\/strong>\u3002\u901a\u8fc7\u8de8\u9886\u57df\u5b9e\u9a8c\u9a8c\u8bc1\uff0c\u5b83\u4e0d\u4ec5\u63d0\u5347\u4e86\u51c6\u786e\u6027\uff0c\u8fd8\u5e94\u7528\u4e8e\u771f\u5b9e\u573a\u666f\uff0c\u63a8\u52a8\u79d1\u5b66\u4ece\u201c\u9ed1\u7bb1\u201d\u5411\u201c\u767d\u7bb1\u201d\u8f6c\u578b\uff0c\u4e3a\u63ed\u793a\u201c\u53d8\u5316\u7684\u6cd5\u5219\u201d\u5f00\u8f9f\u65b0\u8def\u5f84\u3002<\/span><\/span><\/span><\/p>\n


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\u76ee\u524d\u7684\u524d\u6cbf\u7814\u7a76\u4e3b\u8981\u6709\u4e24\u79cd\u65b9\u6cd5\u3002\u7b2c\u4e00\u79cd\u662f\u4e24\u9636\u6bb5\u7a00\u758f\u7b26\u53f7\u56de\u5f52<\/span>\uff08TPSINDy\uff09<\/span>[5]\uff0c\u5b83\u5c06\u63a7\u5236\u51fd\u6570\u53c2\u6570\u5316\u4e3a\u9884\u5b9a\u4e49\u57fa\u672c\u51fd\u6570\u9879\u7684\u53ef\u5b66\u4e60\u7ebf\u6027\u7ec4\u5408\u3002\u867d\u7136\u65b9\u7a0b\u63a8\u65ad\u6548\u7387\u9ad8\uff0c\u4f46\u9884\u5b9a\u4e49\u51fd\u6570\u9879\u7684\u5408\u7406\u6027\u76f4\u63a5\u5f71\u54cd\u7ed3\u679c\uff0c\u9700\u8981\u5145\u5206\u51c6\u786e\u7684\u9886\u57df\u4e13\u4e1a\u77e5\u8bc6\u3002\u53e6\u4e00\u79cd\u65b9\u6cd5\u662f\u4f7f\u7528\u56fe\u795e\u7ecf\u7f51\u7edc<\/span>\uff08graph neural networks, GNNs\uff09<\/span>\u6765\u53c2\u6570\u5316\u63a7\u5236\u51fd\u6570[6]\uff0c\u514b\u670d\u4e86\u8fc7\u5ea6\u4f9d\u8d56\u4e13\u5bb6\u77e5\u8bc6\u7684\u95ee\u9898\u3002\u7136\u800c\uff0c\u7531\u4e8e\u4f7f\u7528\u9057\u4f20\u7f16\u7a0b<\/span>\uff08genetic programming, GP\uff09<\/span>\u5c06\u795e\u7ecf\u7f51\u7edc\u89e3\u6790\u4e3a\u7b26\u53f7\u65b9\u7a0b\uff0c\u5e26\u6765\u4e86\u9ad8\u6210\u672c\u7684\u8fdb\u5316\u641c\u7d22\u6548\u7387\u95ee\u9898\u3002<\/span><\/span><\/p>\n


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LLC\uff1a\u795e\u7ecf\u7f51\u7edc\u4e0e\u7269\u7406\u5148\u9a8c\u7684\u5de7\u5999\u7ed3\u5408<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u9762\u5bf9\u8fd9\u4e9b\u6311\u6218\uff0c\u672c\u7814\u7a76\u5f00\u53d1\u4e86\u4e00\u79cd\u540d\u4e3aLLC<\/span>\uff08Learning Law of Changes\uff09<\/span>\u7684\u901a\u7528\u795e\u7ecf\u7b26\u53f7\u56de\u5f52\u5de5\u5177\uff0c\u65e8\u5728\u4ece\u7f51\u7edc\u52a8\u529b\u5b66\u89c2\u6d4b\u6570\u636e\u4e2d\u81ea\u52a8\u63a8\u65ad\u5e38\u5fae\u5206\u65b9\u7a0b<\/span>\uff08ordinary differential equations, ODEs\uff09<\/span>\u3002<\/span><\/span>\u8fd9\u4e2a\u5de5\u5177\u7684\u6838\u5fc3\u662f\u89e3\u51b3\u9ad8\u7ef4\u81ea\u7531\u53d8\u91cf\u548c\u7b26\u53f7\u63a8\u65ad\u6548\u7387\u4f4e\u4e0b\u7684\u4e24\u5927\u96be\u9898<\/span><\/span><\/strong>\u3002<\/span><\/span><\/span><\/p>\n

LLC\u7684\u7b2c\u4e00\u4e2a\u5173\u952e\u6b65\u9aa4\u662f\u901a\u8fc7\u795e\u7ecf\u7f51\u7edc\u548c\u7269\u7406\u5148\u9a8c\u5b9e\u73b0\u7f51\u7edc\u52a8\u529b\u5b66\u4fe1\u53f7\u7684\u89e3\u8026\u3002\u4e3a\u4e86\u7f13\u89e3\u9ad8\u7ef4\u7f51\u7edc\u52a8\u529b\u5b66\u7684\u7ef4\u5ea6\u8bc5\u5492<\/span>\uff08curse of dimensionality\uff09<\/span>\uff0c\u7814\u7a76\u5f15\u5165\u4e86\u4e00\u4e2a\u7269\u7406\u5148\u9a8c\uff1a\u7f51\u7edc\u72b6\u6001\u7684\u53d8\u5316\u53d7\u81ea\u8eab\u72b6\u6001\u548c\u90bb\u5c45\u72b6\u6001\u7684\u5f71\u54cd\u3002\u56e0\u6b64\uff0c\u5c06\u63a7\u5236\u65b9\u7a0b\u5206\u89e3\u4e3a\u81ea\u8eab\u52a8\u529b\u5b66<\/span><\/span>\"image.png\"<\/span><\/sub><\/span>\uff08self dynamics\uff09<\/span>\u548c\u4ea4\u4e92\u52a8\u529b\u5b66<\/span><\/span>\"image.png\"<\/span><\/sub>\uff08interaction dynamics\uff09<\/span>\u4e24\u4e2a\u8026\u5408\u7ec4\u4ef6\u3002\u7f51\u7edc\u52a8\u529b\u5b66\u7684\u63a7\u5236\u65b9\u7a0b\u53ef\u4ee5\u88ab\u6539\u5199\u6210\u6309\u8282\u70b9\u5212\u5206\u7684\u5f62\u5f0f\uff1a<\/span><\/span><\/span><\/span><\/p>\n

<\/section>\n

\u5176\u4e2d<\/span><\/span><\/span><\/span>\"image.png\"<\/span><\/sub>\u8868\u793a\u8282\u70b9<\/span><\/span><\/em><\/span>i<\/span><\/span><\/span>\u72b6\u6001\u968f\u65f6\u95f4\u7684\u53d8\u5316\u7387\uff1b <\/span><\/span><\/em><\/span>\"image.png\"<\/span><\/sub>\u63cf\u8ff0\u8282\u70b9\u81ea\u8eab\u7684\u52a8\u529b\u5b66\uff1b<\/span><\/span><\/span>\"image.png\"<\/span><\/sub>\u5219\u523b\u753b\u8282\u70b9<\/span><\/span><\/span>i<\/span><\/span><\/span>\u53d7\u76f8\u90bb\u8282\u70b9<\/span><\/span><\/em><\/span>j<\/span><\/span><\/span>\u5f71\u54cd\u7684\u4ea4\u4e92\u52a8\u529b\u5b66\uff1b<\/span><\/span><\/em><\/span>A<\/span><\/span>i,j<\/span><\/sub><\/span><\/span>\u4e3a\u7f51\u7edc\u7684\u90bb\u63a5\u77e9\u9635\u5143\u7d20\uff0c\u6307\u793a\u8282\u70b9<\/span><\/span><\/em><\/span>i<\/span><\/span><\/span>\u4e0e<\/span><\/span><\/em><\/span>j<\/span><\/span><\/span>\u662f\u5426\u5b58\u5728\u8fde\u63a5\u3002<\/span><\/span><\/em><\/span><\/span><\/p>\n

\u7814\u7a76\u8005\u4f7f\u7528\u591a\u5c42\u611f\u77e5\u673a<\/span>\uff08multi-layer perceptron\uff09<\/span>\u53c2\u6570\u5316\u81ea\u8eab\u52a8\u529b\u5b66\u7684\u975e\u7ebf\u6027\uff0c\u5e76\u8bbe\u8ba1\u4e86\u4e00\u4e2a\u7279\u6b8a\u7684\u795e\u7ecf\u7f51\u7edc\u67b6\u6784\u6765\u6355\u6349\u4ea4\u4e92\u52a8\u529b\u5b66\uff0c\u5305\u62ec\u8026\u5408\u975e\u7ebf\u6027\u9879\u548c\u53ef\u5206\u89e3\u9879\u3002\u8fd9\u79cd\u8bbe\u8ba1\u786e\u4fdd\u4e86\u6a21\u578b\u80fd\u81ea\u52a8\u9002\u5e94\u5404\u79cd\u52a8\u529b\u5b66\u573a\u666f\u3002\u901a\u8fc7\u6700\u5c0f\u5316\u635f\u5931\u51fd\u6570\uff1a<\/span><\/span><\/p>\n

<\/section>\n

\u5176\u4e2d\uff0c<\/span><\/span><\/span>N<\/span><\/span><\/span>\u8868\u793a\u8282\u70b9\u6570\uff0c<\/span><\/span><\/span>d<\/span><\/span><\/span>\u8868\u793a\u72b6\u6001\u7ef4\u5ea6\uff0c<\/span><\/span><\/span>|<\/span><\/span>|\u00b7<\/span><\/span><\/span><\/span>|<\/span><\/span>|<\/span><\/span><\/span><\/span>1<\/span><\/span><\/sub><\/span> \u4e3a<\/span><\/span><\/span>\u2113<\/span><\/span>1<\/span><\/sub><\/span><\/span>\u8303\u6570\u3002\u635f\u5931\u51fd\u6570\u7684\u7b2c\u4e00\u90e8\u5206\u8868\u793a\u5e73\u5747\u7edd\u5bf9\u8bef\u5dee\uff0c\u7b2c\u4e8c\u90e8\u5206\u5219\u8868\u793a\u7edd\u5bf9\u8bef\u5dee\u7684\u65b9\u5dee\u3002\u53c2\u6570 <\/span><\/span><\/span>\u03bb<\/span><\/span><\/span> \u662f\u4e00\u4e2a\u7528\u4e8e\u5e73\u8861\u4e24\u90e8\u5206\u7684\u7cfb\u6570\u3002\u8be5\u635f\u5931\u51fd\u6570\u5c06\u5747\u503c\u548c\u65b9\u5dee\u7ed3\u5408\u5728\u4e00\u8d77\uff0c\u65e8\u5728\u4e0d\u4ec5\u6700\u5c0f\u5316\u9884\u6d4b\u503c\u4e0e\u771f\u5b9e\u503c\u4e4b\u95f4\u7684\u5e73\u5747\u5dee\u5f02\uff0c\u540c\u65f6\u63a7\u5236\u8bef\u5dee\u7684\u6ce2\u52a8\u6027\u3002<\/span><\/span><\/span><\/span><\/p>\n

\u7ed3\u5408\u5e73\u5747\u7edd\u5bf9\u8bef\u5dee\u548c\u8bef\u5dee\u65b9\u5dee\u795e\u7ecf\u7f51\u7edc\u88ab\u8bad\u7ec3\u5230\u51c6\u786e\u62df\u5408\u72b6\u6001\u5bfc\u6570\uff0c\u4ece\u800c\u5b9e\u73b0\u4fe1\u53f7\u7684\u7a33\u5b9a\u89e3\u8026\u3002\u5b9e\u9a8c\u663e\u793a\uff0c\u8fd9\u79cd\u65b9\u6cd5\u6bd4\u4f20\u7edf\u56fe\u795e\u7ecf\u7f51\u7edc <\/span>(GNNs) <\/span>\u66f4\u9002\u5408\u7f51\u7edc\u52a8\u529b\u5b66\u5b66\u4e60\uff0c\u907f\u514d\u4e86\u6570\u503c\u79ef\u5206\u7684\u9ad8\u8ba1\u7b97\u590d\u6742\u5ea6\uff0c\u5e76\u63d0\u5347\u4e86\u7a33\u5b9a\u6027\u3002<\/span><\/span><\/p>\n

\u795e\u7ecf\u7f51\u7edc\u8bad\u7ec3\u597d\u540e\uff0cLLC\u5c06\u8fd9\u4e9b\u201c\u9ed1\u7bb1\u201d\u6a21\u578b\u89e3\u6790\u4e3a\u53ef\u89e3\u91ca\u7684\u767d\u7bb1\u65b9\u7a0b\u3002\u8fd9\u91cc\uff0c\u7814\u7a76\u9996\u6b21\u5f15\u5165\u9884\u8bad\u7ec3Transformer\u6a21\u578b\u4f5c\u4e3a\u7b26\u53f7\u56de\u5f52\u5de5\u5177\uff0c\u4ee5\u89e3\u51b3\u4f20\u7edf\u65b9\u6cd5\u5982\u9057\u4f20\u7f16\u7a0b<\/span>\uff08GP\uff09<\/span>\u7684\u4f4e\u6548\u7387\u95ee\u9898\u3002\u9884\u8bad\u7ec3\u6a21\u578b\u5728\u6570\u4ebf\u65b9\u7a0b-\u6570\u636e\u5bf9\u4e0a\u8bad\u7ec3\uff0c\u96c6\u6210\u4e86\u6d77\u91cf\u65b9\u7a0b\u77e5\u8bc6\uff0c\u4ec5\u9700\u5355\u6b21\u524d\u5411\u4f20\u64ad\u5c31\u80fd\u4ece\u8f93\u5165-\u8f93\u51fa\u70b9\u5bf9\u4e2d\u63a8\u65ad\u65b9\u7a0b\uff0c\u968f\u540e\u53ef\u9009\u5730\u4f7f\u7528Broyden\u2013Fletcher\u2013Goldfarb\u2013Shanno<\/span>\uff08BFGS\uff09<\/span>\u7b97\u6cd5\u5fae\u8c03\u5e38\u91cf\u3002\u8fd9\u79cd\u65b9\u6cd5\u5927\u5927\u7f29\u77ed\u4e86\u63a8\u65ad\u65f6\u95f4\uff0c\u907f\u514d\u4e86\u4ece\u96f6\u5f00\u59cb\u7684\u641c\u7d22\u3002\u8fd9\u91cc\u7814\u7a76\u8005\u9009\u62e9\u7684\u662fNSRA\u6a21\u578b\uff0c\u56e0\u4e3a\u5b83\u5728\u9884\u8bad\u7ec3\u6a21\u578b\u4e2d\u8868\u73b0\u6700\u4f73\u3002\u4e3a\u4e86\u9002\u5e94\u65f6\u95f4\u76f8\u5173\u7684\u7f51\u7edc\u52a8\u529b\u5b66\u6570\u636e\uff0c\u4ed6\u4eec\u91c7\u7528K-Means\u91c7\u6837\u7b56\u7565\uff0c\u786e\u4fdd\u6570\u636e\u70b9\u7684\u72ec\u7acb\u6027\u548c\u4ee3\u8868\u6027\uff0c\u4ece\u800c\u63d0\u5347\u56de\u5f52\u51c6\u786e\u6027\u3002\u901a\u8fc7\u8fd9\u4e2a\u6a21\u5757\uff0cLLC\u80fd\u9ad8\u6548\u5730\u5c06\u62df\u5408\u597d\u7684\u795e\u7ecf\u7f51\u7edc\u8f6c\u5316\u4e3a\u7b26\u53f7\u65b9\u7a0b\uff0c\u5f62\u6210\u5b8c\u6574\u7684\u53d1\u73b0\u7ba1\u9053\u3002<\/span><\/span><\/p>\n

<\/section>\n

\u56fe1\uff1aLLC\uff08Learning Law of Changes\uff09\u7684\u6574\u4f53\u6d41\u7a0b\u3002\u89c2\u6d4b\u6570\u636e\u53ef\u4ee5\u4ece\u65b0\u573a\u666f\u7684\u521d\u59cb\u5b9e\u9a8c\u4e2d\u83b7\u53d6\uff0c\u5305\u62ec\u7cfb\u7edf\u72b6\u6001\u968f\u65f6\u95f4\u7684\u53d8\u5316\u548c\u62d3\u6251\u7ed3\u6784\uff0c\u5373<\/span><\/span><\/span>O\uff1d{X<\/span><\/span>(t)<\/span><\/span><\/sub>, A}<\/span><\/span><\/span><\/span>\uff08\u8282\u70b9\u72b6\u6001+\u7f51\u7edc\u62d3\u6251\uff09\u3002\u7136\u540e\uff0c\u4f7f\u7528\u95f4\u9694\u9009\u62e9\u7b56\u7565\u9009\u62e9\u6709\u6548\u95f4\u9694\u6570\u636e\uff0c\u901a\u8fc7\u6709\u9650\u5dee\u5206\u5728<\/span><\/span>X<\/span><\/span>i<\/span><\/span><\/sub>(t)<\/span><\/span><\/span><\/span>\u4e0a\u83b7\u53d6\u5bfc\u6570\uff0c\u5373<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002\u901a\u8fc7\u6574\u5408\u7269\u7406\u5148\u9a8c\uff0c\u795e\u7ecf\u7f51\u7edc<\/span><\/span>\"image.png\"<\/span><\/sub><\/span>\u548c<\/span><\/span>\"image.png\"<\/span><\/sub>\uff0c\u7528\u4e8e\u89e3\u8026\u7f51\u7edc\u52a8\u529b\u5b66\u4fe1\u53f7\u5e76\u5b9e\u73b0\u53d8\u91cf\u51cf\u5c11\uff0c\u76f4\u5230\u6ee1\u8db3\u62df\u5408\u8981\u6c42\u3002\u5728\u83b7\u5f97\u826f\u597d\u62df\u5408\u7684\u795e\u7ecf\u7f51\u7edc\u540e\uff0c\u4f7f\u7528\u7b26\u53f7\u56de\u5f52\u6280\u672f\u9ad8\u6548\u89e3\u6790\u5b83\u4eec\u7684\u8fd1\u4f3c\u767d\u7bb1\u65b9\u7a0b\u3002\u5f53\u7136\uff0c\u5982\u679c\u9700\u8981\u989d\u5916\u89c2\u6d4b\u6570\u636e\u6765\u652f\u6301\u53d1\u73b0\uff0c\u53ef\u4ee5\u91cd\u65b0\u5ba1\u89c6\u5b9e\u9a8c\u8bbe\u8ba1\uff0c\u76f4\u5230\u83b7\u5f97\u6ee1\u610f\u7684\u7f51\u7edc\u52a8\u529b\u5b66\u63a7\u5236\u65b9\u7a0b\uff0c\u4ece\u800c\u6253\u7834\u5faa\u73af\u3002<\/span><\/span><\/span><\/span><\/span><\/p>\n


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

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

<\/span><\/section>\n
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<\/section>\n<\/section>\n<\/section>\n
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LLC\u6a21\u578b\u7684\u6709\u6548\u6027\u9a8c\u8bc1<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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

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


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

\u4e00\u7ef4\u7f51\u7edc\u52a8\u529b\u5b66\u7684\u63a8\u65ad\u9a8c\u8bc1<\/span><\/span><\/h4>\n

\u4e3a\u4e86\u5168\u9762\u8bc4\u4f30LLC\u7684\u6709\u6548\u6027\uff0c\u7814\u7a76\u8005\u5728\u516d\u4e2a\u4ee3\u8868\u6027\u7684\u4e00\u7ef4\u540c\u8d28\u7f51\u7edc\u52a8\u529b\u5b66\u6a21\u578b\u4e0a\u8fdb\u884c\u4e86\u6d4b\u8bd5\uff0c\u5305\u62ec\u751f\u5316\u52a8\u529b\u5b66<\/span>\uff08Biochemical dynamics\uff09<\/span>\u3001\u57fa\u56e0\u8c03\u63a7\u52a8\u529b\u5b66<\/span>\uff08Gene regulatory dynamics\uff09<\/span>\u3001\u4e92\u60e0\u4e92\u52a8\u52a8\u529b\u5b66<\/span>\uff08Mutualistic Interaction dynamics\uff09<\/span>\u3001Lotka-Volterra\u52a8\u529b\u5b66\u3001\u795e\u7ecf\u52a8\u529b\u5b66<\/span>\uff08Neural dynamics\uff09<\/span>\u548c\u6d41\u884c\u75c5\u52a8\u529b\u5b66<\/span>\uff08Epidemic dynamics\uff09<\/span>\u3002\u8fd9\u4e9b\u6a21\u578b\u5e7f\u6cdb\u5e94\u7528\u4e8e\u751f\u7269\u3001\u751f\u6001\u548c\u795e\u7ecf\u79d1\u5b66\u7b49\u9886\u57df\uff0c\u6a21\u62df\u5728Erd\u0151s\u2013R\u00e9nyi\u7f51\u7edc\u4e0a\u751f\u6210\u8fde\u7eed\u72b6\u6001\u6570\u636e\uff0c\u7136\u540e\u4ece\u89c2\u6d4b\u4e2d\u91cd\u5efa\u65b9\u7a0b\u3002\u4e0e\u6700\u5148\u8fdb\u7684TPSINDy\u548cGNN+GP\u65b9\u6cd5\u76f8\u6bd4\uff0cLLC\u5728\u9884\u6d4b\u51c6\u786e\u6027<\/span>\uff08adjusted R\u00b2\u5206\u6570\uff09<\/span>\u548c\u65b9\u7a0b\u53ec\u56de\u7387<\/span>\uff08Recall\uff09<\/span>\u4e0a\u663e\u8457\u4f18\u8d8a\u3002\u5373\u4f7f\u5728\u566a\u58f0\u5e72\u6270\u6216\u62d3\u6251\u4e0d\u5b8c\u6574\u7684\u60c5\u51b5\u4e0b\uff0cLLC\u4e5f\u80fd\u7ef4\u6301\u63a5\u8fd1\u96f6\u7684\u5747\u65b9\u8bef\u5dee<\/span>\uff08mean squared error, MSE\uff09<\/span>\uff0c\u5f97\u76ca\u4e8e\u795e\u7ecf\u7f51\u7edc\u7684\u9c81\u68d2\u62df\u5408\u548c\u635f\u5931\u51fd\u6570\u7684\u65b9\u5dee\u6700\u5c0f\u5316\u3002\u65f6\u95f4\u6548\u7387\u4e0a\uff0cLLC\u4ec5\u9700\u7ea66.5\u5206\u949f\u5b8c\u6210\u56de\u5f52\uff0c\u8fdc\u4f4e\u4e8eGNN+GP\u768412.9\u5206\u949f\uff0c\u540c\u65f6\u65e0\u9700TPSINDy\u90a3\u6837\u7684\u5f3a\u5148\u9a8c\u77e5\u8bc6\u5e93\u3002\u8fd9\u8868\u660eLLC\u5728\u5e73\u8861\u6548\u7387\u548c\u51c6\u786e\u6027\u4e0a\u53d6\u5f97\u4e86\u826f\u597d\u6210\u679c\u3002<\/span><\/span><\/p>\n

<\/section>\n

\u56fe2\uff1a\u63a8\u65ad\u4e00\u7ef4\u540c\u8d28\u7f51\u7edc\u52a8\u529b\u5b66\u7684\u7ed3\u679c\u3002a. \u5728\u516d\u4e2a\u573a\u666f\u4e2d\uff0c\u5bf9\u9884\u6d4b\u51c6\u786e\u6027\uff08adjusted R\u00b2\u5206\u6570\uff09\u548c\u53d1\u73b0\u65b9\u7a0b\uff08Recall\uff09\u7684\u6bd4\u8f83\u3002TPSINDy\u7684\u7ed3\u679c\u9ad8\u5ea6\u4f9d\u8d56\u5176\u51fd\u6570\u9879\u7684\u9009\u62e9\uff0cLLC\u5728\u6240\u6709\u7f51\u7edc\u52a8\u529b\u5b66\u573a\u666f\u4e2d\u663e\u8457\u4f18\u4e8e\u6bd4\u8f83\u65b9\u6cd5\u3002b. \u5404\u79cd\u65b9\u6cd5\u5728\u6240\u6709\u52a8\u529b\u5b66\u4e0a\u7684\u5e73\u5747\u6267\u884c\u65f6\u95f4\u6bd4\u8f83\u3002\u8bf7\u6ce8\u610f\uff0cTPSINDy\u9700\u8981\u5f3a\u5148\u9a8c\uff0c\u5373\u81ea\u8eab\u548c\u4ea4\u4e92\u52a8\u529b\u5b66\u7684\u53ef\u5206\u89e3\u6027\uff0c\u4ee5\u53ca\u9884\u5b9a\u4e49\u7684\u6b63\u4ea4\u57fa\u672c\u51fd\u6570\u9879\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5176\u4ed6\u65b9\u6cd5\u57fa\u4e8e\u76f8\u540c\u6c34\u5e73\u7684\u5047\u8bbe\uff0c\u5373\u4ec5\u9700\u53ef\u5206\u89e3\u6027\u3002\u901a\u8fc7\u7ed3\u5408\u57fa\u4e8eTransformer\u7684\u9884\u8bad\u7ec3\u7b26\u53f7\u56de\u5f52\uff0cLLC\u5728\u6548\u7387\u548c\u51c6\u786e\u6027\u4e4b\u95f4\u5b9e\u73b0\u4e86\u826f\u597d\u5e73\u8861\u3002c. \u5728LV\u573a\u666f\u4e2d\uff0c\u7531\u53d1\u73b0\u7684\u63a7\u5236\u65b9\u7a0b\u4ea7\u751f\u7684\u9884\u6d4b\u7ed3\u679c\u4e0e\u771f\u5b9e\u7ed3\u679c\u4e4b\u95f4\u7684\u5f52\u4e00\u5316\u4f30\u8ba1\u8bef\u5dee\uff08Normalized Estimation Error, NED\uff09\u3002d. \u5404\u79cd\u65b9\u6cd5\u53d1\u73b0\u7684\u63a7\u5236\u65b9\u7a0b\u4e2d\u62df\u5408\u7cfb\u6570\u7684\u6bd4\u8f83\u3002e. \u5355\u4e2a\u8282\u70b9\u7684\u72b6\u614b\u9884\u6d4b\u66f2\u7ebf\u7684\u6bd4\u8f83\u3002<\/span><\/span><\/p>\n

\u591a\u7ef4\u548c\u5f02\u8d28\u7f51\u7edc\u52a8\u529b\u5b66\u7684\u6311\u6218\u5e94\u5bf9<\/span><\/span><\/h4>\n

LLC\u8fdb\u4e00\u6b65\u6269\u5c55\u5230\u591a\u7ef4\u7cfb\u7edf\uff0c\u5982FitzHugh\u2013Nagumo<\/span>\uff08FHN\uff09<\/span>\u795e\u7ecf\u52a8\u529b\u5b66\u548c\u6355\u98df\u8005-\u730e\u7269<\/span>\uff08predator-prey, PP\uff09<\/span>\u7cfb\u7edf\u3002FHN\u662f\u4e00\u4e2a\u4e8c\u7ef4\u6a21\u578b\uff0c\u6355\u6349\u795e\u7ecf\u5143\u653e\u7535\u884c\u4e3a\uff0c\u5728Barab\u00e1si\u2013Albert\u7f51\u7edc\u4e0a\u6a21\u62df\u8111\u529f\u80fd\u533a\u4e92\u52a8\u3002\u901a\u8fc7\u8bbe\u7f6e\u63a9\u7801<\/span>\uff08masks\uff09<\/span>\u5904\u7406\u5c40\u90e8\u89c2\u6d4b\uff0cLLC\u6709\u6548\u89e3\u8026\u81ea\u8eab\u548c\u4ea4\u4e92\u52a8\u529b\u5b66\uff0c\u751f\u6210\u7684\u65b9\u7a0b\u6bd4TPSINDy\u66f4\u63a5\u8fd1\u771f\u5b9e\u5f62\u5f0f\uff0c\u5e76\u5728\u957f\u671f\u9884\u6d4b\u4e2d\u4fdd\u6301\u5468\u671f\u6027\u3002\u540c\u6837\uff0c\u5728\u5f02\u8d28PP\u7cfb\u7edf\u4e2d\uff0c\u8282\u70b9\u5206\u4e3a\u6355\u98df\u8005\u548c\u730e\u7269\u89d2\u8272\uff0cLLC\u901a\u8fc7\u591a\u7ec4\u795e\u7ecf\u7f51\u7edc\u5904\u7406\u4e0d\u540c\u7c7b\u578b\u4ea4\u4e92\uff0c\u6210\u529f\u6355\u6349\u73af\u72b6\u5e73\u8861\u72b6\u6001\uff0c\u800cTPSINDy\u867d\u5f62\u5f0f\u5339\u914d\u4f46\u9884\u6d4b\u8f68\u8ff9\u504f\u5dee\u5927\u3002\u8fd9\u7a81\u663e\u4e86LLC\u65e0\u9700\u5f3a\u5148\u9a8c\u7684\u4f18\u52bf\uff0c\u5e76\u5728\u771f\u5b9e\u62d3\u6251\u5982C. elegans\u548cDrosophila\u7f51\u7edc\u4e0a\u8868\u73b0\u51fa\u8272\u3002<\/span><\/span><\/p>\n

<\/section>\n

\u56fe3\uff1a\u63a8\u65adFitzHugh\u2013Nagumo\uff08FHN\uff09\u52a8\u529b\u5b66\u7684\u7ed3\u679c\u3002a. \u795e\u7ecf\u7f51\u7edc\u5bf9\u4e00\u4e2a\u8282\u70b9\u7b2c\u4e00\u7ef4\u5ea6<\/span><\/span><\/span>\"image.png\"<\/span><\/sub>\u7684\u62df\u5408\u7ed3\u679c\u3002b. \u4e00\u4e2a\u8282\u70b9\u7b2c\u4e00\u7ef4\u5ea6\u81ea\u8eab\u52a8\u529b\u5b66\u7684\u89e3\u8026\u7ed3\u679c<\/span><\/span>\"image.png\"<\/span><\/sub><\/span>\u3002c. \u4e00\u4e2a\u8282\u70b9\u7b2c\u4e00\u7ef4\u5ea6\u4ea4\u4e92\u52a8\u529b\u5b66\u7684\u89e3\u8026\u7ed3\u679c<\/span><\/span>\"image.png\"<\/span><\/sub>\u3002d. \u795e\u7ecf\u7f51\u7edc\u5bf9\u4e00\u4e2a\u8282\u70b9\u7b2c\u4e8c\u7ef4\u5ea6<\/span><\/span>\"image.png\"<\/span><\/sub><\/span>\u7684\u62df\u5408\u7ed3\u679c\u3002e. \u5404\u79cd\u65b9\u6cd5\u63a8\u65ad\u7684\u63a7\u5236\u65b9\u7a0b\u7684\u6bd4\u8f83\u3002f. \u7531\u63a8\u65ad\u65b9\u7a0b\u751f\u6210\u7684\u8f68\u8ff9\u4e0e\u771f\u5b9e\u65b9\u7a0b\u751f\u6210\u7684\u8f68\u8ff9\u4e4b\u95f4\u7684\u5f52\u4e00\u5316\u6b27\u6c0f\u8ddd\u79bb\uff08Normalized Euclidean Distance, NED\uff09\u7684\u6bd4\u8f83\uff0c\u6a2a\u8f74\u8868\u793a\u8282\u70b9\u7d22\u5f15\u3002g. \u5728Barab\u00e1si\u2013Albert\u7f51\u7edc\u4e0a\uff0c\u7531\u63a8\u65ad\u65b9\u7a0b\u548c\u771f\u5b9e\u65b9\u7a0b\u751f\u6210\u7684\u8f68\u8ff9\u7684\u6bd4\u8f83<\/span>\u3002<\/span><\/span><\/span><\/p>\n

<\/section>\n

\u56fe4\uff1a\u63a8\u65ad\u6355\u98df\u8005-\u730e\u7269\uff08PP\uff09\u7cfb\u7edf\u7684\u7ed3\u679c\u3002a. \u5404\u79cd\u65b9\u6cd5\u63a8\u65ad\u7684\u63a7\u5236\u65b9\u7a0b\u7684\u6bd4\u8f83\u3002b. \u968f\u65f6\u95f4\u53d8\u5316\u7684\u6355\u98df\u8005\uff08\u65b9\u5757\uff09\u548c\u730e\u7269\u7fa4\uff08\u70b9\uff09\u7684\u771f\u5b9e\u4f4d\u7f6e\u3002c. \u7531TPSINDy\u63a8\u65ad\u7684\u63a7\u5236\u65b9\u7a0b\u751f\u6210\u7684\u9884\u6d4b\u4f4d\u7f6e\u3002d. \u7531LLC\u63a8\u65ad\u7684\u63a7\u5236\u65b9\u7a0b\u751f\u6210\u7684\u9884\u6d4b\u4f4d\u7f6e\u3002<\/span><\/span><\/p>\n

\u6df7\u6c8c\u7cfb\u7edf\u52a8\u529b\u5b66\u7684\u6df1\u5165\u5256\u6790<\/span><\/span><\/h4>\n

\u7814\u7a76\u8fd8\u9488\u5bf9\u4e09\u7ef4\u6df7\u6c8c\u7cfb\u7edf\u5982Lorenz\u548cR\u00f6ssler\u52a8\u529b\u5b66\u8fdb\u884c\u4e86\u6d4b\u8bd5\uff0c\u8003\u5bdf\u521d\u59cb\u6761\u4ef6\u654f\u611f\u6027\u3002\u65e0\u8bba\u521d\u59cb\u503c\u56fa\u5b9a\u6216\u4ece\u5206\u5e03\u91c7\u6837\uff0cLLC\u63a8\u65ad\u7684\u65b9\u7a0b\u9ad8\u5ea6\u76f8\u4f3c\uff0c\u80fd\u57281000\u65f6\u95f4\u6b65\u5185\u51c6\u786e\u9884\u6d4b\u5438\u5f15\u5b50\u8f68\u8ff9\uff0c\u968f\u540e\u663e\u73b0\u8774\u8776\u6548\u5e94<\/span>\uff08butterfly effect\uff09<\/span>\u3002\u5728R\u00f6ssler\u7cfb\u7edf\u4e2d\uff0cLLC\u6062\u590d\u7684\u5206\u5c94\u56fe<\/span>\uff08bifurcation diagram\uff09<\/span>\u7cbe\u786e\u518d\u73b0\u4ece\u5468\u671f1\u5230\u6df7\u6c8c\u7684\u8fc7\u6e21\uff0c\u8bc1\u660e\u4e86\u5176\u5728\u5206\u6790\u672a\u77e5\u52a8\u529b\u5b66\u5c5e\u6027\u5982\u5468\u671f\u500d\u589e<\/span>\uff08period-doubling\uff09<\/span>\u65b9\u9762\u7684\u6f5c\u529b\u3002<\/span><\/span><\/p>\n

<\/section>\n

\u56fe5\uff1a\u63a8\u65ad\u6df7\u6c8c\u7cfb\u7edf\u52a8\u529b\u5b66\u7684\u7ed3\u679c\u3002a. \u5728\u4e0d\u540c\u521d\u59cb\u6761\u4ef6\u4e0b\uff0c\u6211\u4eec\u7684LLC\u63a8\u65ad\u7684\u63a7\u5236\u65b9\u7a0b\u7684\u6bd4\u8f83\uff0c\u5bf9\u8026\u5408Lorenz\u7cfb\u7edf\u3002b. \u7531\u4e0d\u540c\u521d\u59cb\u6761\u4ef6\u4e0bLLC\u63a8\u65ad\u7684\u65b9\u7a0b\u4ea7\u751f\u7684\u540c\u4e00\u521d\u59cb\u503c\u7684\u5438\u5f15\u5b50\u9884\u6d4b\u72b6\u6001\u7684\u6bd4\u8f83\u3002c. \u6bcf\u4e2a\u65b9\u6cd5\u5728R\u00f6ssler\u7cfb\u7edf\u4e0a\u63a8\u65ad\u7684\u65b9\u7a0b\u4e0e\u771f\u5b9e\u65b9\u7a0b\u4e4b\u95f4\u7cfb\u6570\u8bef\u5dee\u7684\u6bd4\u8f83\u3002d. \u7531TPSINDy\u3001LLC\u548cLLC+TPSINDy\u63a8\u65ad\u7684\u63a7\u5236\u65b9\u7a0b\u751f\u6210\u7684\u540c\u4e00\u5438\u5f15\u5b50\u7684\u72b6\u6001\u6bd4\u8f83\u3002e. \u901a\u8fc7Poincar\u00e9\u622a\u9762\u65b9\u6cd5\u5f97\u5230\u7684R\u00f6ssler\u7cfb\u7edf\u7684\u5206\u5c94\u56fe\uff0c\u6a2a\u8f74\u8868\u793a\u53c2\u6570c\uff08\u8303\u56f4\u4ece1\u52306\uff09\uff0c\u7eb5\u8f74\u8868\u793a\u5438\u5f15\u5b50\u7b2c\u4e8c\u7ef4\u5ea6\u72b6\u6001<\/span><\/span>(X<\/span><\/span>i,2<\/span><\/sub>) <\/span><\/span><\/span><\/span>\u3002\u53d1\u73b0\u7684\u65b9\u7a0b\u5c55\u73b0\u51fa\u4e0e\u771f\u5b9e\u65b9\u7a0b\u76f8\u540c\u7684\u5468\u671f\u500d\u589e\u548c\u6df7\u6c8c\u73b0\u8c61\u3002f. \u5728\u5468\u671f1\u65f6\u7684\u6781\u9650\u73af\u6bd4\u8f83\uff0c\u5373c=2.5\u3002g. \u5728c=5.7\u65f6\u7684\u6df7\u6c8c\u6bd4\u8f83\u3002<\/span><\/span><\/span><\/p>\n

\u771f\u5b9e\u4e16\u754c\u7cfb\u7edf\u7684\u5b9e\u9645\u5e94\u7528<\/span><\/span><\/h4>\n

LLC\u5728\u771f\u5b9e\u7cfb\u7edf\u4e2d\u8868\u73b0\u51fa\u8272\uff0c\u4f8b\u5982\u5168\u7403COVID-19\u4f20\u64ad\u6570\u636e\u548c\u884c\u4eba\u8fd0\u52a8\u8f68\u8ff9\u3002\u5728\u822a\u7a7a\u7f51\u7edc\u4e0a\u5206\u6790\u65e9\u671f\u75ab\u60c5\uff0cLLC\u53d1\u73b0\u7684\u65b9\u7a0b\u5f3a\u8c03\u81ea\u8eab\u52a8\u529b\u5b66\u7684\u4e3b\u5bfc\u4f5c\u7528\uff0c\u4ea4\u4e92\u9879\u7c7b\u4f3c\u4e8e\u751f\u6001\u4e92\u60e0\u52a8\u529b\u5b66\uff0c\u80fd\u5728\u4e0d\u540c\u8282\u70b9\u5ea6\u4e0b\u5b9e\u73b0\u51c6\u786e\u9884\u6d4b\uff0c\u800cTPSINDy\u6613\u8fc7\u4f30\u6216\u4f4e\u4f30\u3002\u540c\u6837\uff0c\u5728\u884c\u4eba\u52a8\u529b\u5b66\u4e2d\uff0cLLC\u6355\u6349\u5230\u201c\u9886\u5730\u6548\u5e94\u201d<\/span>\uff08territorial effect\uff09<\/span>\uff0c\u5305\u62ec\u8ddd\u79bb\u56e0\u7d20\uff0c\u5bfc\u81f4\u4eba\u7fa4\u805a\u96c6\u548c\u6392\u961f\u73b0\u8c61\uff0c\u4e0e\u771f\u5b9e\u8f68\u8ff9\u9ad8\u5ea6\u4e00\u81f4\uff0c\u4f18\u4e8e\u4f20\u7edf\u793e\u4f1a\u529b\u6a21\u578b<\/span>\uff08social force models, SFM\uff09<\/span>\u3002\u8fd9\u4e9b\u5e94\u7528\u9a8c\u8bc1\u4e86LLC\u5728\u672a\u77e5\u53d8\u5316\u6a21\u5f0f\u4e0b\u7684\u5b9e\u7528\u6027\u3002<\/span><\/span><\/p>\n

<\/section>\n

\u56fe6\uff1a\u63a8\u65ad\u771f\u5b9e\u7cfb\u7edf\u52a8\u529b\u5b66\u7684\u7ed3\u679c\uff0c\u5305\u62ec\u5168\u7403COVID-19\u4f20\u64ad\u548c\u884c\u4eba\u52a8\u529b\u5b66\u3002a\u2013d. \u5404\u79cd\u56fd\u5bb6\u6216\u5730\u533a\u968f\u65f6\u95f4\u53d8\u5316\u7684\u75c5\u4f8b\u6570\u6bd4\u8f83\uff0c\u7531TPSINDy\u3001LLCeach+TPSINDy\u3001LLCeach\u548cLLCtotal\u751f\u6210\u3002e. \u56db\u4e2a\u4ee3\u8868\u6027\u56fd\u5bb6\u6216\u5730\u533a\u7684\u5404\u79cd\u65b9\u6cd5\u63a8\u65ad\u7684\u63a7\u5236\u65b9\u7a0b\u7684\u6bd4\u8f83\u3002\u8bf7\u6ce8\u610f\uff0cLLCeach+TPSINDy\u4e2d\u7684\u6a21\u677f\u662f\u6211\u4eecLLC\u4e3a\u6bcf\u4e2a\u8282\u70b9\u751f\u6210\u7684\u65b9\u7a0b\u7684\u5f52\u7eb3\u3002f. \u968f\u65f6\u95f4\u53d8\u5316\u7684\u771f\u5b9e\u884c\u4eba\u8fd0\u52a8\u8f68\u8ff9\uff0c\u5176\u4e2dT\u4e4b\u524d\u7684\u6570\u636e\u7528\u4e8e\u5b66\u4e60\u3002g. \u4f7f\u7528\u4e3b\u6d41\u793e\u4f1a\u529b\u6a21\u578b\uff08SFM\uff09\u5bf9\u884c\u4eba\u52a8\u529b\u5b66\u7684\u63a8\u65ad\u7ed3\u679c\u3002h. \u4f7f\u7528\u6211\u4eec\u7684LLC\u5bf9\u884c\u4eba\u52a8\u529b\u5b66\u7684\u63a8\u65ad\u7ed3\u679c\u3002<\/span><\/span><\/p>\n


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\u8ba8\u8bba\uff1a\u5de5\u5177\u7684\u6f5c\u529b\u4e0e\u672a\u6765\u5c55\u671b<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u603b\u4e4b\uff0cLLC\u63d0\u4f9b\u4e86\u4e00\u79cd\u9ad8\u6548\u51c6\u786e\u7684\u89e3\u51b3\u65b9\u6848\uff0c\u901a\u8fc7\u4fe1\u53f7\u89e3\u8026\u548c\u9884\u8bad\u7ec3\u7b26\u53f7\u56de\u5f52\uff0c\u63ed\u793a\u4e86\u4ece\u751f\u7269\u5230\u6df7\u6c8c\u7cfb\u7edf\u7684\u52a8\u529b\u5b66\u65b9\u7a0b\u3002\u5b83\u5728\u566a\u58f0\u548c\u4e0d\u5b8c\u6574\u6570\u636e\u4e0b\u7684\u9c81\u68d2\u6027\uff0c\u4ee5\u53ca\u5728\u591a\u7ef4\u5f02\u8d28\u7cfb\u7edf\u4e2d\u7684\u901a\u7528\u6027\uff0c\u5c55\u793a\u4e86\u5176\u63a8\u52a8\u79d1\u5b66\u53d1\u73b0\u7684\u6f5c\u529b\u3002\u7136\u800c\uff0c\u6311\u6218\u4ecd\u5b58\uff0c\u5982\u672a\u77e5\u62d3\u6251\u7ed3\u6784\u548c\u5916\u90e8\u6270\u52a8\u3002\u672a\u6765\uff0c\u53ef\u6269\u5c55\u5230\u975e\u786e\u5b9a\u7cfb\u7edf<\/span>\uff08non-deterministic systems\uff09<\/span>\u548c\u9ad8\u9636\u4ea4\u4e92<\/span>\uff08higher-order interactions\uff09<\/span>\uff0c\u751a\u81f3\u4e0e\u5927\u8bed\u8a00\u6a21\u578b<\/span>\uff08large language models, LLMs\uff09<\/span>\u7ed3\u5408\uff0c\u52a9\u529bAI\u4ee3\u7406\u51b3\u7b56\u3002\u8fd9\u9879\u5de5\u4f5c\u5982\u4e00\u4e2a\u201c\u6570\u5b57\u57f9\u517b\u76bf\u201d\uff0c\u4e3a\u5e94\u5bf9\u65b0\u75ab\u60c5\u6216\u6c14\u5019\u53d8\u5316\u94fa\u5e73\u9053\u8def\u3002<\/span><\/span><\/p>\n


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

\u53c2\u8003\u6587\u732e<\/span><\/span><\/span><\/span><\/p>\n

1. Makke, N. & Chawla, S. Interpretable scientific discovery with symbolic regression: a review. Artif. Intell. Rev. 57, 2 (2024).<\/span><\/span><\/span><\/p>\n

2. La Cava, W. et al. Contemporary symbolic regression methods and their relative performance. Adv. Neural Inf. Process. Syst. 2021,1 (2021).<\/span><\/span><\/p>\n

3. Lipshtat, A., Alimi, R. & Ben-Horin, Y. Commuting in metapopulation epidemic modeling. Sci. Rep. 11, 15198 (2021).<\/span><\/span><\/p>\n

4. Udrescu, S.-M. & Tegmark, M. AI Feynman: a physics-inspired method for symbolic regression. Sci. Adv. 6, 2631 (2020).<\/span><\/span><\/p>\n

5. Gao, T. T. & Yan, G. Autonomous inference of complex network dynamics from incomplete and noisy data. Nat. Comput. Sci. 2,160\u2013168 (2022).<\/span><\/span><\/p>\n

6. Zang, C. & Wang, F. Neural dynamics on complex networks. In Proc.26th ACM SIGKDD International Conference<\/span><\/span><\/span><\/p>\n


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\u590d\u6742\u7f51\u7edc\u52a8\u529b\u5b66\u8bfb\u4e66\u4f1a<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n


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

\u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u5408\u80a5\u5de5\u4e1a\u5927\u5b66\u7269\u7406\u7cfb\u6559\u6388\u674e\u660e\u3001\u540c\u6d4e\u5927\u5b66\u526f\u6559\u6388\u5f20\u6bc5\u8d85\u3001\u5317\u4eac\u5e08\u8303\u5927\u5b66\u7279\u8058\u526f\u7814\u7a76\u5458\u53f2\u8d35\u5143\u4e0e\u5728\u8bfb\u535a\u58eb\u751f\u90b1\u4ef2\u666e\u3001\u5f20\u7ae0\u5171\u540c\u53d1\u8d77<\/span><\/span><\/span>\u300c\u590d\u6742\u7f51\u7edc\u52a8\u529b\u5b66\u300d\u8bfb\u4e66\u4f1a<\/a><\/span>\u3002\u672c\u6b21\u8bfb\u4e66\u4f1a\u5c06\u63a2\u8ba8\uff1a\u540c\u6b65\u76f8\u53d8\u7684\u4e34\u754c\u6027\u3001\u5982\u4f55\u666e\u9002\u5730\u523b\u753b\u591a\u7a33\u6001\u4e0e\u4e34\u754c\u70b9\u3001\u5982\u4f55\u8bc6\u522b\u5e76\u9884\u6d4b\u4e34\u754c\u8f6c\u53d8\u3001\u5982\u4f55\u901a\u8fc7\u5c40\u90e8\u5e72\u9884\u6765\u8c03\u63a7\u7cfb\u7edf\u4fdd\u6301\u6216\u56de\u5230\u671f\u671b\u7a33\u6001\u3001\u7206\u70b8\u903e\u6e17\u4e34\u754c\u884c\u4e3a\u7684\u5173\u952e\u7279\u5f81\u3001\u4e0d\u540c\u7c7b\u578b\u7684\u7ea7\u8054\u8fc7\u7a0b\u5bf9\u903e\u6e17\u76f8\u53d8\u7684\u5f71\u54cd\u6709\u4f55\u5f02\u540c\u3001\u9ad8\u9636\u76f8\u4e92\u4f5c\u7528\u7684\u5f71\u54cd\u80fd\u5426\u7b49\u6548\u4e3a\u82e5\u5e72\u7b80\u5355\u673a\u5236\u7684\u53e0\u52a0\u3001\u5982\u4f55\u6709\u6548\u5730\u4fc3\u8fdb\u4eba\u7c7b\u4e2a\u4f53\u95f4\u7684\u5408\u4f5c\u7b49\u95ee\u9898\u3002<\/span><\/span>\u8bfb\u4e66\u4f1a\u5df2\u5b8c\u7ed3\uff0c\u73b0\u5728\u62a5\u540d\u53ef\u52a0\u5165\u793e\u7fa4\u5e76\u89e3\u9501\u56de\u653e\u89c6\u9891\u6743\u9650\u3002<\/span><\/span><\/span><\/span><\/section>\n
\"\u56fe\u7247\"<\/span><\/a><\/p>\n


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

\u8be6\u60c5\u8bf7\u89c1\uff1a\u590d\u6742\u7f51\u7edc\u4e0a\u7684\u81ea\u7ec4\u7ec7\u4e0e\u96c6\u4f53\u884c\u4e3a\uff1a\u4ece\u6269\u6563\u3001\u76f8\u53d8\u5230\u535a\u5f08 | \u8bfb\u4e66\u4f1a\u542f\u52a8<\/a><\/span><\/span><\/p>\n<\/section>\n


<\/span><\/span><\/strong><\/strong><\/strong><\/section>\n
\u63a8\u8350\u9605\u8bfb<\/span><\/span><\/strong><\/strong><\/strong><\/section>\n
<\/strong><\/section>\n
1. Phys. Rep. \u91cd\u78c5\u7efc\u8ff0\uff1a\u7f51\u7edc\u53cd\u5e94-\u6269\u6563\u7cfb\u7edf\u89e3\u6790\u4f20\u67d3\u75c5\u4f20\u64ad\u7684\u590d\u6742\u52a8\u529b\u5b66<\/span><\/a><\/span><\/strong>
<\/span><\/strong><\/span><\/section>\n
2. \u7a00\u758f\u6709\u5411\u7f51\u7edc\u4e0a\u590d\u6742\u7cfb\u7edf\u7684\u52a8\u529b\u5b66\u5e73\u5747\u573a\u7406\u8bba<\/span><\/a><\/span><\/strong>
<\/span><\/strong><\/span><\/section>\n
3. <\/span>\u590d\u6742\u7f51\u7edc\u4e2d\u7684\u52a8\u529b\u5b66\uff1a\u4ece\u7406\u8bba\u7a81\u7834\u5230\u5730\u7403\u751f\u6001\u7684\u8de8\u5b66\u79d1\u5bf9\u8bdd<\/span><\/a><\/span><\/strong><\/section>\n
<\/strong><\/section>\n
4. <\/span>\u6d8c\u73b0\u52a8\u529b\u5b66\u5982\u4f55\u7528\u6765\u5206\u6790\u590d\u6742\u7cfb\u7edf\uff1f | \u65b0\u8bfe\u4e0a\u7ebf<\/span><\/a><\/span><\/strong><\/span>
<\/span><\/strong><\/strong><\/span><\/section>\n
<\/strong>5. <\/span><\/strong>\u96c6\u667a\u5b66\u56ed\u7cbe\u54c1\u8bfe\u7a0b\u514d\u8d39\u5f00\u653e\uff0c\u89e3\u9501\u7cfb\u7edf\u79d1\u5b66\u4e0e AI \u65b0\u4e16\u754c<\/span><\/a><\/span><\/strong><\/section>\n

6. \u9ad8\u8003\u5206\u6570\u53ea\u662f\u5f20\u5165\u573a\u5238\uff0c\u4f60\u7684\u79d1\u7814\u5192\u9669\u5728\u8fd9\u91cc\u542f\u822a\uff01<\/span><\/a><\/span><\/strong><\/p>\n

7.<\/span><\/span><\/strong><\/strong><\/span> <\/span><\/span><\/strong><\/strong><\/span>\u7cfb\u7edf\u79d1\u5b66\u524d\u6cbf\u5341\u8bb2\uff1a\u7406\u89e3\u81ea\u7136\u3001\u751f\u547d\u4e0e\u793e\u4f1a\u7684\u590d\u6742\u6027<\/a><\/span><\/strong><\/strong><\/span><\/p>\n


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


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

\u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfb\u4e66\u4f1a<\/span><\/span><\/section>\n<\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

\u5bfc\u8bed \u53d1\u73b0\u590d\u6742\u7f51\u7edc\u52a8\u529b\u5b66\u65b9\u7a0b\u662f\u5f53\u4ee3\u79d1\u5b66\u4e2d\u7684\u4e00\u4e2a\u57fa\u672c\u6311\u6218\uff0c\u5229\u7528\u4e30\u5bcc\u7684\u6570\u636e\u53ef\u4ee5\u63ed\u793a\u5404\u79cd\u9886\u57df\u4e2d\u590d\u6742\u73b0\u8c61\u7684\u5f62\u6210\u548c\u6f14\u5316\u80cc\u540e\u7684\u9690\u85cf\u6a21\u5f0f\uff0c\u5e76\u8f85\u52a9\u51b3\u7b56\u3002\u8fd1\u65e5\uff0c \u300a\u81ea\u7136 \u00b7 \u901a\u8baf\u300b\u6742\u5fd7\u53d1\u8868\u8bba\u6587\u63d0\u51fa\u4e86\u4e00\u79cd\u901a\u7528\u7684\u8ba1\u7b97\u5de5\u5177LLC\uff08Learning Law of Changes\uff09\uff0c\u80fd\u591f\u901a\u8fc7\u7ed3\u5408\u6df1\u5ea6\u5b66\u4e60\u7684\u51fa\u8272\u62df\u5408\u80fd\u529b…<\/p>\n","protected":false},"author":0,"featured_media":63934,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"special":[],"_links":{"self":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/63961"}],"collection":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/types\/post"}],"replies":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=63961"}],"version-history":[{"count":0,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/63961\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/media\/63934"}],"wp:attachment":[{"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=63961"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=63961"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=63961"},{"taxonomy":"special","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fspecial&post=63961"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}