{"id":50265,"date":"2024-05-11T19:03:19","date_gmt":"2024-05-11T11:03:19","guid":{"rendered":"https:\/\/swarma.org\/?p=50265"},"modified":"2024-05-11T19:03:19","modified_gmt":"2024-05-11T11:03:19","slug":"%e6%97%a0%e5%bf%83%e6%8f%92%e6%9f%b3%ef%bc%9a%e8%8b%8f%e8%81%94%e6%95%b0%e5%ad%a6%e5%ae%b6%e6%9f%af%e5%b0%94%e8%8e%ab%e5%93%a5%e6%b4%9b%e5%a4%ab%e4%b8%8e%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9c%e7%9a%84","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=50265","title":{"rendered":"\u65e0\u5fc3\u63d2\u67f3\uff1a\u82cf\u8054\u6570\u5b66\u5bb6\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u4e0e\u795e\u7ecf\u7f51\u7edc\u7684\u65b0\u751f"},"content":{"rendered":"
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\u82cf\u8054\u6570\u5b66\u5bb6\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\uff08Andrey N. Kolmogorov\uff0c1903-1987\uff09\u3002\u56fe\u6e90\uff1ahttps:\/\/wolffund.org.il\/<\/span><\/section>\n<\/section>\n
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\u4eca\u5e74\u4e94\u4e00\u5047\u671f\uff0c\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7269\u7406\u5b66\u5bb6Max Tegmark\u548c\u5176\u535a\u58eb\u751f\u5218\u5b50\u9e23\u7b49\u4eba\u5728arxiv\u4e0a\u6302\u51fa\u7684\u4e00\u7bc7\u5173\u4e8e\u673a\u5668\u5b66\u4e60\u7684\u6587\u7ae0\u5f15\u53d1\u5173\u6ce8\u3002\u4ed6\u4eec\u63d0\u51fa\u4e86\u4e00\u79cd\u53eb\u505aKAN<\/span>\uff08Kolmogorov-Arnold Network\uff09<\/span>\u7684\u65b0\u6846\u67b6\uff0c\u79f0\u5176\u5728\u51c6\u786e\u6027\u548c\u53ef\u89e3\u91ca\u6027\u65b9\u9762\u7684\u8868\u73b0\u5747\u4f18\u4e8e\u591a\u5c42\u611f\u77e5\u5668<\/span>\uff08MLP\uff09<\/span>\u3002<\/span><\/section>\n
\u4eca\u5929\uff0c\u6211\u4eec\u5c31\u6765\u8c08\u8c08\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b-\u963f\u8bfa\u5fb7\u53e0\u52a0\u5b9a\u7406\u7684\u6e90\u8d77\u548c\u53d1\u5c55\u3002\u5728\u4e4b\u524d\u7684\u6587\u7ae0\u201c\u4eba\u7269 | \u6240\u7f57\u95e8\u8bfa\u592b\uff1a\u5927\u8bed\u8a00\u6a21\u578b\u7684\u5148\u77e5<\/a>\u201d\uff0c\u7b14\u8005\u66fe\u7ecf\u7b80\u8981\u4ecb\u7ecd\u8fc7\u8fc7\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b<\/span>\uff08A. N. Kolmogorov\uff0c1903-1987\uff09<\/span>\u548c\u963f\u8bfa\u5fb7<\/span>\uff08Vladimir Arnold\uff0c1937-2010\uff09<\/span>\u7684\u5176\u4ed6\u5de5\u4f5c\u3002
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\u4e07\u80fd\u7684\u82cf\u8054\u6570\u5b66\u5bb6\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u5bf9\u8ba1\u7b97\u673a\u79d1\u5b66\u6709\u4e24\u5927\u8d21\u732e\u3002\u9996\u5148\uff0c\u4ed6\u548c\u7f8e\u56fd\u6570\u5b66\u5bb6\u6240\u7f57\u95e8\u8bfa\u592b\u548c\u8521\u5ef7\u72ec\u7acb\u53d1\u5c55\u7684\u6240\u7f57\u95e8\u8bfa\u592b-\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b-\u8521\u5ef7\u7406\u8bba<\/span>\uff08\u5927\u591a\u6570\u65f6\u95f4\u88ab\u66f4\u7b80\u5355\u5730\u79f0\u4e3a\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u590d\u6742\u6027\uff0c\u6216\u7b97\u6cd5\u4fe1\u606f\u8bba\uff09<\/span>\u6b63\u5728\u6210\u4e3a\u5927\u8bed\u8a00\u6a21\u578b\u7684\u7406\u8bba\u57fa\u7840\u548c\u89e3\u91ca\u5de5\u5177\u3002\u8ffd\u968f\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u505a\u590d\u6742\u6027\u7814\u7a76\u7684\u5b66\u751f\u5217\u6587<\/span>\uff08Leonid Levin\uff0c1948-\uff09<\/span>\uff0c\u72ec\u7acb\u4e8e\u5e93\u514b<\/span>\uff08Stephen Cook\uff0c1939-\uff09<\/span>\uff0c\u57281970\u5e74\u4ee3\u521d\u671f\u5f97\u51fa\u4e86NP-\u5b8c\u5168\u6027\u7684\u7ed3\u679c\uff0c2000\u5e74\u540e\u8fd9\u4e2a\u539f\u4ee5\u5e93\u514b\u4e3a\u540d\u7684\u5b9a\u7406\uff0c\u5728\u8ba1\u7b97\u7406\u8bba\u7684\u6559\u79d1\u4e66\u91cc\u591a\u88ab\u6539\u79f0\u4e3a\u5e93\u514b-\u5217\u6587\u5b9a\u7406\u3002\u5217\u6587\u867d\u6ca1\u6709\u50cf\u5e93\u514b\u90a3\u6837\u5f97\u5230\u8ba1\u7b97\u673a\u79d1\u5b66\u7684\u6700\u9ad8\u5956\u56fe\u7075\u5956\uff0c\u4f46\u5f97\u4e86ACM\u548cIEEE\u8054\u5408\u9881\u53d1\u7684\u9ad8\u5fb7\u7eb3<\/span>\uff08Knuth\uff09<\/span>\u5956\uff0c\u7b97\u662f\u8865\u507f\u3002<\/span><\/o:p><\/section>\n
\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u7684\u53e6\u4e00\u91cd\u8981\u8d21\u732e\u5728\u6570\u5b66\u754c\u5f71\u54cd\u5e7f\u6cdb\uff0c\u4f46\u5f88\u665a\u624d\u88ab\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6\u548c\u4eba\u5de5\u667a\u80fd\u5b66\u8005\u8d4f\u8bc6\uff0c\u5c3d\u7ba1\u8fd9\u9879\u5de5\u4f5c\u51fa\u73b0\u66f4\u65e9\u3002\u4ed6\u548c\u5b66\u751f\u5f17\u62c9\u57fa\u7c73\u5c14\u00b7\u963f\u8bfa\u5fb7\u57281956-1957\u5e74\u95f4\u5171\u540c\u8bc1\u660e\u7684\u8868\u793a\u5b9a\u7406\u6216\u79f0\u53e0\u52a0<\/span>\uff08superposition\uff09<\/span>\u5b9a\u7406\uff0c\u540e\u6765\u6210\u4e3a\u795e\u7ecf\u7f51\u7edc\u7684\u7406\u8bba\u57fa\u7840\u3002\u795e\u7ecf\u7f51\u7edc\u590d\u5174\u7684\u6570\u5b66\u4fdd\u969c\u662f\u901a\u7528\u903c\u8fd1\u5b9a\u7406<\/span>\uff08universal approximation theorem\uff09<\/span>\uff0c\u5176\u6e90\u5934\u5c31\u662f\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b-\u963f\u8bfa\u5fb7\u53e0\u52a0\u3002\u5c31\u50cf\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u7684\u5f88\u591a\u5de5\u4f5c\uff0c\u90fd\u662f\u4ed6\u5148\u5f00\u5934\u6307\u660e\u65b9\u5411\uff0c\u5e76\u4e14\u7ed9\u51fa\u8bc1\u660e\u7684\u601d\u8def\u6216\u8005\u8bc1\u660e\u7684\u901f\u5199\u7248\uff0c\u7136\u540e\u7531\u5b66\u751f\u4eec\u7cbe\u5316\u4e3a\u5b8c\u7f8e\u7684\u7d20\u63cf\u3002<\/span><\/o:p><\/section>\n
\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u7684\u53e6\u4e00\u91cd\u8981\u5de5\u4f5cKAM\u7406\u8bba\uff0c\u4e5f\u662f\u548c\u963f\u8bfa\u5fb7\u5408\u4f5c\u5b8c\u6210\u7684\u3002\u963f\u8bfa\u5fb7\u548c\u4ee5\u8272\u5217\u903b\u8f91\u5b66\u5bb6\u8c22\u4e50\u8d6b\u5206\u4eab<\/span>\uff08Saharon Shelah\uff09<\/span>\u4e862001\u5e74\u7684\u6c83\u5c14\u592b\u5956\uff0c\u548c\u53e6\u4e00\u4f4d\u4fc4\u56fd\u6570\u5b66\u7269\u7406\u5b66\u5bb6\u6cd5\u6377\u8036\u592b\u5206\u4eab\u4e862008\u5e74\u7684\u90b5\u9038\u592b\u5956\u3002\u4e18\u6210\u6850\u516c\u5141\u5730\u8bf4\uff0c\u5728\u4e9a\u5386\u5c71\u5fb7\u7f57\u592b\u548c\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u7b49\u9886\u5bfc\u4e0b\u7684\u4fc4\u7f57\u65af\u6570\u5b66\u5b66\u6d3e\uff0c\u5f53\u65f6\u5df2\u7ecf\u63a5\u8fd1\u7f8e\u56fd\u6570\u5b66\u7684\u603b\u4f53\u6c34\u5e73\u3002\u5176\u4e2d\u72b9\u592a\u88d4\u6570\u5b66\u5bb6\u7fa4\u4f53\u7684\u6545\u4e8b\uff0c\u73b0\u5728\u8bfb\u6765\uff0c\u52b1\u5fd7\u4e14\u4ee4\u4eba\u550f\u5618\u3002<\/span><\/section>\n
\u4ee5ChatGPT\u4ee3\u8868\u7684\u5927\u8bed\u8a00\u6a21\u578b\u5f15\u53d1\u7684\u8ba8\u8bba\uff0c\u591a\u805a\u7126\u4e8e\u6570\u636e\u4e0e\u7b97\u529b\u7b49\u5de5\u7a0b\u95ee\u9898\uff0c\u7406\u8bba\u65b9\u9762\u7684\u7814\u7a76\u5219\u4e0d\u90a3\u4e48\u70ed\u70c8\u3002\u5927\u6a21\u578b\u5411\u8ba1\u7b97\u7406\u8bba\u63d0\u51fa\u4e86\u65b0\u95ee\u9898\uff0c\u800c\u8ba1\u7b97\u7406\u8bba\u4e5f\u53ef\u5e2e\u52a9\u5927\u6a21\u578b\u793a\u7b2c\u4e00\u6027\u539f\u7406\uff0c\u4ece\u800c\u627e\u5230\u8fb9\u754c\u548c\u65b9\u5411\u3002\u5728\u5f53\u4e0b\u7684\u7406\u8bba\u6ca1\u6cd5\u89e3\u91ca\u5de5\u7a0b\u5b9e\u8df5\u65f6\uff0c\u5de5\u7a0b\u5e08\u4eec\u4e5f\u4f1a\u8f6c\u5411\u5386\u53f2\u53bb\u5bfb\u627e\u524d\u8f88\u4eec\u88ab\u57cb\u6ca1\u7684\u601d\u60f3\uff0c\u529b\u56fe\u4e3a\u4f55\u53bb\u4f55\u4ece\u63d0\u4f9b\u65b9\u5411\u6027\u7684\u6d1e\u89c1\u3002<\/span><\/section>\n

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\u795e\u7ecf\u7f51\u7edc\u4f5c\u4e3a\u5927\u8111\u542f\u53d1<\/span>\uff08Brain-inspired\uff09<\/span>\u7684\u8ba1\u7b97\u6a21\u578b\uff0c\u59cb\u4e8e\u9ea6\u5361\u6d1b\u514b<\/span>\uff08Warren McCulloch, 1898-1969\uff09<\/span>\u548c\u76ae\u8328<\/span>\uff08Walter Pitts\uff0c1923-1969\uff09<\/span>1943\u5e74\u7684\u5de5\u4f5c\u3002\u4e24\u4f4d\u4f5c\u8005\u4e4b\u4e00\u7684\u76ae\u8328\u662f\u81ea\u5b66\u6210\u624d\u7684\u903b\u8f91\u5b66\u5bb6\uff0c\u800c\u9ea6\u5361\u6d1b\u514b\u662f\u795e\u7ecf\u5fc3\u7406\u5b66\u5bb6\uff0c\u662f\u76ae\u8328\u8001\u5e08\u8f88\u7684\u4eba\u7269\uff0c\u4ed6\u4eec\u63d0\u51fa\u795e\u7ecf\u5143\u6709\u4e00\u4e2a\u9608\u503c\uff0c\u5373\u523a\u6fc0\u5fc5\u987b\u8d85\u8fc7\u8fd9\u4e2a\u9608\u503c\u624d\u80fd\u4ea7\u751f\u8109\u51b2\u3002\u867d\u7136\u4e8c\u4eba\u6ca1\u6709\u5f15\u7528\u56fe\u7075\u548c\u4e18\u5947<\/span>\uff08Alonzo Church\uff09<\/span>\u57281936-1937\u5e74\u7684\u8bba\u6587\uff0c\u4f46\u5728\u6587\u4e2d\u660e\u786e\u63d0\u51fa\uff0c\u53ea\u8981\u7ed9\u4ed6\u4eec\u7684\u7f51\u7edc\u63d0\u4f9b\u65e0\u7a77\u5b58\u50a8\uff0c\u4ed6\u4eec\u7684\u7f51\u7edc\u53ef\u4ee5\u6a21\u62df\u03bb-\u6f14\u7b97\u548c\u56fe\u7075\u673a\u3002<\/span><\/section>\n

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\u968f\u540e\u76f8\u5173\u7684\u7814\u7a76\u4e0d\u65ad\u3002\u4e18\u5947\u7684\u5b66\u751f\u514b\u83b1\u5c3c<\/span>\uff08Stephen Cole Kleene\uff0c1909-1994\uff09<\/span>1956\u5e74\u8fdb\u4e00\u6b65\u7814\u7a76\u4e86McCulloch-Pitts \u7f51\u7edc\u7684\u8868\u8fbe\u80fd\u529b\u3002\u5404\u79cd\u6fc0\u6d3b\u51fd\u6570\u7684\u9009\u62e9\u662f\u95e8\u827a\u672f\u3002McCulloch-Pitts \u7f51\u7edc\u662f\u79bb\u6563\u7684\uff0c\u66f4\u7279\u5b9a\u5730\u8bf4\uff0c\u662f\u5e03\u5c14\u503c\u7684\uff0c\u975e0\u53731\u3002\u73b0\u4ee3\u7684\u795e\u7ecf\u7f51\u7edc\u6fc0\u6d3b\u51fd\u6570\u5927\u591a\u662f\u975e\u7ebf\u6027\u7684\uff0c\u4e0d\u5c40\u9650\u4e8e\u5e03\u5c14\u503c\u3002\u53ef\u4ee5\u8bf4\uff0cMcCulloch-Pitts \u7f51\u7edc\u66f4\u50cf\u662f\u5e03\u5c14\u7535\u8def\uff0c\u800c\u4e0d\u662f\u73b0\u4ee3\u610f\u4e49\u7684\u795e\u7ecf\u7f51\u7edc\u3002<\/span><\/o:p><\/section>\n
\u5f17\u5170\u514b\u00b7\u7f57\u68ee\u5e03\u62c9\u7279(Frank Rosenblatt)\u63d0\u51fa\u7684\u201c\u611f\u77e5\u673a\u201d<\/span>\uff08Perceptron\uff09<\/span>\u662f1950\u5e74\u4ee3\u795e\u7ecf\u7f51\u7edc\u7684\u6807\u5fd7\u6027\u5de5\u4f5c\u3002\u7f57\u68ee\u5e03\u62c9\u7279\u662f\u5fc3\u7406\u5b66\u51fa\u8eab\uff0c1957\u5e74\u5728\u4e00\u53f0IBM 704\u673a\u4e0a\u5b9e\u73b0\u4e86\u5355\u5c42\u201c\u611f\u77e5\u673a\u201d\u795e\u7ecf\u7f51\u7edc\uff0c\u8bc1\u660e\u4e86\u611f\u77e5\u673a\u53ef\u4ee5\u5904\u7406\u7ebf\u6027\u53ef\u5206\u7684\u6a21\u5f0f\u8bc6\u522b\u95ee\u9898\uff0c\u968f\u540e\u53c8\u505a\u4e86\u82e5\u5e72\u5fc3\u7406\u5b66\u5b9e\u9a8c\uff0c\u529b\u56fe\u8bc1\u660e\u611f\u77e5\u673a\u6709\u5b66\u4e60\u80fd\u529b\u3002\u7f57\u68ee\u5e03\u62c9\u72791962\u5e74\u51fa\u4e86\u672c\u4e66\u300a\u795e\u7ecf\u52a8\u529b\u5b66\u539f\u7406\uff1a\u611f\u77e5\u673a\u548c\u5927\u8111\u673a\u5236\u7684\u7406\u8bba\u300b<\/span>\uff08Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms<\/em>\uff09<\/span>\uff0c\u603b\u7ed3\u4e86\u4ed6\u7684\u6240\u6709\u7814\u7a76\u6210\u679c\uff0c\u4e00\u65f6\u6210\u4e3a\u795e\u7ecf\u7f51\u7edc\u6d3e\u7684\u5fc5\u8bfb\u4e66\u3002\u7f57\u68ee\u5e03\u62c9\u7279\u7684\u540d\u58f0\u8d8a\u6765\u8d8a\u5927\uff0c\u5f97\u5230\u7684\u7814\u7a76\u7ecf\u8d39\u4e5f\u8d8a\u6765\u8d8a\u591a\u3002\u7f8e\u56fd\u56fd\u9632\u90e8\u548c\u6d77\u519b\u90fd\u8d44\u52a9\u4e86\u4ed6\u7684\u7814\u7a76\u3002\u7f57\u68ee\u5e03\u62c9\u7279\u4e00\u6539\u5f80\u65e5\u7684\u5bb3\u7f9e\uff0c\u6210\u4e86\u660e\u661f\uff0c\u9891\u9891\u5728\u5a92\u4f53\u51fa\u955c\uff0c\u5f00\u8dd1\u8f66\uff0c\u5f39\u94a2\u7434\uff0c\u5230\u5904\u663e\u6446\u3002\u51e0\u4e4e\u6bcf\u4e2a\u65f6\u4ee3\u90fd\u6709\u5a92\u4f53\u9700\u8981\u7684\u5177\u6709\u5a31\u4e50\u6f5c\u8d28\u7684\u79d1\u6280\u4ee3\u8868\u4eba\u7269\u3002\u5a92\u4f53\u5bf9\u7f57\u68ee\u5e03\u62c9\u7279\u4e5f\u8868\u73b0\u51fa\u4e86\u8fc7\u5ea6\u7684\u70ed\u60c5\u3002\u6bd5\u7adf\uff0c\u80fd\u591f\u6784\u5efa\u4e00\u53f0\u53ef\u4ee5\u6a21\u62df\u5927\u8111\u7684\u673a\u5668\uff0c\u5f53\u7136\u503c\u4e00\u4e2a\u5934\u7248\u5934\u6761\u3002\u8fd9\u4f7f\u5f97\u53e6\u4e00\u6d3e\u7684\u4eba\u76f8\u5f53\u4e0d\u723d\u3002<\/span><\/o:p><\/section>\n
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\u7f57\u68ee\u5e03\u62c9\u7279\u548c\u4ed6\u7684\u611f\u77e5\u673a\uff08Frank Rosenblatt\uff0c1928-1971\uff09\u3002\u56fe\u6e90\uff1a\u7ef4\u57fa\u767e\u79d1<\/span><\/p>\n

\u660e\u65af\u57fa<\/span>\uff08Marvin Minsky\uff09<\/span>\u662f\u4eba\u5de5\u667a\u80fd\u7684\u5960\u57fa\u4eba\u4e4b\u4e00\uff0c\u4e5f\u662f1956\u5e74\u8fbe\u7279\u8305\u65af\u4f1a\u8bae\u7684\u7ec4\u7ec7\u8005\u4e4b\u4e00<\/a>\uff0c\u8fd9\u4e2a\u4f1a\u8bae\u5b9a\u4e49\u4e86\u201c\u4eba\u5de5\u667a\u80fd\u201d\u8fd9\u4e2a\u8bcd\uff0c\u5e76\u628a\u795e\u7ecf\u7f51\u7edc\u4e5f\u7eb3\u5165\u4eba\u5de5\u667a\u80fd\u7684\u7814\u7a76\u8303\u56f4\u3002\u65e9\u671f\u660e\u65af\u57fa\u662f\u795e\u7ecf\u7f51\u7edc\u7684\u652f\u6301\u8005\u3002\u4ed61954\u5e74\u5728\u666e\u6797\u65af\u987f\u5927\u5b66\u7684\u535a\u58eb\u8bba\u6587\u9898\u76ee\u662f\u300a\u795e\u7ecf-\u6a21\u62df\u5f3a\u5316\u7cfb\u7edf\u7684\u7406\u8bba\u53ca\u5176\u5728\u5927\u8111\u6a21\u578b-\u95ee\u9898\u4e0a\u7684\u5e94\u7528\u300b\uff0c\u5b9e\u9645\u4e0a\u5c31\u662f\u4e00\u7bc7\u5173\u4e8e\u795e\u7ecf\u7f51\u7edc\u7684\u8bba\u6587\u3002\u4f46\u4ed6\u540e\u6765\u6539\u53d8\u4e86\u7acb\u573a\uff0c\u8ba4\u4e3a\u795e\u7ecf\u7f51\u7edc\u5e76\u4e0d\u662f\u89e3\u51b3\u4eba\u5de5\u667a\u80fd\u95ee\u9898\u7684\u6709\u6548\u5de5\u5177\u3002\u4ed6\u665a\u5e74\u63a5\u53d7\u91c7\u8bbf\u65f6\u5f00\u73a9\u7b11\u8bf4\uff0c\u90a3\u7bc7300\u591a\u9875\u7684\u535a\u58eb\u8bba\u6587\u4ece\u6765\u6ca1\u6709\u6b63\u5f0f\u53d1\u8868\u8fc7\uff0c\u5927\u6982\u53ea\u5370\u4e86\u4e09\u672c\uff0c\u4ed6\u81ea\u5df1\u4e5f\u8bb0\u4e0d\u6e05\u5185\u5bb9\u4e86\u3002\u8c8c\u4f3c\u60f3\u6781\u529b\u5f00\u8131\u81ea\u5df1\u548c\u795e\u7ecf\u7f51\u7edc\u5b66\u79d1\u7684\u5173\u7cfb\u3002<\/span><\/o:p><\/section>\n
\u660e\u65af\u57fa\u548c\u7f57\u68ee\u5e03\u62c9\u7279\u662f\u4e2d\u5b66\u540c\u5b66\uff0c\u4ed6\u4eec\u5747\u5728\u7ebd\u7ea6\u7684\u5e03\u6717\u514b\u65af<\/span>\uff08Bronx\uff09<\/span>\u79d1\u5b66\u9ad8\u4e2d\u5c31\u8bfb\u3002\u8fd9\u6240\u5b66\u6821\u5927\u6982\u662f\u5168\u4e16\u754c\u667a\u529b\u6700\u5bc6\u96c6\u7684\u9ad8\u4e2d\u4e4b\u4e00\uff0c\u6bd5\u4e1a\u751f\u91cc\u51fa\u8fc79\u4e2a\u8bfa\u8d1d\u5c14\u5956\u30016\u4e2a\u666e\u5229\u7b56\u5956\u548c2\u4e2a\u56fe\u7075\u5956\u3002\u660e\u65af\u57fa\u662f1945\u5e74\u7684\u6bd5\u4e1a\u751f\uff0c\u800c\u7f57\u68ee\u5e03\u62c9\u7279\u662f1946\u5e74\u7684\u6bd5\u4e1a\u751f\u3002\u4ed6\u4eec\u5f7c\u6b64\u8ba4\u8bc6\uff0c\u4e92\u76f8\u5ac9\u5992\u3002<\/span><\/o:p><\/section>\n
\u4e00\u6b21\u4f1a\u8bae\u4e0a\uff0c\u660e\u65af\u57fa\u548c\u7f57\u68ee\u5e03\u62c9\u7279\u5927\u5435\u4e86\u4e00\u67b6\u3002\u968f\u540e\uff0c\u660e\u65af\u57fa\u548c\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7684\u6559\u6388\u4f69\u73c0\u7279<\/span>\uff08Seymour Papert\uff09<\/span>\u5408\u4f5c\uff0c\u8bc1\u660e\u7f57\u68ee\u5e03\u62c9\u7279\u7684\u611f\u77e5\u673a\u7f51\u7edc\u4e0d\u80fd\u89e3\u51b3XOR<\/span>\uff08\u5f02\u6216\uff09<\/span>\u95ee\u9898\u3002\u5f02\u6216\u662f\u4e00\u4e2a\u57fa\u672c\u7684\u903b\u8f91\u95ee\u9898\uff0c\u5982\u679c\u8fde\u8fd9\u4e2a\u95ee\u9898\u90fd\u89e3\u51b3\u4e0d\u4e86\uff0c\u90a3\u610f\u5473\u7740\u795e\u7ecf\u7f51\u7edc\u7684\u8ba1\u7b97\u80fd\u529b\u5b9e\u5728\u6709\u9650\u3002\u660e\u65af\u57fa\u548c\u4f69\u73c0\u7279\u628a\u5408\u4f5c\u6210\u679c\u5199\u6210\u4e66\uff1a\u300a\u611f\u77e5\u673a\uff1a\u8ba1\u7b97\u51e0\u4f55\u5b66\u5bfc\u8bba\u300b<\/span>\uff08Perceptrons: An Introduction to Computational Geometry<\/em>\uff09<\/span>\uff0c\u8fd9\u4e66\u5f71\u54cd\u5de8\u5927\uff0c\u51e0\u4e4e\u5426\u5b9a\u4e86\u6574\u4e2a\u795e\u7ecf\u7f51\u7edc\u7814\u7a76\u3002<\/span><\/section>\n
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\u660e\u65af\u57fa\uff08Marvin Minsky\uff0c1927-2016\uff09\u3002\u56fe\u6e90\uff1a\u7ef4\u57fa\u767e\u79d1<\/span><\/section>\n

\u5176\u5b9e\u7f57\u68ee\u5e03\u62c9\u7279\u6b64\u524d\u4e5f\u5df2\u9884\u611f\u5230\u611f\u77e5\u673a\u5b58\u5728\u5c40\u9650\uff0c\u7279\u522b\u662f\u5728\u201c\u7b26\u53f7\u5904\u7406\u201d\u65b9\u9762\uff0c\u5e76\u4ee5\u81ea\u5df1\u795e\u7ecf\u5fc3\u7406\u5b66\u5bb6\u7684\u7ecf\u9a8c\u6307\u51fa\uff0c\u67d0\u4e9b\u5927\u8111\u53d7\u5230\u4f24\u5bb3\u7684\u4eba\u4e5f\u4e0d\u80fd\u5904\u7406\u7b26\u53f7\u3002\u4f46\u611f\u77e5\u673a\u7684\u7f3a\u9677\u88ab\u660e\u65af\u57fa\u4ee5\u4e00\u79cd\u654c\u610f\u7684\u65b9\u5f0f\u5448\u73b0\u51fa\u6765\uff0c\u5bf9\u7f57\u68ee\u5e03\u62c9\u7279\u662f\u4e2a\u81f4\u547d\u6253\u51fb\u3002\u6700\u7ec8\uff0c\u653f\u5e9c\u8d44\u52a9\u673a\u6784\u4e5f\u9010\u6e10\u505c\u6b62\u5bf9\u795e\u7ecf\u7f51\u7edc\u7814\u7a76\u7684\u652f\u6301\u30021971\u5e74\uff0c\u7f57\u68ee\u5e03\u62c9\u7279\u572843\u5c81\u751f\u65e5\u90a3\u5929\u5212\u8239\u65f6\u6eba\u4ea1\uff0c\u6709\u8bf4\u6cd5\u8ba4\u4e3a\u8fd9\u662f\u81ea\u6740\u3002<\/span><\/p>\n

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\u8981\u60f3\u7406\u89e3AlphaFold3\u8fdb\u6b65\u7684\u610f\u4e49\uff0c\u9700\u8981\u4e86\u89e3\u751f\u7269\u4f53\u5305\u542b\u4e86\u591a\u4e2a\u5c42\u6b21\u7684\u590d\u6742\u6027\u3002\u751f\u7269\u7684\u8fd0\u884c\u84dd\u56fe\u4eceDNA\u7ecf\u7531\u8f6c\u5f55\u5f97\u5230RNA\uff0c\u518d\u7531RNA\u7ffb\u8bd1\u751f\u6210\u86cb\u767d\u8d28\uff0c\u86cb\u767d\u5728\u751f\u6210\u540e\u4f1a\u8fdb\u884c\u4fee\u9970<\/span>\uff08\u86cb\u767d\u8d28\u4ea7\u751f\u540e\u53d1\u751f\u7684\u5316\u5b66\u53d8\u5316\uff09<\/span>\uff0c\u5e76\u7ecf\u7531\u4e0e\u5c0f\u5206\u5b50\u914d\u4f53<\/span>\uff08ligand\uff09<\/span>\u548c\u79bb\u5b50\uff0c\u4ee5\u53ca\u86cb\u767d\u8d28\u4e4b\u95f4\u7684\u4e92\u4f5c\u5171\u540c\u5f71\u54cd\uff0c\u6267\u884c\u7279\u5b9a\u7684\u529f\u80fd\u3002<\/span><\/o:p><\/section>\n
\u5927\u536b\u00b7\u5e0c\u5c14\u4f2f\u72791900\u5e74\u5728\u7b2c\u4e8c\u5c4a\u56fd\u9645\u6570\u5b66\u5bb6\u5927\u4f1a\u63d0\u51fa\u4e8623\u4e2a\u5f85\u89e3\u6570\u5b66\u95ee\u9898\uff0c\u8fd9\u4e9b\u95ee\u9898\u6307\u5f15\u4e86\u540e\u7eed\u7684\u6570\u5b66\u53d1\u5c55\u3002\u5e0c\u5c14\u4f2f\u7279\u63d0\u51fa\u7684\u7b2c13\u4e2a\u95ee\u9898\uff0c\u76f8\u8f83\u4e8e\u5176\u4ed6\u95ee\u9898\uff0c\u5e76\u4e0d\u662f\u7279\u522b\u5f15\u4eba\u6ce8\u76ee\uff0c\u5373\u4f7f\u5728\u6570\u5b66\u5bb6\u7fa4\u4f53\u4e2d\uff0c\u4e5f\u8fdc\u975e\u5e7f\u4e3a\u4eba\u77e5\u3002\u6211\u4eec\u7528\u7ebf\u4e0a\u641c\u7d22\u5f15\u64ceGoogle Ngram\u6bd4\u8f83\u4e00\u4e0b\u7b2c10\u548c\u7b2c13\u95ee\u9898\uff0c\u53ef\u77e5\u5927\u6982\u3002<\/span><\/o:p><\/section>\n
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Google Ngram\u7684\u7ed3\u679c<\/span><\/section>\n
\u5e0c\u5c14\u4f2f\u7279\u7b2c13\u95ee\u9898\u662f\u8fd9\u6837\u8bf4\u7684\uff1a7\u6b21\u65b9\u7a0b\u7684\u89e3\uff0c\u80fd\u5426\u7528\u4e24\u4e2a\u53d8\u91cf\u7684\u51fd\u6570\u7684\u7ec4\u5408\u8868\u793a\uff1f<\/span>\uff08Impossibility of the solution of the general equation of the 7-th degree by means of functions of only two arguments\uff09<\/span>\u3002\u5e0c\u5c14\u4f2f\u7279\u7684\u731c\u6d4b\u662f\u4e0d\u80fd\u3002<\/span><\/o:p><\/section>\n

\u6211\u4eec\u77e5\u90535\u6b21\u4ee5\u4e0a\u7684\u65b9\u7a0b\u662f\u6ca1\u6709\u6c42\u6839\u516c\u5f0f\u7684\u3002\u4f46\u4e00\u51435\u6b21\u548c6\u6b21\u65b9\u7a0b\u53ef\u4ee5\u5206\u522b\u53d8\u6362\u4e3a\uff1a<\/span><\/o:p><\/p>\n

x<\/span>5<\/span><\/sup> + ax +<\/span> 1<\/span> =<\/span> 0,<\/span><\/span><\/o:p><\/span><\/section>\n
x<\/span>6<\/span><\/sup> + ax<\/span>2<\/span><\/sup> + bx +1<\/span> =<\/span> 0<\/span><\/span><\/o:p><\/span><\/section>\n

\u7231\u5c14\u5170\u6570\u5b66\u5bb6\u54c8\u5bc6\u5c14\u987f1836\u5e74\u8bc1\u660e7\u6b21\u65b9\u7a0b\u53ef\u4ee5\u901a\u8fc7\u53d8\u6362\u7b80\u5316\u4e3a\uff1a<\/span><\/o:p><\/p>\n

x<\/span>7<\/span><\/sup>+ax<\/span>3<\/span><\/sup>+bx<\/span>2<\/span><\/sup>+cx+1=0<\/span><\/span><\/section>\n
\u89e3\u8868\u793a\u4e3a\u7cfb\u6570a\uff0cb\uff0cc\u7684\u51fd\u6570\uff0c\u5373x=f(a, b, c)\u3002\u5e0c\u5c14\u4f2f\u7279\u7b2c13\u95ee\u9898\u5c31\u662f\u95ee\u8fd9\u4e2a\u4e09\u5143\u51fd\u6570\u662f\u5426\u53ef\u4ee5\u8868\u793a\u4e3a\u4e8c\u5143\u51fd\u6570\u7684\u7ec4\u5408\u3002<\/span><\/o:p><\/section>\n

\u5728\u7f57\u68ee\u5e03\u62c9\u7279\u505a\u611f\u77e5\u673a<\/span>\uff08Perceptron\uff09<\/span>\u7684\u540c\u65f6\uff0c\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u548c\u963f\u8bfa\u5fb7\u6b63\u5728\u7814\u7a76\u201c\u53e0\u52a0\u201d\u95ee\u9898\u3002\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b1956\u5e74\u9996\u5148\u8bc1\u660e\u4efb\u610f\u591a\u5143\u51fd\u6570\u53ef\u7528\u6709\u9650\u4e2a\u4e09\u5143\u51fd\u6570\u53e0\u52a0\u6784\u6210\u3002\u963f\u8bfa\u5fb7\u5728\u6b64\u57fa\u7840\u4e0a\u8bc1\u660e\u4e24\u5143\u8db3\u77e3\u3002\u4ed6\u4eec\u7684\u6210\u679c\u88ab\u79f0\u4e3a\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b-\u963f\u8bfa\u5fb7\u8868\u793a\u5b9a\u7406\uff0c\u6216\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u53e0\u52a0\u5b9a\u7406\uff0c\u6709\u65f6\u4e5f\u88ab\u79f0\u4e3a\u963f\u8bfa\u5fb7-\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u53e0\u52a0<\/span>\uff08AK\u53e0\u52a0\uff09<\/span>\uff0c\u56e0\u4e3a\u662f\u963f\u8bfa\u5fb7\u5b8c\u6210\u4e86\u6700\u540e\u7684\u4e34\u95e8\u4e00\u811a\u3002\u672c\u6587\u4e2d\u4ee5\u4e0b\u7edf\u79f0KA\u53e0\u52a0\u5b9a\u7406\u6216KA\u8868\u793a\u5b9a\u7406\u3002\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u7684\u672c\u610f\u4e0d\u5b8c\u5168\u662f\u4e3a\u4e86\u89e3\u51b3\u5e0c\u5c14\u4f2f\u7279\u7b2c13\u95ee\u9898\uff0c\u4f46\u53e0\u52a0\u5b9a\u7406\u4e8b\u5b9e\u4e0a\u6784\u6210\u4e86\u5bf9\u5e0c\u5c14\u4f2f\u7279\u5bf9\u7b2c13\u95ee\u9898\u539f\u6765\u731c\u6d4b\u7684<\/span>\uff08\u57fa\u672c\uff09<\/span>\u5426\u5b9a\u3002\u518d\u540e\u963f\u8bfa\u5fb7\u548c\u65e5\u672c\u6570\u5b66\u5bb6\u5fd7\u6751\u4e94\u90ce\u5408\u4f5c\u5728\u8fd9\u4e2a\u95ee\u9898\u4e0a\u8fdb\u4e00\u6b65\u63a8\u8fdb\u3002<\/span><\/p>\n

<\/section>\n
1973\u5e74\uff0c\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u5728\u51c6\u5907\u81ea\u5df1\u7684\u62a5\u544a\u3002\u56fe\u6e90\uff1a\u7ef4\u57fa\u767e\u79d1<\/span><\/section>\n

\u81f3\u4e8e\u662f\u5426\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u548c\u963f\u8bfa\u5fb7\u201c\u53e0\u52a0\u201d\u662f\u5e0c\u5c14\u4f2f\u7279\u7b2c13\u95ee\u9898\u7684\u5f7b\u5e95\u89e3\u51b3\uff0c\u6570\u5b66\u754c\u6709\u4e0d\u540c\u770b\u6cd5\uff0c\u6709\u4e9b\u6570\u5b66\u5bb6\u4eec\u8ba4\u4e3a\u5e0c\u5c14\u4f2f\u7279\u539f\u6765\u8bf4\u7684\u662f\u4ee3\u6570\u51fd\u6570\uff0c\u800c\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u548c\u963f\u8bfa\u5fb7\u8bc1\u660e\u7684\u662f\u8fde\u7eed\u51fd\u6570\u3002\u5e0c\u5c14\u4f2f\u7279\u7684\u539f\u8bdd\u662f\u201c\u8fde\u7eed\u51fd\u6570\u201d\uff0c\u4f46\u8003\u8651\u5230\u9ece\u66fc-\u514b\u83b1\u56e0-\u5e0c\u5c14\u4f2f\u7279\u7684\u4f20\u7edf\uff0c\u6570\u5b66\u5bb6\u4eec\u8ba4\u4e3a\u5e0c\u5c14\u4f2f\u7279\u7684\u672c\u610f\u662f\u4ee3\u6570\u51fd\u6570\u3002\u5e0c\u5c14\u4f2f\u7279\u7b2c13\u95ee\u9898\u7684\u7814\u7a76\u5e76\u6ca1\u6709\u56e0\u4e3aKA\u53e0\u52a0\u5b9a\u7406\u5b8c\u7ed3\uff0c\u800c\u662f\u8fd8\u5728\u7ee7\u7eed\uff0c\u8fd9\u8d85\u51fa\u4e86\u672c\u6587\u8303\u56f4\u3002\u65e0\u8bba\u5982\u4f55\uff0c\u53e0\u52a0\u5b9a\u7406\uff0c\u6b6a\u6253\u6b63\u7740\uff0c\u4e3a\u540e\u7eed\u795e\u7ecf\u7f51\u7edc\u7814\u7a76\u5960\u5b9a\u4e86\u7406\u8bba\u57fa\u7840\u3002<\/strong><\/span><\/p>\n

<\/section>\n
\u963f\u8bfa\u5fb7\uff08Vladimir Arnold\uff0c1937-2010\uff09\u3002\u56fe\u6e90\uff1a\u7ef4\u57fa\u767e\u79d1<\/span><\/section>\n
\u628aKA\u53e0\u52a0\u5b9a\u7406\u548c\u795e\u7ecf\u7f51\u7edc\u8054\u7cfb\u8d77\u6765\u7684\u662f\u4e00\u4f4d\u6570\u5b66\u5bb6\u51fa\u8eab\u7684\u4f01\u4e1a\u5bb6\u3002\u8d6b\u514b-\u5c3c\u5c14\u68ee<\/span>\uff08Robert Hecht-Nielsen\uff0c1947-2019\uff09<\/span>\u4e00\u76f4\u662f\u795e\u7ecf\u7f51\u7edc\u7684\u575a\u4fe1\u8005\u3002\u4ed61986\u5e74\u521b\u529e\u4e86\u4ee5\u81ea\u5df1\u540d\u5b57\u547d\u540d\u7684\u8f6f\u4ef6\u516c\u53f8HNC\uff0c\u4e13\u4e8b\u4fe1\u7528\u5361\u53cd\u6b3a\u8bc8\u3002\u4ed6\u628a\u516c\u53f8\u7684\u6838\u5fc3\u6280\u672f\u8d4c\u5728\u795e\u7ecf\u7f51\u7edc\uff0c\u5f53\u65f6\u6b63\u662f\u795e\u7ecf\u7f51\u7edc\u7684\u4f4e\u6f6e\u671f\uff0c\u9700\u8981\u6781\u5927\u7684\u52c7\u6c14\u548c\u8fdc\u89c1\u30022002\u5e74\u516c\u53f8\u88ab\u6700\u5927\u7684\u4fe1\u7528\u8bc4\u7ea7\u516c\u53f8Fair Issac\u4ee58.1\u4ebf\u7f8e\u91d1\u6536\u8d2d\u30021987\u5e74\uff0c\u8d6b\u514b-\u5c3c\u5c14\u68ee\u5728\u7b2c\u4e00\u5c4a\u795e\u7ecf\u7f51\u7edc\u5927\u4f1a<\/span>\uff08ICNN\uff0c\u540e\u6539\u540d\u4e3a\u795e\u7ecf\u7f51\u7edc\u8054\u5408\u5927\u4f1aIJCNN\uff09<\/span>\u4e0a\u53d1\u8868\u6587\u7ae0\uff0c\u8bc1\u660e\u53ef\u4ee5\u7528\u4e09\u5c42\u795e\u7ecf\u7f51\u7edc\u5b9e\u73b0\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b-\u963f\u8bfa\u5fb7\u53e0\u52a0\u3002\u8fd9\u7bc7\u6587\u7ae0\u5f88\u77ed\uff0c\u53ea\u6709\u4e09\u9875\u7eb8\uff0c\u4f46\u5374\u4ee4\u4eba\u8111\u6d1e\u5927\u5f00\u3002\u8fd9\u4e2a\u7ed3\u679c\u5728\u7406\u8bba\u4e0a\u7ed9\u795e\u7ecf\u7f51\u7edc\u7814\u7a76\u5e26\u6765\u4e86\u6170\u85c9\uff0c\u5e76\u6fc0\u53d1\u4e86\u4e00\u7cfb\u5217\u6709\u8da3\u7684\u6570\u5b66\u548c\u7406\u8bba\u8ba1\u7b97\u673a\u79d1\u5b66\u7684\u7814\u7a76\u3002<\/span><\/o:p><\/section>\n

KA\u53e0\u52a0\u5b9a\u7406\u5982\u4e0b\uff1a<\/span><\/o:p><\/p>\n

<\/section>\n

\u5373\uff0c\u4efb\u610f\u591a\u5143\u7684\u8fde\u7eed\u51fd\u6570\u90fd\u53ef\u4ee5\u8868\u793a\u4e3a\u82e5\u5e72\u4e00\u5143\u51fd\u6570\u548c\u52a0\u6cd5\u7684\u53e0\u52a0\u3002\u52a0\u6cd5\u662f\u552f\u4e00\u7684\u4e8c\u5143\u51fd\u6570\u3002\u7b80\u5355\u7684\u51fd\u6570\u90fd\u53ef\u4ee5\u901a\u8fc7\u52a0\u6cd5\u548c\u4e00\u5143\u51fd\u6570\u53e0\u52a0\u800c\u6210\uff0c\u8fd9\u4e2a\u9053\u7406\u5e76\u4e0d\u96be\u7406\u89e3\uff0c\u5982\u4e0b\u6240\u793a\uff0c\u51cf\u3001\u4e58\u3001\u9664\u53ef\u7531\u52a0\u6cd5\u53e0\u52a0\u800c\u6210\uff1a<\/span><\/o:p><\/p>\n

a \u2212 b = a + (\u2212b),<\/o:p><\/span><\/section>\n
a \u00b7 b = 1 \/4 ((a + b)<\/span> 2<\/span><\/sup> \u2212 (a \u2212 b) <\/span>2<\/span><\/sup>) <\/span>\uff0c<\/o:p><\/span><\/span><\/section>\n
a \/b = a \u00b7 1\/ b<\/span><\/section>\n
\u6240\u6709\u7684\u521d\u7b49\u8fd0\u7b97\u90fd\u53ef\u4ee5\u901a\u8fc7\u4e00\u5143\u8fd0\u7b97\u548c\u52a0\u6cd5\u5b8c\u6210\u3002\u5728\u8fd9\u4e2a\u610f\u4e49\u4e0a\uff0c\u52a0\u6cd5\u662f\u901a\u7528\u7684<\/span>\uff08universal\uff09<\/span>\uff0c\u7528\u52a0\u6cd5\u53e0\u52a0\u505a\u5176\u4ed6\u8fd0\u7b97\u65f6\u5e76\u4e0d\u9700\u589e\u52a0\u989d\u5916\u7684\u7ef4\u5ea6\u3002<\/span><\/section>\n

\u8d6b\u514b-\u5c3c\u5c14\u68ee\u6307\u51fa\uff0cKA\u53e0\u52a0\u5b9a\u7406\u53ef\u4ee5\u901a\u8fc7\u4e24\u5c42\u7f51\u7edc\u5b9e\u73b0\uff0c\u6bcf\u5c42\u5b9e\u73b0\u53e0\u52a0\u4e2d\u7684\u4e00\u4e2a\u52a0\u53f7\u3002\u4ed6\u5e72\u8106\u5c31\u628a\u8fd9\u4e2a\u5b9e\u73b0\u7f51\u7edc\u79f0\u4e3a\u201c\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u7f51\u7edc\u201d\u3002\u6cd5\u56fd\u6570\u5b66\u5bb6\u5361\u6c49<\/span>\uff08Jean-Pierre Kahane\uff0c1926-2017<\/span>\uff09\uff0c\u57281975\u5e74\u6539\u8fdb\u4e86KA\u53e0\u52a0\u5b9a\u7406\uff0c\u5982\u4e0b\uff1a<\/span><\/p>\n

<\/section>\n

\u5176\u4e2d\uff0ch\u88ab\u8fdb\u4e00\u6b65\u9650\u5236\u4e3a\u4e25\u683c\u5355\u8c03\u51fd\u6570\uff0clp\u662f\u5c0f\u4e8e1\u7684\u6b63\u5e38\u91cf\u3002<\/span><\/o:p><\/p>\n

<\/section>\n
<\/section>\n
<\/section>\n
\u8d6b\u514b-\u5c3c\u5c14\u68ee\uff08Robert Hecht-Nielsen\uff0c1947-2019\uff09\u548c1987\u5e74\u8bba\u6587\u53ca\u63d2\u56fe<\/span><\/span><\/section>\n
\u53e6\u4e00\u4f4d\u4e5f\u662f\u6570\u5b66\u51fa\u8eab\u7684\u5de5\u5b66\u6559\u6388\u8d5b\u672c\u79d1<\/span>\uff08George Cybenko\uff09<\/span>\uff0c\u7a0d\u540e\u57281988\u5e74\u8bc1\u660e\u4e86\u6709\u4e24\u4e2a\u9690\u5c42\u4e14\u5177sigmoid\u6fc0\u6d3b\u51fd\u6570\u7684\u795e\u7ecf\u7f51\u7edc\u53ef\u4ee5\u903c\u8fd1\u4efb\u610f\u8fde\u7eed\u51fd\u6570\u3002\u8d5b\u672c\u79d1\u7684\u6587\u7ae0\u66f4\u5177\u7ec6\u8282\u548c\u8bc1\u660e\u3002\u867d\u7136\u8d5b\u672c\u79d1\u6ca1\u6709\u5f15\u7528Hecht-Nielsen-1986\uff0c\u4f46\u5f15\u7528\u4e86Kolmogorov-1957\u3002\u6211\u4eec\u4e0d\u77e5\u9053\u4ed6\u662f\u5426\u53d7\u5230\u4e86\u8d6b\u514b-\u5c3c\u5c14\u68ee\u7684\u542f\u53d1\u3002\u800c\u51e0\u4e4e\u540c\u65f6\u7684\u72ec\u7acb\u5de5\u4f5c\u5982Hornik-1989\uff0c\u90fd\u662f\u65e2\u5f15\u7528Kolmogorov\u4e5f\u5f15\u7528Hecht-Nielsen\uff0c\u53ef\u89c1\u8d6b\u514b-\u5c3c\u5c14\u68ee\u5bf9\u63ed\u793a\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b-\u963f\u8bfa\u5fb7\u53e0\u52a0\u7684\u8ba1\u7b97\u673a\u79d1\u5b66\u6db5\u4e49\u662f\u5148\u77e5\u5148\u89c9\u3002\u8fd9\u4e9b\u76f8\u5173\u7684\u7ed3\u8bba\u53ca\u5404\u79cd\u53d8\u4f53\u548c\u63a8\u5e7f\u88ab\u7edf\u79f0\u4e3a\u201c\u901a\u7528\u903c\u8fd1\u5b9a\u7406\u201d<\/span>\uff08Universal Approximation Theorem\uff09<\/span>\u3002\u9664\u4e86\u8fde\u7eed\u51fd\u6570\uff0c\u4e5f\u6709\u4eba\u529b\u56fe\u8bc1\u660e\u975e\u8fde\u7eed\u51fd\u6570\u4e5f\u53ef\u4ee5\u7528\u4e09\u5c42\u795e\u7ecf\u7f51\u7edc\u903c\u8fd1<\/span>(Ismailov,2022)<\/span>\u3002\u65e0\u8bba\u5982\u4f55\uff0c\u5b66\u8fc7\u6570\u5b66\u7684\u603b\u80fd\u4ece\u7b2c\u4e00\u6027\u539f\u7406\u51fa\u53d1\u6765\u8003\u8651\u95ee\u9898\u3002\u6b64\u540e\uff0c\u660e\u65af\u57fa\u5bfc\u81f4\u7684\u795e\u7ecf\u7f51\u7edc\u5371\u673a\u7b97\u662f\u7ffb\u7bc7\u4e86\u3002<\/span><\/o:p><\/section>\n
<\/section>\n
George Cybenko\u3002\u56fe\u6e90\uff1ahttps:\/\/engineering.dartmouth.edu\/<\/span><\/section>\n
\u8d6b\u514b-\u5c3c\u5c14\u68ee\u7b49\u7684\u8bc1\u660e\u662f\u5b58\u5728\u6027\u7684\u800c\u4e0d\u662f\u6784\u9020\u6027\u7684\u3002\u8ba1\u7b97\u673a\u79d1\u5b66\u65e2\u662f\u79d1\u5b66\u4e5f\u662f\u5de5\u7a0b\uff0c\u4e00\u4e2a\u601d\u8def\u662f\u4e0d\u662f\u53ef\u884c\uff0c\u5fc5\u987b\u5f97\u4f34\u4e4b\u4ee5\u7b97\u6cd5\u548c\u5b9e\u73b0\uff0c\u5e76\u4e14\u8981\u6709\u53ef\u9760\u7684\u590d\u6742\u6027\u5206\u6790\u548c\u5de5\u7a0b\u6f14\u793a\u3002<\/span><\/o:p><\/section>\n
1974\u5e74\uff0c\u54c8\u4f5b\u5927\u5b66\u7684\u4e00\u7bc7\u7edf\u8ba1\u5b66\u535a\u58eb\u8bba\u6587\u8bc1\u660e\u4e86\u795e\u7ecf\u7f51\u7edc\u7684\u5c42\u6570\u589e\u52a0\u5230\u4e09\u5c42\uff0c\u5e76\u4e14\u5229\u7528\u201c\u53cd\u5411\u4f20\u64ad\u201d<\/span>\uff08back-propagation\uff09<\/span>\u5b66\u4e60\u65b9\u6cd5\uff0c\u5c31\u53ef\u4ee5\u89e3\u51b3XOR\u95ee\u9898\u3002\u8fd9\u7bc7\u8bba\u6587\u7684\u4f5c\u8005\u6c83\u6ce2\u65af<\/span>\uff08Paul Werbos\uff09<\/span>\uff0c\u540e\u6765\u5f97\u4e86IEEE\u795e\u7ecf\u7f51\u7edc\u5b66\u4f1a\u7684\u5148\u9a71\u5956\u3002\u6c83\u6ce2\u65af\u8fd9\u7bc7\u6587\u7ae0\u521a\u53d1\u8868\u65f6\uff0c\u5e76\u6ca1\u6709\u5728\u795e\u7ecf\u7f51\u7edc\u5708\u5b50\u91cc\u5f15\u8d77\u591a\u5c11\u91cd\u89c6\uff0c\u4e3b\u8981\u539f\u56e0\u662f\u6c83\u6ce2\u65af\u7684\u5bfc\u5e08\u662f\u793e\u4f1a\u5b66\u548c\u653f\u6cbb\u5b66\u9886\u57df\u7684\uff0c\u4ed6\u60f3\u89e3\u51b3\u7684\u95ee\u9898\u4e5f\u662f\u7edf\u8ba1\u5b66\u548c\u793e\u4f1a\u79d1\u5b66\u7684\u3002\u628a\u201c\u53cd\u5411\u4f20\u64ad\u201d\u653e\u5230\u591a\u5c42\u795e\u7ecf\u7f51\u7edc\u4e0a\u5c31\u6210\u4e86\u201c\u6df1\u5ea6\u5b66\u4e60\u201d\u3002<\/span><\/o:p><\/section>\n
\u6df1\u5ea6\u5b66\u4e60\u9996\u5148\u5728\u8bed\u97f3\u548c\u56fe\u50cf\u8bc6\u522b\u4e0a\u53d6\u5f97\u7a81\u7834\uff0c\u540e\u6765\u5728\u5f3a\u5316\u5b66\u4e60\u7684\u5e2e\u52a9\u4e0b\uff0c\u53c8\u5728\u535a\u5f08\u548c\u81ea\u7136\u8bed\u8a00\u5904\u7406\u65b9\u9762\u5f97\u5230\u60ca\u4eba\u7684\u6210\u529f\u3002\u4f46\u6df1\u5ea6\u5b66\u4e60\u7684\u673a\u5236\u4e00\u76f4\u6ca1\u6709\u5f97\u5230\u6ee1\u610f\u7684\u89e3\u91ca\u3002<\/span><\/o:p><\/section>\n
Tomaso Poggio\u662f\u82f1\u5e74\u65e9\u901d\u7684\u8ba1\u7b97\u673a\u89c6\u89c9\u7814\u7a76\u8005\u9a6c\u5c14<\/span>\uff08David Marr\uff09<\/span>\u7684\u5408\u4f5c\u8005\u3002\u9a6c\u5c14\u57281980\u5e74\u53bb\u4e16\u540e\uff0cPoggio\u7684\u5174\u8da3\u9010\u6e10\u8f6c\u5411\u673a\u5668\u5b66\u4e60\u548c\u795e\u7ecf\u7f51\u7edc\u7684\u7406\u8bba\u3002\u8bb8\u591a\u6570\u5b66\u5bb6\u4e5f\u88ab\u62c9\u5165\uff0c\u4f8b\u5982\u83f2\u5c14\u5179\u5956\u5f97\u4e3b\u3001\u957f\u5bff\u591a\u4ea7\u4e14\u5174\u8da3\u5e7f\u6cdb\u7684\u6570\u5b66\u5bb6\u65af\u6885\u5c14<\/span>\uff08Stephen Smale\uff09<\/span>\u3002\u4e0d\u8fc7\uff0c Poggio\u5e76\u4e0d\u8ba4\u53efKA\u53e0\u52a0\u5b9a\u7406\u53ef\u4ee5\u4f5c\u4e3a\u901a\u7528\u903c\u8fd1\u7f51\u7edc\u7684\u6570\u5b66\u57fa\u7840\u3002\u4ed6\u65e9\u57281989\u5e74\u5c31\u8457\u6587\u6307\u51fa\u5b9e\u73b0KA\u53e0\u52a0\u7684\u7f51\u7edc\u662f\u4e0d\u5e73\u6ed1\u7684<\/span>\uff08\u89c1Girosi & Poggio, 1989\uff09<\/span>\u3002\u5728\u540e\u7eed\u7684\u5de5\u4f5c\u4e2d\uff0c\u4ed6\u8fd8\u6307\u51fa\u7ef4\u5ea6\u707e\u96be<\/span>\uff08CoD\uff09<\/span>\u53d6\u51b3\u4e8e\u7ef4\u5ea6\/\u5e73\u6ed1\u5ea6\u3002\u5e73\u6ed1\u5ea6\u4e0d\u597d\uff0c\u81ea\u7136\u4f1a\u5bfc\u81f4\u7ef4\u5ea6\u707e\u96be\u3002KA\u53e0\u52a0\u5b9a\u7406\u867d\u6709\u7406\u8bba\u4ef7\u503c\uff0c\u4f46\u76f4\u63a5\u5e94\u7528\u5230\u5b9e\u9645\u7684\u795e\u7ecf\u7f51\u7edc\u4e0a\uff0c\u4f1a\u78b0\u5230\u5e73\u6ed1\u6027\u95ee\u9898\uff0c\u4e8e\u662f\u5de5\u7a0b\u4e0a\u9c9c\u6709\u8fdb\u5c55\u3002KA\u53e0\u52a0\u5b9a\u7406\u4ece\u6b64\u88ab\u9057\u5fd8\u3002<\/span><\/o:p><\/section>\n
\u5230\u4e86\u4eca\u5e745\u6708\uff0cKA\u53e0\u52a0\u5b9a\u7406\u53c8\u91cd\u65b0\u5f97\u5230\u91cd\u89c6\u3002\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7684\u7269\u7406\u5b66\u5bb6\u548c\u79d1\u666e\u4f5c\u5bb6Max Tegmark\uff0c\u5728\u4e3b\u4e1a\u5b87\u5b99\u5b66\u4e4b\u5916\u4e00\u76f4\u559c\u6b22\u8ddf\u8e2a\u4eba\u5de5\u667a\u80fd\u5c24\u5176\u662f\u673a\u5668\u5b66\u4e60\u3002\u4ed6\u4f5c\u4e3a\u901a\u8baf\u4f5c\u8005\u3001\u5218\u5b50\u9e23\u4f5c\u4e3a\u7b2c\u4e00\u4f5c\u8005\u7684\u6700\u65b0\u6587\u7ae0\u5c31\u529b\u56fe\u590d\u6d3bKA\u53e0\u52a0\u5b9a\u7406\u3002\u4ed6\u4eec\u8ba4\u4e3a\uff0c\u539f\u59cb\u7684\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u7f51\u7edc\u53ea\u6709\u4e24\u5c42\uff0c\u5982\u679c\u628a\u5c42\u6570\u52a0\u6df1\uff0c\u628a\u5bbd\u5ea6\u62d3\u5e7f\uff0c\u4e5f\u8bb8\u53ef\u4ee5\u514b\u670d\u5e73\u6ed1\u6027\u7684\u95ee\u9898\u3002\u4ed6\u4eec\u628a\u63a8\u5e7f\u7684\u7f51\u7edc\u79f0\u4e3aKAN<\/span>\uff08Kolmogorov-Arnold Network\uff09<\/span>\uff0c\u800c\u4e8b\u5b9e\u4e0a\uff0c\u8d6b\u514b-\u5c3c\u5c14\u68ee\u65e9\u5c31\u628a\u5b9e\u73b0KA\u53e0\u52a0\u5b9a\u7406\u7684\u7f51\u7edc\u79f0\u4e3aKolmogorov Network\u3002<\/span><\/o:p><\/section>\n
<\/section>\n
\u7269\u7406\u5b66\u5bb6Max Tegmark\u3002\u56fe\u6e90\uff1a\u7ef4\u57fa\u767e\u79d1<\/span><\/section>\n
\u5b9e\u73b0\u53e0\u52a0\u5b9a\u7406\u7684\u7f51\u7edc\u90fd\u53ef\u56fe\u793a\u5982\u4e0b\uff1a<\/span><\/section>\n
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KAN\u7684\u4f5c\u8005\u4eec\u81ea\u79f0\u7684\u6700\u5173\u952e\u7684\u521b\u65b0\u70b9\u4e4b\u4e00\uff0c\u662f\u628a\u8ba1\u7b97\u653e\u5728\u7f51\u7edc\u7684\u8fb9<\/span>\uff08edge\uff09<\/span>\u4e0a\u800c\u4e0d\u662f\u70b9<\/span>\uff08node\uff09<\/span>\u4e0a\uff0c\u70b9\u4e0a\u53ea\u6267\u884c\u52a0\u6cd5\u8fd0\u7b97\uff0c\u8fd9\u6837\u53ef\u4ee5\u7f29\u5c0f\u7f51\u7edc\u7684\u89c4\u6a21\u3002\u56e0\u4e3a\u53e0\u52a0\u5b9a\u7406\u53ea\u9700\u8981\u52a0\u6cd5<\/span>\uff08\u4e8c\u5143\u7684\uff09<\/span>\u548c\u4e00\u5143\u51fd\u6570\uff0c\u5b9e\u73b0\u4e00\u5143\u51fd\u6570\u7684\u70b9\u53ef\u4ee5\u574d\u584c\u6210\u4e3a\u8fb9\uff0c\u8fb9\u5c31\u662f\u70b9\uff0c\u70b9\u5c31\u662f\u8fb9\uff0c\u8fd9\u4e00\u70b9\u5e76\u4e0d\u4ee4\u4eba\u60ca\u5947\u3002\u5982\u4e0a\u56fe\uff0c\u70b9h\u53ef\u4ee5\u770b\u6210\u662f\u8fde\u63a5\u8f93\u5165x\u548cg\u4e4b\u524d\u52a0\u6cd5\u7684\u8fb9\u3002KAN\u4e2d\u7684\u52a0\u6cd5\u53ea\u80fd\u5728\u70b9\u4e0a\u5b9e\u73b0\uff0c\u56e0\u4e3a\u52a0\u6cd5\u662f\u4e8c\u5143\u7684\uff0c\u9700\u8981\u4e24\u4e2a\u8f93\u5165\u3002\u5728\u8fd9\u4e2a\u610f\u4e49\u4e0a\uff0cKAN\u548c\u591a\u5c42\u611f\u77e5\u5668<\/span> (MLP) <\/span>\u6ca1\u6709\u672c\u8d28\u533a\u522b\u3002\u6b63\u5e38\u7684\u7f51\u7edc\u4e2d\uff0c\u8fb9\u6570\u662f\u70b9\u6570\u7684\u5e73\u65b9\u7ea7\uff0c\u7f51\u7edc\u89c4\u6a21\u7684\u7f29\u5c0f\u662f\u5426\u771f\u6b63\u5e26\u6765\u8ba1\u7b97\u91cf\u7684\u51cf\u5c11\uff0c\u4ecd\u9700\u7406\u8bba\u548c\u5b9e\u9a8c\u7684\u5171\u540c\u63a2\u7d22\u3002<\/span><\/section>\n
KAN\u7684\u53e6\u4e00\u521b\u65b0\u4e4b\u4e00\u662f\u5b66\u4e60\u6fc0\u6d3b\u51fd\u6570\u800c\u4e0d\u662f\u5b66\u4e60\u6743\u91cd\u3002\u4f46\u5b66\u4e60\u51fd\u6570\u6bd4\u5b66\u4e60\u6743\u91cd\u8981\u56f0\u96be\u5f97\u591a\u3002KAN\u4e2d\uff0c\u6fc0\u6d3b\u51fd\u6570\u662f\u7528B-spline\u6765\u62df\u5408\u7684\uff0cB-spline\u7684\u6240\u6709\u52a3\u52bf\u81ea\u7136\u4e5f\u4f1a\u88ab\u5e26\u8fdbKAN\u4e2d\u3002\u5982\u679c\u5b66\u4e60\u6fc0\u6d3b\u51fd\u6570\u7684\u6210\u672c\u5f88\u9ad8\uff0c\u7f51\u7edc\u5c31\u4e27\u5931\u4e86\u901a\u7528\u6027<\/span>\uff08universality\uff09<\/span>\u3002\u76ee\u524d\u9700\u8981\u5728\u5de5\u7a0b\u5b9e\u8df5\u4e2d\u8fdb\u4e00\u6b65\u8bc1\u660eKAN\u7684\u4f18\u52bf\u3002 \u4f46\u65e0\u8bba\u65e0\u4f55\uff0cKAN\u7684\u5de5\u4f5c\u9f13\u52b1\u6211\u4eec\u4ece\u7b2c\u4e00\u6027\u539f\u7406\u51fa\u53d1\u53bb\u5bfb\u627e\u65b0\u7684\u8def\u5f84\uff0c\u4e3a\u89e3\u91ca\u63d0\u4f9b\u6570\u5b66\u4f9d\u636e\u3002<\/strong><\/span><\/section>\n

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\u4e00\u6bb5\u5386\u53f2\u548c\u4e00\u4e2a\u601d\u60f3\u5b9e\u9a8c\uff1a\u7b2c\u8c37\u673a\u5668<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u542f\u8499\u8fd0\u52a8\u540e\u7684\u601d\u6f6e\u5728\u7406\u6027\u4e3b\u4e49\u548c\u7ecf\u9a8c\u4e3b\u4e49\u4e4b\u95f4\u4e00\u62e8\u4e00\u62e8\u5730\u8fed\u4ee3\u6f14\u8fdb\u3002\u79d1\u5b66\u4ea6\u5982\u6b64\u3002\u54e5\u767d\u5c3c\u6beb\u65e0\u7591\u95ee\u662f\u7406\u6027\u4e3b\u4e49\u8005\uff0c\u4ed6\u7684\u5de5\u5177\u662f\u6570\u5b66\u3002\u4e4b\u540e\u7684\u5f00\u666e\u52d2\u4e5f\u662f\u7406\u6027\u4e3b\u4e49\u8005\u3002\u4f46\u4ed6\u4eec\u4e4b\u95f4\u7684\u7b2c\u8c37\u5e03\u62c9\u8d6b\u5374\u662f\u5f7b\u5934\u5f7b\u5c3e\u7684\u7ecf\u9a8c\u4e3b\u4e49\u8005\uff0c\u4ed6\u7684\u5de5\u5177\u662f\u81ea\u5236\u7684\u5404\u79cd\u6d4b\u91cf\u4eea\u5668\uff0c\u4ed6\u90a3\u65f6\u8fd8\u6ca1\u6709\u671b\u8fdc\u955c\u3002\u5176\u8eab\u540e\u7559\u4e0b24\u672c\u89c2\u6d4b\u6570\u636e\uff0c\u5176\u4e2d\u5305\u542b777\u9897\u6052\u661f\u7684\u661f\u8868\u3002\u8fd9\u6279\u6570\u636e\u88ab\u5f00\u666e\u52d2\u7ee7\u627f\u3002\u6fd2\u6b7b\u7684\u8d35\u65cf\u7b2c\u8c37\u5e03\u62c9\u8d6b\u5df2\u7ecf\u548c\u5c0f\u809a\u9e21\u80a0\u7684\u5f00\u666e\u52d2\u51e0\u4e4e\u95f9\u7ffb\u4e86\u3002\u8fd9\u6bb5\u5386\u53f2\u7684\u7b80\u5316\u7248\u5c31\u662f\uff1a\u54e5\u767d\u5c3c\u63a8\u7ffb\u4e86\u6258\u52d2\u5bc6\u7684\u5730\u5fc3\u8bf4\uff0c\u4ee3\u4e4b\u4ee5\u66f4\u7b80\u5355\u7684\u65e5\u5fc3\u8bf4\u3002\u4f46\u7b2c\u8c37\u5e03\u62c9\u8d6b\u5e76\u6ca1\u6709\u8d70\u5f97\u90a3\u4e48\u8fdc\uff0c\u6309\u7167\u5251\u6865\u79d1\u5b66\u53f2\u5bb6\u970d\u65af\u91d1\u7684\u8bf4\u6cd5\uff1a\u7b2c\u8c37\u5728\u54e5\u767d\u5c3c\u521b\u65b0\u7684\u5730\u65b9\u4fdd\u5b88\uff0c\u800c\u5728\u54e5\u767d\u5c3c\u4fdd\u5b88\u7684\u5730\u65b9\u521b\u65b0\u3002\u4ed6\u63d0\u51fa\u4e86\u51c6\u65e5\u5fc3\u8bf4\uff1a\u6240\u6709\u884c\u661f\u90fd\u7ed5\u7740\u592a\u9633\u8f6c\uff0c\u4f46\u592a\u9633\u5e26\u7740\u884c\u661f\u4eec\u56f4\u7740\u5730\u7403\u8f6c\u3002\u5f00\u666e\u52d2\u4fee\u6b63\u4e86\u7b2c\u8c37\u7684\u5de5\u4f5c\uff0c\u63d0\u51fa\u4e86\u884c\u661f\u8fd0\u52a8\u4e09\u5b9a\u5f8b\uff0c\u5de5\u5177\u5c31\u662f\u6570\u5b66\u3002\u5f00\u666e\u52d2\u7684\u5de5\u4f5c\u5f97\u5230\u4f3d\u5229\u7565\u7684\u80af\u5b9a\u3002\u4f3d\u5229\u7565\u53c8\u662f\u4e00\u4e2a\u7ecf\u9a8c\u4e3b\u4e49\u8005\uff0c\u4ed6\u6709\u4e86\u66f4\u65b0\u7684\u5de5\u5177\uff1a\u671b\u8fdc\u955c\u30021609\u5e74\uff0c\u5f00\u666e\u52d2\u5728\u300a\u65b0\u5929\u6587\u5b66\u300b\u4e2d\u9996\u5148\u53d1\u8868\u5934\u4e24\u6761\u5b9a\u5f8b\u7684\u65f6\u5019\uff0c\u4f3d\u5229\u7565\u521a\u9020\u51fa3\u500d\u7684\u671b\u8fdc\u955c\u3002<\/span><\/section>\n
\u7b2c\u8c37\u5e03\u62c9\u8d6b\u81ea\u5236\u7684\u89c2\u6d4b\u8bbe\u5907\uff0c\u6309\u4eca\u5929\u7684\u6807\u51c6\u770b\uff0c\u90fd\u662f\u4e9b\u7b80\u964b\u7684\u73a9\u5177\u3002\u4f46\uff0c\u6211\u4eec\u5047\u8bbe\u4ed6\u9664\u4e86\u90a3\u4e9b\u8bbe\u5907\u4e4b\u5916\uff0c\u8fd8\u6709\u4e00\u53f0\u73b0\u4ee3\u7684\u5148\u8fdb\u7684\u673a\u5668\u5b66\u4e60\u88c5\u5907\u3002\u4e3a\u4e86\u65b9\u4fbf\uff0c\u6211\u4eec\u6682\u4e14\u79f0\u4e4b\u4e3a\u7b2c\u8c37\u673a\uff0c\u8fd9\u53f0\u5b66\u4e60\u673a\u53ef\u4ee5\u5b9e\u73b0\u4eca\u5929\u7684\u6df1\u5ea6\u5b66\u4e60\u548c\u5f3a\u5316\u5b66\u4e60\u7b49\u5404\u79cd\u673a\u5668\u5b66\u4e60\u7b97\u6cd5\u3002\u4ed6\u53ef\u4ee5\u628a\u4ed6\u90a324\u5927\u672c\u8bb0\u5f55\u6570\u636e\u5582\u7ed9\u8fd9\u53f0\u7b2c\u8c37\u673a\u5bf9\u4e4b\u52a0\u4ee5\u8bad\u7ec3\uff0c\u8bad\u7ec3\u597d\u7684\u7b2c\u8c37\u673a\u5c31\u53ef\u4ee5\u8f93\u51fa\u4efb\u610f\u4e00\u4e2a\u6052\u661f\u548c\u884c\u661f\u7684\u8f68\u9053\u6570\u636e\u3002\u82e5\u5982\u6b64\uff0c\u65e5\u5fc3\u8bf4\u8fd8\u662f\u5730\u5fc3\u8bf4\u8fd8\u91cd\u8981\u5417?\u53cd\u6b63\u7b2c\u8c37\u673a\u80fd\u8f93\u51fa\u6211\u4eec\u9700\u8981\u7684\u6570\u636e\u3002\u7b2c\u8c37\u673a\u7684\u8868\u73b0\uff0c\u5f53\u65f6\u7684\u4eba\u80af\u5b9a\u96be\u4ee5\u89e3\u91ca\uff0c\u4f30\u8ba1\u4f1a\u79f0\u4e4b\u4e3a\u201c\u6d8c\u73b0\u201d\u3002\u90a3\u6211\u4eec\u53ef\u4ee5\u8bf4\u8fd9\u4e2a\u7b2c\u8c37\u673a\u5b8c\u6210\u7684\u529f\u80fd\u662f\u79d1\u5b66\u5417\uff1f\u5b83\u7684\u884c\u4e3a\u53ef\u4ee5\u89e3\u91ca\u5417\uff1f\u5fc5\u987b\u5217\u51fa\u65b9\u7a0b\u624d\u80fd\u79f0\u4e4b\u4e3a\u89e3\u91ca\u5417\uff1f\u591a\u4e48\u7b80\u5355\u7684\u65b9\u7a0b\u624d\u80fd\u79f0\u4e4b\u4e3a\u89e3\u91ca\uff1f\u8fd9\u751a\u81f3\u53ef\u4ee5\u5f71\u54cd\u6211\u4eec\u505a\u79d1\u5b66\u7684\u65b9\u5f0f\uff0c\u6309\u7167\u7b2c\u8c37\u673a\u7684\u673a\u5236\uff0c\u6211\u4eec\u53ea\u662f\u8d1f\u8d23\u4e0d\u505c\u5730\u6536\u96c6\u6570\u636e\uff0c\u7136\u540e\u5582\u7ed9\u673a\u5668\uff0c\u4e0d\u65ad\u5b8c\u5584\u9884\u6d4b\u673a\u5236\u3002\u5982\u679c\u725b\u987f\u4e4b\u524d\u5c31\u6709\u4e86\u6df1\u5ea6\u5b66\u4e60\uff0c\u90a3\u4e48\u662f\u4e0d\u662f\u5c31\u4e0d\u4f1a\u6709\u725b\u987f\u5b9a\u5f8b\u4e43\u81f3\u76f8\u5bf9\u8bba\u4e86\uff1f\u53ea\u6709\u6570\u503c\u89e3\u800c\u6ca1\u6709\u89e3\u6790\u89e3\u7684\u65b9\u7a0b\u6240\u63cf\u8ff0\u7684\u4e16\u754c\u5c31\u4e0d\u80fd\u88ab\u7406\u89e3\u5417\uff1f\u4e0d\u53d7\u9650\u5236\u7684\u4e09\u4f53\u95ee\u9898\u4e0d\u80fd\u7b97\u88ab\u7406\u89e3\u5417\uff1f <\/span><\/section>\n

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\u963f\u8bfa\u5fb7\u9664\u4e86\u5bf9\u6570\u5b66\u7684\u6280\u672f\u5185\u5bb9\u505a\u51fa\u6df1\u523b\u8d21\u732e\uff0c\u5bf9\u6570\u5b66\u54f2\u5b66\u548c\u6570\u5b66\u53f2\u4e5f\u6709\u5f88\u591a\u6709\u8da3\u7684\u8bba\u65ad\u3002\u4ed6\u66fe\u8bf4\uff1a\u6570\u5b66\u5c31\u662f\u7269\u7406\u5b66\u4e2d\u90a3\u4e9b\u5b9e\u9a8c\u6210\u672c\u5f88\u4f4e\u7684\u90e8\u5206<\/span>\uff08Mathematics is the part of physics where experiments are cheap\uff09<\/span>\u3002\u6211\u4eec\u4e5f\u53ef\u4ee5\u5957\u7528\u6b64\u8bdd\u8bf4\uff0c\u8ba1\u7b97\u79d1\u5b66\u5176\u5b9e\u624d\u662f\u7269\u7406\u5b66\u4e2d\u5b9e\u9a8c\u6210\u672c\u5f88\u4f4e\u7684\u90e8\u5206\u3002\u6211\u4eec\u53ef\u4ee5\u95ee\uff1a\u4eba\u7c7b\u80fd\u591f\u5b66\u4e60\u7684\u4e1c\u897f\u6bd4\u7406\u6027\u66f4\u591a\u3001\u66f4\u5c11\u8fd8\u662f\u4e00\u6837\uff1f\u901a\u7528\u903c\u8fd1\u5b9a\u7406\u4f3c\u4e4e\u6307\u5411\u7406\u6027\u4e0d\u4f1a\u6bd4\u53ef\u5b66\u4e60\u7684\u66f4\u591a\u3002\u6211\u4eec\u53ef\u4ee5\u66f4\u52a0\u6df1\u5165\u5730\u63a2\u8ba8KA\u53e0\u52a0\u5b9a\u7406\u7684\u54f2\u5b66\u8574\u610f\u3002<\/span><\/o:p><\/section>\n
\u963f\u8bfa\u5fb7\u5728\u63a5\u53d7\u91c7\u8bbf\u65f6\u8bf4\uff1a\u201c\u5e03\u5c14\u5df4\u57fa\u5b66\u6d3e<\/span>\uff08Bourbakists\uff09<\/span>\u58f0\u79f0\u6240\u6709\u4f1f\u5927\u7684\u6570\u5b66\u5bb6\u2014\u2014\u7528\u72c4\u5229\u514b\u96f7<\/span>\uff08Peter Gustav Lejeune Dirichlet\uff09<\/span>\u7684\u8bdd\u6765\u8bb2\u2014\u2014\u662f\u2018\u7528\u6e05\u6670\u7684\u601d\u60f3\u4ee3\u66ff\u76f2\u76ee\u7684\u8ba1\u7b97\u2019\u3002\u5e03\u5c14\u5df4\u57fa\u5ba3\u8a00\u4e2d\u7684\u8fd9\u53e5\u8bdd\uff0c\u7ffb\u8bd1\u6210\u4fc4\u8bed\u53d8\u6210\u4e86\u2018\u7528\u76f2\u76ee\u7684\u8ba1\u7b97\u4ee3\u66ff\u6e05\u6670\u7684\u601d\u60f3\u2019\uff0c\u5ba3\u8a00\u7684\u8bd1\u5ba1\u662f\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\uff0c\u4ed6\u7cbe\u901a\u6cd5\u8bed\u3002\u6211\u53d1\u73b0\u8fd9\u4e00\u9519\u8bef\u540e\u5927\u5403\u4e00\u60ca\uff0c\u5c31\u53bb\u627e\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u8ba8\u8bba\u3002\u4ed6\u7b54\u9053\uff1a\u6211\u4e0d\u89c9\u5f97\u7ffb\u8bd1\u6709\u4ec0\u4e48\u95ee\u9898\uff0c\u7ffb\u8bd1\u628a\u5e03\u5c14\u5df4\u57fa\u98ce\u683c\u63cf\u8ff0\u5f97\u6bd4\u4ed6\u4eec\u81ea\u5df1\u8bf4\u7684\u66f4\u51c6\u786e\u3002\u9057\u61be\u7684\u662f\uff0c\u5e9e\u52a0\u83b1<\/span>\uff08Henri Poincar\u00e9\uff09<\/span>\u6ca1\u5728\u6cd5\u56fd\u521b\u5efa\u4e00\u4e2a\u5b66\u6d3e\u3002\u201d\u82cf\u8054\u6570\u5b66\u5bb6\u7684\u6bd2\u820c\u548c\u5e7d\u9ed8\u522b\u5177\u4e00\u683c\u3002<\/span><\/o:p><\/section>\n
\u5f53\u4f11\u8c1f\u8bf4\u725b\u987f\u53d1\u73b0\u4e86\u7269\u7684\u5b9a\u5f8b\u65f6\uff0c\u4ed6\u672c\u610f\u662f\u60f3\u628a\u725b\u987f\u62c9\u5230\u7ecf\u9a8c\u4e3b\u4e49\u7684\u9635\u8425\u3002\u4f46\u725b\u987f\u6452\u5f03\u4e86\u673a\u68b0\u8bba\u3002\u6211\u4eec\u529b\u56fe\u8bf4\u4e18\u5947-\u56fe\u7075\u8bba\u9898\u4e3a\u5fc3\u7684\u5b9a\u5f8b<\/span>\uff08\u5b9e\u9645\u4e0a\u662f\u201c\u8bba\u9898\u201d\uff09<\/span>\u3002\u6211\u4eec\u5bf9\u4eba\u5de5\u667a\u80fd\u7684\u53ef\u89e3\u91ca\u6027\u8981\u6c42\u662f\u57fa\u4e8e\u673a\u68b0\u8bba\u7684\u3002\u53ef\u89e3\u91ca\u6027\u7684\u76ee\u7684\u662f\u5728\u6cd5\u5ead\u4e0a\u7ed9\u8001\u767e\u59d3<\/span>\uff08average person\uff09<\/span>\u89e3\u91ca\uff0c\u800c\u4e0d\u662f\u5728\u6700\u806a\u660e\u7684\u7cbe\u82f1\u5316\u4e4b\u95f4\u5f62\u6210\u5171\u8bc6\u3002\u8fc7\u53bb\uff0c\u79d1\u5b66\u7684\u53ef\u89e3\u91ca\u6027\uff0c\u662f\u79d1\u5b66\u6210\u529f\u7684\u6807\u5fd7\u3002\u8fd8\u539f\u8bba<\/span>\uff08reductionist\uff09<\/span>\uff0c\u4f5c\u4e3a\u73b0\u4ee3\u79d1\u5b66\u7684\u4f20\u7edf\uff0c\u5c31\u662f\u628a\u4e00\u4e2a\u96be\u89e3\u7684\u5927\u95ee\u9898\u8fd8\u539f\u4e3a\u66f4\u5c0f\u7684\u3001\u66f4\u539f\u59cb\u4e14\u66f4\u6613\u89e3\u91ca\u7684\u5c0f\u95ee\u9898\u3002\u673a\u68b0\u8bba\u5c31\u662f\u628a\u6240\u6709\u8fd0\u52a8\u90fd\u8fd8\u539f\u4e3a\u67d0\u79cd\u78b0\u649e\uff0c\u4e07\u6709\u5f15\u529b\u4e0d\u9700\u8981\u78b0\u649e\uff0c\u5728\u673a\u68b0\u8bba\u7684\u89d2\u5ea6\u5c31\u662f\u4e0d\u53ef\u89e3\u91ca\u7684\u3002\u800c\u5927\u6a21\u578b\u5219\u662f\u628a\u4e00\u5806\u4ee5\u524d\u4e0d\u77e5\u662f\u4e0d\u662f\u53ef\u89e3\u7684\u5c0f\u95ee\u9898\u5806\u5728\u4e00\u8d77\u6253\u5305\u89e3\u51b3\uff0c\u5f53\u6574\u4f53\u95ee\u9898\u88ab\u89e3\u51b3\u540e\uff0c\u91cc\u9762\u7684\u5c0f\u95ee\u9898\u5c31\u4e0d\u88ab\u4eba\u4eec\u8ba4\u4e3a\u662f\u91cd\u8981\u4e86\u3002<\/span><\/section>\n
\u6211\u4eec\u53ef\u4ee5\u8bf4\u7406\u6027\u4e3b\u4e49\u662f\u201c\u6e05\u6670\u7684\u601d\u60f3\u201d\uff0c\u800c\u9ed1\u76d2\u5b50\u662f\u201c\u76f2\u76ee\u7684\u8ba1\u7b97\u201d\u3002\u4f46\u7269\u7406\u5b9a\u5f8b\u4e00\u5b9a\u6bd4\u9ed1\u76d2\u5b50\u66f4\u52a0\u7ecf\u6d4e\u5417\uff1f\u5e38\u8bf4\u98de\u673a\u4e0d\u662f\u5b66\u9e1f\uff0c\u800c\u662f\u4f9d\u9760\u6d41\u4f53\u529b\u5b66\uff0c\u90a3\u662f\u56e0\u4e3a\u5236\u9020\u8d85\u7ea7\u5927\u9e1f\u7684\u6210\u672c\u592a\u9ad8\u3002\u5982\u679c\u9020\u9e1f\u7684\u6210\u672c\u4f4e\u4e8e\u6c42\u89e3\u6d41\u4f53\u529b\u5b66\u65b9\u7a0b\uff0c\u9020\u9e1f\u4e5f\u8bb8\u4e0d\u662f\u4e00\u4e2a\u5f88\u574f\u7684\u9009\u62e9\u3002Max Tegmark\u7b49\u8bd5\u56fe\u7528\u7b26\u53f7\u56de\u5f52<\/span>\uff08symbolic regression\uff09<\/span>\u4ece\u6570\u636e\u4e2d\u627e\u51fa\u7269\u7406\u5b9a\u5f8b<\/span>\uff08Undrescu & Tegmark\uff09<\/span>\u3002\u4ed6\u7684\u6570\u636e\u96c6\u4e4b\u4e00\u662f\u8d39\u66fc\u300a\u7269\u7406\u5b66\u8bb2\u4e49\u300b\u4e2d\u5217\u51fa\u7684\u516c\u5f0f\u3002\u5982\u679c\u6211\u4eec\u628a\u5b66\u4e60\u770b\u4f5c\u662f\u538b\u7f29\uff0c\u201c\u76f2\u76ee\u8ba1\u7b97\u201d\u6709\u65f6\u53ef\u80fd\u6bd4\u201c\u6e05\u6670\u601d\u60f3\u201d<\/span>\uff08\u5982\u7b26\u53f7\u56de\u5f52\u6216\u89e3\u6790\u89e3\uff09<\/span>\u7684\u538b\u7f29\u6bd4\u66f4\u9ad8\u3002\u79d1\u5b66\u95ee\u9898\u5728\u67d0\u79cd\u610f\u4e49\u4e0a\u6210\u4e86\u7ecf\u6d4e\u5b66\u95ee\u9898\u3002\u4e5f\u6b63\u662f\u5728\u8fd9\u4e2a\u610f\u4e49\u4e0a\uff0c\u8ba1\u7b97\u673a\u79d1\u5b66\u6bd4\u7269\u7406\u5b66\u66f4\u52a0\u63a5\u8fd1\u7b2c\u4e00\u6027\u539f\u7406\uff0c\u7269\u7406\u5b66\u4e0d\u8fc7\u662f\u8ba1\u7b97\u673a\u79d1\u5b66\u7684\u7b26\u53f7\u56de\u5f52\u3002<\/span><\/section>\n
\u4f5c\u8005\u7b80\u4ecb\uff1a<\/strong><\/span><\/section>\n
\u5c3c\u514b\uff0c\u4e4c\u9547\u667a\u5e93\u7406\u4e8b\u957f\u3002\u66fe\u83b7\u5434\u6587\u4fca\u4eba\u5de5\u667a\u80fd\u79d1\u6280\u8fdb\u6b65\u5956\u3002\u4e2d\u6587\u8457\u4f5c\u5305\u62ec\u300a\u4eba\u5de5\u667a\u80fd\u7b80\u53f2\u300b\u300a\u7406\u89e3\u56fe\u7075\u300b\u300aUNIX\u5185\u6838\u5256\u6790\u300b\u548c\u300a\u54f2\u5b66\u8bc4\u4e66\u300b\u7b49\u3002<\/span><\/section>\n

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[1]Liu, Z., Wang, Y., Vaidya, S., Ruehle, F., Halverson, J., Solja\u010di\u0107, M., … & Tegmark, M. (2024). KAN: Kolmogorov-Arnold Networks. arXiv preprint arXiv:2404.19756.<\/span><\/p>\n

[2]Cucker, F. & Smale, S. (2001), On the Mathematical Foundations of Learning, BULLETIN OF THE AMS, Volume 39, Number 1, Pages 1\u201349<\/span><\/p>\n

[3]Cybenko, G. (1988), Continuous Valued Neural Networks with Two Hidden Layers are Sufficient, Technical Report, Department of Computer Science, Tufts University. <\/span><\/p>\n

[4]Cybenko, G. (1989),  Approximation by Superpositions of a Sigmoidal Function, Mathematics of Control, Signals, and Systems, 2(4), 303\u2013314.<\/span><\/p>\n

[5]Girosi, F. & Poggio, T. (1989), Representation properties of networks: Kolmogorov\u2019s theorem is irrelevant, Neural Computation\uff0c1 (1989), 465\u2013469. <\/span><\/p>\n

[6]Hecht-Nielsen, R. (1987), Kolmogorov\u2019s mapping neural network existence theorem, Proc. I987 IEEE Int. Conf. on Neural Networks, IEEE Press, New York, 1987, vol. 3, 11\u201314. <\/span><\/p>\n

[7]Hilbert, D. (1900). Mathematische probleme. Nachr. Akad. Wiss. Gottingen, 290- 329. (\u201c\u6570\u5b66\u95ee\u9898\u201d\uff0c\u300a\u6570\u5b66\u4e0e\u6587\u5316\u300b\u5317\u4eac\u5927\u5b66\u51fa\u7248\u793e, 1990\uff09<\/span><\/p>\n

[8]Hornik, K. et al (1989), Multilayer Feedforward Networks are Universal Approximators, Neural Networks, Vol. 2, pp. 35Y-366, 1989.<\/span><\/p>\n

[9]Ismailov, V. (2022), A three layer neural network can represent any multivariate function, 2012.03016.pdf (arxiv.org)<\/span><\/p>\n

[10]Kahane, J. P. (1975), Sur le theoreme de superposition de Kolmogorov. J. Approx. Theory 13, 229-234.<\/span><\/p>\n

[11]Kleene, S.C. (1956), Representation of Events in Nerve Nets and Finite Automata. Automata Studies, Editors: C.E. Shannon and J. McCarthy, Princeton University Press, p. 3-42, Princeton, N.J.<\/span><\/p>\n

[12]Kolmogorov, A.N. (1957), On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition, (Russian)\uff0c Dokl. Akad. Nauk SSSR 114, 953\u2013956. <\/span><\/p>\n

[13]Kolmogorov, A.N. (1988), How I became a mathematician, (\u59da\u82b3\u7b49\u7f16\u8bd1\uff0c\u6211\u662f\u600e\u4e48\u6210\u4e3a\u6570\u5b66\u5bb6\u7684\uff0c\u5927\u8fde\u7406\u5de5\u5927\u5b66\u51fa\u7248\u793e\uff0c2023)<\/span><\/p>\n

[14]W. S. McCulloch, W. Pitts. (1943), A Logical Calculus of Ideas Immanent in Nervous Activity. Bulletin of Mathematical Biophysics, Vol. 5, p. 115-133.<\/span><\/p>\n

[15]Poggio, T. & Smale, S. (2003), The Mathematics of Learning: Dealing with Data, NOTICES OF THE AMS<\/span><\/p>\n

[16]Togelius, J. & Georgios N. Yannakakis, Choose Your Weapon: Survival Strategies for Depressed AI Academics, [2304.06035] (arxiv.org)<\/span><\/p>\n

[17]Udrescu, Silviu-Marian & M. Tegmark (2020), AI Feynman: a Physics-Inspired Method for Symbolic Regression, arxiv<\/span><\/p>\n

[18]\u5c3c\u514b (2021), \u4eba\u5de5\u667a\u80fd\u7b80\u53f2, \u7b2c2\u7248.<\/span><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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AI+Science \u662f\u8fd1\u5e74\u5174\u8d77\u7684\u5c06\u4eba\u5de5\u667a\u80fd\u548c\u79d1\u5b66\u76f8\u7ed3\u5408\u7684\u4e00\u79cd\u8d8b\u52bf\u3002\u4e00\u65b9\u9762\u662f AI for Science\uff0c\u673a\u5668\u5b66\u4e60\u548c\u5176\u4ed6 AI \u6280\u672f\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u79d1\u5b66\u7814\u7a76\u4e2d\u7684\u95ee\u9898\uff0c\u4ece\u9884\u6d4b\u5929\u6c14\u548c\u86cb\u767d\u8d28\u7ed3\u6784\uff0c\u5230\u6a21\u62df\u661f\u7cfb\u78b0\u649e\u3001\u8bbe\u8ba1\u4f18\u5316\u6838\u805a\u53d8\u53cd\u5e94\u5806\uff0c\u751a\u81f3\u50cf\u79d1\u5b66\u5bb6\u4e00\u6837\u8fdb\u884c\u79d1\u5b66\u53d1\u73b0\uff0c\u88ab\u79f0\u4e3a\u79d1\u5b66\u53d1\u73b0\u7684\u201c\u7b2c\u4e94\u8303\u5f0f\u201d\u3002\u53e6\u4e00\u65b9\u9762\u662f Science for AI\uff0c\u79d1\u5b66\u5c24\u5176\u662f\u7269\u7406\u5b66\u4e2d\u7684\u89c4\u5f8b\u548c\u601d\u60f3\u542f\u53d1\u673a\u5668\u5b66\u4e60\u7406\u8bba\uff0c\u4e3a\u4eba\u5de5\u667a\u80fd\u7684\u53d1\u5c55\u63d0\u4f9b\u5168\u65b0\u7684\u89c6\u89d2\u548c\u65b9\u6cd5\u3002<\/span><\/section>\n

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\u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u65af\u5766\u798f\u5927\u5b66\u8ba1\u7b97\u673a\u79d1\u5b66\u7cfb\u535a\u58eb\u540e\u7814\u7a76\u5458\u5434\u6cf0\u9716\uff08Jure Leskovec \u6559\u6388\u6307\u5bfc\uff09\u3001\u54c8\u4f5b\u91cf\u5b50\u8ba1\u5212\u7814\u7a76\u5458\u6248\u9e3f\u4e1a\u3001\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7269\u7406\u7cfb\u535a\u58eb\u751f\u5218\u5b50\u9e23\uff08Max Tegmark \u6559\u6388\u6307\u5bfc\uff09\uff0c\u5171\u540c\u53d1\u8d77\u4ee5\u201cAI+Science<\/a>\u201d\u4e3a\u4e3b\u9898\u7684\u8bfb\u4e66\u4f1a\uff0c\u63a2\u8ba8\u8be5\u9886\u57df\u7684\u91cd\u8981\u95ee\u9898\uff0c\u5171\u5b66\u5171\u7814\u76f8\u5173\u6587\u732e\u3002\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><\/section>\n

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\u6570\u5341\u5e74\u6765\uff0c\u4eba\u5de5\u667a\u80fd\u7684\u7406\u8bba\u53d1\u5c55\u548c\u6280\u672f\u5b9e\u8df5\u4e00\u76f4\u4e0e\u79d1\u5b66\u63a2\u7d22\u76f8\u4f34\u800c\u751f\uff0c\u5c24\u5176\u5728\u4ee5\u5927\u6a21\u578b\u4e3a\u4ee3\u8868\u7684\u4eba\u5de5\u667a\u80fd\u6280\u672f\u5e94\u7528\u96c6\u4e2d\u7206\u53d1\u7684\u5f53\u4e0b\uff0c\u4eba\u5de5\u667a\u80fd\u6b63\u5728\u52a0\u901f\u7269\u7406\u3001\u5316\u5b66\u3001\u751f\u7269\u7b49\u57fa\u7840\u79d1\u5b66\u7684\u9769\u65b0\uff0c\u800c\u8fd9\u4e9b\u5b66\u79d1\u4e5f\u5728\u53cd\u8fc7\u6765\u542f\u53d1\u4eba\u5de5\u667a\u80fd\u6280\u672f\u521b\u65b0\u3002\u5728\u6b64\u8fc7\u7a0b\u4e2d\uff0c\u6570\u5b66\u4f5c\u4e3a\u517c\u5177\u7406\u8bba\u5c5e\u6027\u4e0e\u5de5\u5177\u5c5e\u6027\u7684\u91cd\u8981\u57fa\u7840\u5b66\u79d1\uff0c\u4e0e\u4eba\u5de5\u667a\u80fd\u5173\u7cfb\u751a\u5bc6\uff0c\u76f8\u8f85\u76f8\u6210\u3002\u4e00\u65b9\u9762\uff0c\u4eba\u5de5\u667a\u80fd\u5728\u89e3\u51b3\u6570\u5b66\u9886\u57df\u7684\u8bf8\u591a\u5de5\u7a0b\u95ee\u9898\u3001\u7406\u8bba\u95ee\u9898\u4e43\u81f3\u5723\u676f\u96be\u9898\u4e0a\u5c61\u521b\u8bb0\u5f55\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u6570\u5b66\u6301\u7eed\u4e3a\u4eba\u5de5\u667a\u80fd\u6784\u7b51\u7406\u8bba\u57fa\u77f3\u5e76\u62d3\u5c55\u5176\u672a\u6765\u7a7a\u95f4\u3002\u8fd9\u4e24\u4e2a\u5173\u952e\u9886\u57df\u7684\u4ea4\u53c9\u878d\u5408\uff0c\u6b63\u5728\u63ed\u5f00\u4e0b\u4e2a\u65f6\u4ee3\u7684\u79d1\u5b66\u4e4b\u5e55\u3002<\/span><\/p>\n


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\u4e3a\u4e86\u63a2\u7d22\u6570\u5b66\u4e0e\u4eba\u5de5\u667a\u80fd\u6df1\u5ea6\u878d\u5408\u7684\u53ef\u80fd\u6027\uff0c\u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u540c\u6d4e\u5927\u5b66\u7279\u8058\u7814\u7a76\u5458\u9648\u5c0f\u6768\u3001\u6e05\u534e\u5927\u5b66\u4ea4\u53c9\u4fe1\u606f\u5b66\u9662\u52a9\u7406\u6559\u6388\u8881\u6d0b\u3001\u5357\u6d0b\u7406\u5de5\u5927\u5b66\u526f\u6559\u6388\u590f\u514b\u6797\u4e09\u4f4d\u8001\u5e08\uff0c\u5171\u540c\u53d1\u8d77\u201c\u4eba\u5de5\u667a\u80fd\u4e0e\u6570\u5b66\u201d\u8bfb\u4e66\u4f1a\uff0c\u5e0c\u671b\u4ece AI\u3000for\u3000Math\uff0cMath\u3000for\u3000AI \u4e24\u4e2a\u65b9\u9762\u6df1\u5165\u63a2\u8ba8\u4eba\u5de5\u667a\u80fd\u4e0e\u6570\u5b66\u7684\u5bc6\u5207\u8054\u7cfb\u3002\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><\/p>\n


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

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\u8be6\u60c5\u8bf7\u89c1\uff1a<\/span><\/p>\n

\u4eba\u5de5\u667a\u80fd\u4e0e\u6570\u5b66\u8bfb\u4e66\u4f1a\u542f\u52a8\uff1aAI for Math\uff0cMath for AI<\/a><\/span><\/p>\n

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

\n

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

1. <\/strong>\u73b0\u4ee3\u6982\u7387\u8bba\u4e4b\u7236\uff1a\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u7684\u201c\u968f\u673a\u201d\u4eba\u751f<\/a><\/strong><\/span><\/strong><\/strong><\/section>\n
2. <\/strong>\u627e\u5230\u4f60\u6df7\u4e71\u751f\u6d3b\u4e2d\u7684\u79e9\u5e8f\u2014\u2014Kolmogorov\u590d\u6742\u5ea6<\/a><\/strong><\/span><\/strong><\/strong><\/section>\n
3. <\/strong>Science\u91cd\u78c5\uff1a\u975e\u5e73\u8861\u7cfb\u7edf\u4e2d\uff0c\u79e9\u5e8f\u662f\u5982\u4f55\u6d8c\u73b0\u7684\uff1f<\/a><\/strong><\/span><\/strong><\/strong><\/section>\n
4. <\/strong><\/span>\u5f20\u6c5f\uff1a\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u6280\u672f\u57fa\u7840\u2014\u2014\u4ece\u53ef\u5fae\u5206\u7f16\u7a0b\u5230\u56e0\u679c\u63a8\u7406 | \u96c6\u667a\u5b66\u56ed\u5168\u65b0\u8bfe\u7a0b<\/a><\/strong><\/strong><\/span><\/section>\n
<\/strong><\/strong>5.<\/strong> \u9f99\u5e74\u5927\u8fd0\u8d77\uff0c\u5b66\u4e60\u6b63\u5f53\u65f6\uff01\u89e3\u9501\u96c6\u667a\u5168\u7ad9\u5185\u5bb9\uff0c\u5f00\u542f\u65b0\u5e74\u5b66\u4e60\u8ba1\u5212<\/a><\/strong><\/strong><\/strong><\/strong><\/span><\/section>\n
6. <\/strong>\u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/strong><\/span><\/a><\/section>\n


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

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

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

\u82cf\u8054\u6570\u5b66\u5bb6\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\uff08Andrey N. Kolmogorov\uff0c1903-1987\uff09\u3002\u56fe\u6e90\uff1ahttps:\/\/wolffund.org.il\/ \u7f16\u8005\u6309 \u5927\u6a21\u578b\u5411\u8ba1\u7b97\u7406\u8bba\u63d0\u51fa\u4e86\u65b0\u95ee\u9898\uff0c\u800c\u8ba1\u7b97\u7406\u8bba\u4e5f\u53ef\u5e2e\u52a9\u5927\u6a21\u578b\u63ed\u793a\u7b2c\u4e00\u6027\u539f\u7406\uff0c\u4ece\u800c\u627e\u5230\u8fb9\u754c\u548c\u65b9\u5411\u3002\u4f8b\u5982\uff0c\u82cf\u8054\u6570\u5b66\u5bb6\u67ef\u5c14\u83ab\u54e5\u6d1b\u592b\u548c\u5b66\u751f\u963f\u8bfa\u5fb7\u572820\u4e16\u7eaa50\u5e74\u4ee3…<\/p>\n","protected":false},"author":0,"featured_media":50247,"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\/50265"}],"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=50265"}],"version-history":[{"count":0,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/posts\/50265\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=\/wp\/v2\/media\/50247"}],"wp:attachment":[{"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=50265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=50265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=50265"},{"taxonomy":"special","embeddable":true,"href":"https:\/\/swarma.org\/index.php?rest_route=%2Fwp%2Fv2%2Fspecial&post=50265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}