{"id":60673,"date":"2025-06-12T22:06:46","date_gmt":"2025-06-12T14:06:46","guid":{"rendered":"https:\/\/swarma.org\/?p=60673"},"modified":"2025-06-12T22:06:46","modified_gmt":"2025-06-12T14:06:46","slug":"%e8%8c%83%e7%95%b4%e5%ad%a6%ef%bc%9a%e4%ba%ba%e5%b7%a5%e6%99%ba%e8%83%bd%e9%87%8c%e8%95%b4%e5%90%ab%e7%9a%84%e6%b7%b1%e5%88%bb%e6%95%b0%e5%ad%a6","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=60673","title":{"rendered":"\u8303\u7574\u5b66\uff1a\u4eba\u5de5\u667a\u80fd\u91cc\u8574\u542b\u7684\u6df1\u523b\u6570\u5b66"},"content":{"rendered":"
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\u672c\u6587\u6839\u636e\u4f5c\u8005\u57282025\u5e744\u6708\u5728\u6cb3\u5357\u5e08\u8303\u5927\u5b66\u6570\u5b66\u7cfb\u548c\u8ba1\u7b97\u673a\u7cfb\u7684\u62a5\u544a\u64b0\u5199\uff0c<\/span>\u8c08\u8ba1\u7b97\u673a\u548c\u4eba\u5de5\u667a\u80fd\u7406\u8bba\u4f7f\u7528\u8303\u7574\u5b66\u7684\u4e00\u4e9b\u4f8b\u5b50\u3002\u5e0c\u671b\u501f\u52a9\u8fd9\u4e9b\u4f8b\u5b50\u9f13\u52b1\u66f4\u591a\u8ba1\u7b97\u673a\u79d1\u5b66\u4e0e\u6570\u5b66\u7684\u4ea4\u6d41\uff0c\u76f8\u4e92\u4fc3\u8fdb\u66f4\u5927\u7684\u521b\u65b0\u53d1\u5c55\uff0c\u5e0c\u671b\u4eba\u5de5\u667a\u80fd\u6709\u4e00\u5929\u7528\u4e0a\u66f4\u6df1\u523b\u7684\u6570\u5b66\u3002<\/span><\/span><\/strong><\/span><\/strong><\/p>\n


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\u4eba\u4eba\u53ef\u5b66\u7684\u8303\u7574\u8bba\u7b2c\u4e8c\u5b63\u2014\u2014\u8de8\u5b66\u79d1\u7684\u79d1\u5b66\u65b9\u6cd5\u8bba | \u7cbe\u54c1\u5165\u95e8\u7cfb\u5217\u8bfe<\/span><\/a><\/span><\/section>\n
\u7ad9\u5728\u8303\u7574\u8bba\u89c6\u89d2\u770b\u673a\u5668\u5b66\u4e60  | \u300c\u8303\u7574\u8bba\u4e0e\u673a\u5668\u5b66\u4e60\u300d\u7cfb\u5217\u8bfe\u7a0b\u4e0a\u7ebf<\/span><\/a><\/span><\/section>\n<\/section>\n
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    1. \u8ba1\u7b97\u673a\u548c\u4eba\u5de5\u667a\u80fd<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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    194<\/span><\/span><\/span>3-<\/span><\/span><\/span>19<\/span><\/span><\/span>45<\/span><\/span><\/span>\u5e74<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u7f8e\u56fd<\/span><\/span><\/span>\u5bbe\u5915\u6cd5\u5c3c\u4e9a\u5927\u5b66\u7535\u673a\u5de5\u7a0b\u5b66\u9662<\/span><\/span><\/span>\u7269\u7406\u5b66\u5bb6<\/span><\/span><\/span>\u83ab\u5947\u5229<\/span>\uff08<\/span><\/span><\/span>John William Mauchly<\/span><\/span><\/span>\uff09<\/span><\/span><\/span>\u548c<\/span><\/span><\/span>\u5de5\u7a0b\u5e08<\/span><\/span><\/span>\u57c3\u514b\u7279<\/span>\uff08<\/span><\/span><\/span>John Presper Eckert Jr.<\/span><\/span><\/span>\uff09<\/span><\/span><\/span>\u5efa\u6210<\/span><\/span><\/span>\u4e86<\/span><\/span><\/span>\u4e16\u754c\u4e0a\u7b2c\u4e00\u53f0<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u91cd\u7ea630\u5428<\/span><\/span><\/span>\u53ef<\/span><\/span><\/span>\u8bbe\u5b9a\u7a0b\u5e8f<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u901a\u7528\u7535\u5b50\u8ba1\u7b97\u673aENIAC<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u5f53<\/span><\/span><\/span>\u5076\u7136\u542c\u8bf4\u4e86\u8fd9\u4e2a\u9879\u76ee\u4e4b\u540e\uff0c<\/span><\/span><\/span>\u6570\u5b66\u5bb6\u51af\u00b7\u8bfa\u4f9d\u66fc<\/span>\uff08John<\/span><\/span><\/span><\/span>von Neumann\uff09<\/span><\/span><\/span>\u540e\u6765\u4ee5\u987e\u95ee\u7684\u5f62\u5f0f\u53c2\u4e0e\u5230<\/span><\/span><\/span>ENIAC\u7684\u7814\u5236\u8ba1\u5212<\/span><\/span><\/span>\u4e4b\u4e2d\u3002<\/span><\/span><\/span><\/section>\n
    \u636e\u6807\u51c6\u7684\u8bfe\u672c<\/span>\uff08\u5982 <\/span><\/span><\/span>\u53c2\u8003\u6587\u732e<\/span><\/span><\/span>[11<\/span><\/span><\/span>]<\/span><\/span><\/span>[14<\/span><\/span><\/span>]<\/span><\/span><\/span>[<\/span><\/span><\/span>27<\/span><\/span><\/span>][<\/span><\/span><\/span>39<\/span><\/span><\/span>]<\/span><\/span><\/span>)<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>1943\u5e74\uff0c\u5fc3\u7406\u5b66\u5bb6<\/span><\/span><\/span>Warren<\/span><\/span><\/span><\/span>McCulloch\u548c\u6570\u5b66\u5bb6<\/span><\/span><\/span>Walter<\/span><\/span><\/span><\/span>Pitts\u6700\u65e9\u63d0\u51fa\u4e86\u4e00\u79cd\u57fa\u4e8e\u7b80\u5355\u903b\u8f91\u8fd0\u7b97\u7684\u4eba\u5de5\u795e\u7ecf\u7f51\u7edc\u30021950\u5e74\u56fe\u7075<\/span>\uff08Alan Turing\uff09<\/span>\u53d1\u8868<\/span><\/span><\/span>\u9898\u76ee\u4e3a<\/span><\/span><\/span>\u300a<\/span><\/span><\/span>\u8ba1\u7b97\u673a<\/span><\/span><\/span>\u5668<\/span><\/span><\/span>\u548c\u667a\u80fd<\/span><\/span><\/span>\u300b\u7684\u8bba\u6587<\/span><\/span><\/span>\u30021956\u5e74<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u5728<\/span><\/span><\/span>Dartmouth\u4f1a\u8bae<\/span><\/span><\/span>\u4e0a<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>McCarthy\u63d0\u51fa\u4e86\u4eba\u5de5\u667a\u80fd\u7684\u5b9a\u4e49<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd\u5c31\u662f\u8981\u8ba9\u673a\u5668\u7684\u884c\u4e3a\u770b\u8d77\u6765\u5c31\u50cf\u662f\u4eba\u6240\u8868\u73b0\u51fa\u7684\u667a\u80fd\u884c\u4e3a\u4e00\u6837\u3002<\/span><\/span><\/span>\u7f8e\u56fd\u5fc3\u7406\u5b66\u5bb6\u548c\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6<\/span><\/span><\/span>\u5f17\u5170\u514b\u00b7\u7f57\u68ee\u5e03\u62c9\u7279<\/span><\/span><\/span>\uff08Frank Rosenblatt\uff09<\/span><\/span><\/span>\u5219<\/span><\/span><\/span>\u5728<\/span><\/span><\/span>1958\u5e74\u63d0\u51fa\u4e86\u4e00\u79cd\u79f0\u4e3a\u611f\u77e5\u5668<\/span><\/span><\/span><\/span>\uff08Perceptron\uff09<\/span><\/span><\/span>\u7684\u3001<\/span><\/span><\/span>\u4ee5\u6a21\u62df\u4eba\u7c7b\u611f\u77e5\u80fd\u529b<\/span><\/span><\/span>\u4e3a\u7279\u5f81<\/span><\/span><\/span>\u7684\u795e\u7ecf\u7f51\u7edc\u6a21\u578b\u30021983\u5e74\uff0c\u7269\u7406\u5b66\u5bb6<\/span><\/span><\/span>John Joseph Hopfield<\/span><\/span><\/span>\u63d0\u51fa\u4e86\u4e00\u79cd\u7528\u4e8e\u8054\u60f3\u8bb0\u5fc6<\/span>\uff08Associative Memory\uff09<\/span>\u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u79f0\u4e3aHopfield\u7f51\u7edc\u30021984\u5e74<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u52a0\u62ff\u5927\u8ba1\u7b97\u673a\u79d1\u5b66\u5bb6Geoffrey <\/span><\/span><\/span>Hinton\u63d0\u51fa\u4e00\u79cd\u968f\u673a\u5316\u7248\u672c\u7684Hopfield\u7f51\u7edc\uff0c\u5373\u73bb\u5c14\u5179\u66fc\u673a<\/span>\uff08Boltzmann<\/span><\/span><\/span><\/span>Machine\uff09<\/span>\u3002<\/span><\/span><\/span>\u56e0\u4e3a\u8fd9\u4e9b\u5de5\u4f5c<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>Hopfield<\/span><\/span><\/span>\u548c<\/span><\/span><\/span>Hinton<\/span><\/span><\/span>\u83b7\u5f972024\u5e74<\/span><\/span><\/span>\u8bfa\u8d1d\u5c14<\/span><\/span><\/span>\u7269\u7406\u5b66<\/span><\/span><\/span>\u5956<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>2017<\/span><\/span><\/span>\u5e74<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u5728NeurlPS\u4f1a\u8bae<\/span><\/span><\/span>\u4e0a\uff0c<\/span><\/span><\/span>\u8c37\u6b4c<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>8<\/span><\/span><\/span>\u4eba<\/span><\/span><\/span>\u56e2\u961f<\/span><\/span><\/span>\u5728\u6587\u7ae0\u201c<\/span><\/span><\/span>Attention Is All You Need\u201d<\/span><\/span><\/span> (<\/span><\/span><\/span>arXiv<\/span><\/span><\/span>:<\/span><\/span><\/span>1706.03762<\/span><\/span><\/span>)<\/span><\/span><\/span>\u4e2d<\/span><\/span><\/span>\u53d1\u8868<\/span><\/span><\/span>\u4e86<\/span><\/span><\/span>Transformer <\/span><\/span><\/span>\u2014\u2014<\/span><\/span><\/span>\u4e00\u4e2a<\/span><\/span><\/span>\u521b\u65b0\u7684<\/span><\/span><\/span>\u751f\u6210\u5f0f\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\uff08<\/span><\/span><\/span>Generative AI<\/span><\/span><\/span>\uff09<\/span><\/span><\/span>\u67b6\u6784<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u4e0d\u540c\u4e8e\u4f20\u7edf\u7684\u4eba\u5de5\u667a\u80fd\u4ec5\u5bf9\u8f93\u5165\u6570\u636e\u8fdb\u884c\u5904\u7406\u548c\u5206\u6790\uff0c\u751f\u6210\u5f0f\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u4f7f\u7528<\/span><\/span><\/span>\u6df1\u5ea6\u5b66\u4e60<\/span><\/span><\/span>\u901a<\/span><\/span><\/span>\u8fc7<\/span><\/span><\/span>\u8bad\u7ec3<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u4f18\u5316\u548c\u751f\u6210\u6765<\/span><\/span><\/span>\u5b66\u4e60\u5e76\u6a21\u62df\u4e8b\u7269\u7684\u5185\u5728\u89c4\u5f8b\uff0c\u6839\u636e\u8f93\u5165<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u8d44\u6599\u751f\u6210\u5177\u6709\u903b\u8f91\u6027\u548c\u8fde\u8d2f\u6027\u7684\u65b0\u5185\u5bb9\u3002<\/span><\/span><\/span>\u7531\u6b64\uff0c<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u8fdb\u5165<\/span><\/span><\/span>\u4e86\u201c<\/span><\/span><\/span>\u5927\u578b\u8bed\u8a00\u6a21\u578b<\/span><\/span><\/span>\u201d<\/span> <\/span><\/span><\/span>(LLM) <\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u65f6\u4ee3\u30022024\u5e74<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>EpochAI<\/span><\/span><\/span>[43]<\/span><\/sup><\/strong><\/span>\u63d0\u51fa<\/span><\/span><\/span>FrontierMath, <\/span><\/span><\/span>\u8fd9\u662f<\/span><\/span><\/span>\u4e00\u4e2a\u7528\u4e8e\u8bc4\u4f30\u4eba\u5de5\u667a\u80fd\u6570\u5b66\u63a8\u7406\u80fd\u529b\u7684\u57fa\u51c6\u6d4b\u8bd5<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u516d\u4e2a<\/span><\/span><\/span>\u5f53\u4eca\u4e16\u754c\u6700<\/span><\/span><\/span>\u9886\u5148\u7684\u6a21\u578b : o1<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>Claude 3.5<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>Sonnet\u3001GPT-4o<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>Grok\u548cGemini 1.5 Pro<\/span><\/span><\/span>\u5728\u8fd9\u4e2a\u6d4b\u8bd5\u4e0b\u7684<\/span><\/span><\/span>\u6210\u529f\u7387<\/span><\/span><\/span>\u90fd<\/span><\/span><\/span>\u4f4e\u4e8e2<\/span><\/span><\/span>\uff05\u3002\u8fd9\u4e2a\u6d4b\u8bd5\u662f\u57fa\u4e8e\u4e09\u767e\u591a\u4e2a<\/span><\/span><\/span>\u6db5\u76d6\u4e86\u73b0\u4ee3\u6570\u5b66\u7684\u5927\u591a\u6570\u4e3b\u8981\u5206\u652f\u7684\u95ee\u9898<\/span><\/span><\/span>\u800c\u6765\u7684<\/span><\/span><\/span>\u3002 <\/span><\/span><\/span><\/section>\n
    \u4e0b\u56fe\u663e\u793a<\/span><\/span><\/span>EpochAI\u7684\u6570\u5b66\u4e16\u754c <\/span><\/span><\/span><\/section>\n
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    EpochAI\u7684\u6570\u5b66\u4e16\u754c\u4e28<\/span><\/span><\/span>\u56fe\u6e90<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>Epoch AI<\/span><\/span><\/span><\/span><\/section>\n
    \u8fd9\u4e9b\u95ee\u9898\u6709\u4e24\u4e2a\u7279\u5f81\uff1a(1\uff09\u89e3\u51b3\u65b9\u6848\u662f\u81ea\u52a8\u53ef\u9a8c\u8bc1\u7684\uff0c\u4ece\u800c\u5b9e\u73b0\u9ad8\u6548\u7684\u8bc4\u4f30\uff1b(2\uff09\u6240\u6709\u95ee\u9898\u90fd\u662f\u65b0\u7684\u4e14\u672a\u53d1\u8868\u7684\uff0c\u4ee5\u9632\u6b62\u6570\u636e\u6c61\u67d3<\/span><\/span><\/span>\u2014\u2014<\/span><\/span><\/span>\u8fd9\u662f\u8bf4LLM\u5229\u7528\u8ba1\u7b97\u673a\u7684\u8d85\u7ea7\u901f\u5ea6\u5bf9\u5173\u8054\u7684\u5927\u6570\u636e\u8fdb\u884c\u5b9e\u65f6<\/span><\/span><\/span>\u91c7\u6398<\/span><\/span><\/span>, <\/span><\/span><\/span>\u5bf9\u6ca1\u6709<\/span><\/span><\/span>\u6536\u5f55\u5728\u5b83<\/span><\/span><\/span>\u4eec\u6570\u636e\u5e93\u7684<\/span><\/span><\/span>\u5168\u65b0<\/span><\/span><\/span>\u5167\u5bb9<\/span><\/span><\/span>, LLM<\/span><\/span><\/span>\u662f\u6ca1\u6709\u529e\u6cd5\u7684<\/span><\/span><\/span>, <\/span><\/span><\/span>\u66f4\u4f55\u51b5\u4eba\u7c7b\u81ea\u5df2\u4e5f\u4e0d\u77e5\u9053\u7684\u731c\u60f3\u7684\u7b54\u6848\u5462<\/span><\/span><\/span>! <\/span><\/span><\/span>\u8bb2<\/span><\/span><\/span>\u4e2a\u7b11\u8bdd<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>\u7f51\u4e0a\u7684<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u5de5\u5177\u53ef\u4ee5\u8c08\u604b\u7231, \u4f46\u8ddd\u79bb\u53ef\u4ee5\u521b\u65b0\u6570\u5b66<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u7269\u7406\u5b66\u548c\u5176\u4ed6\u57fa\u7840\u7406\u8bba\u79d1\u5b66\u8fd8<\/span><\/span><\/span>\u8fdc<\/span><\/span><\/span>, \u5927\u5bb6\u662f\u4ecd\u7136\u6709\u5f88\u591a\u673a\u4f1a\u7684\u3002<\/span><\/span><\/span><\/span><\/section>\n
    \u6211\u4eec\u53ef\u4ee5\u8bf4: <\/span><\/span><\/span>\u4eba\u5de5\u667a<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u8d77\u98de<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\u662f\u59cb\u4e8e<\/span><\/span><\/span>OpenAI<\/span><\/span><\/span>\u4f7f\u7528<\/span><\/span><\/span>\u82f1\u4f1f\u8fbe<\/span><\/span><\/span>\u5efa\u9020\u7684\u5fae<\/span><\/span><\/span>\u8d85\u7ea7\u8ba1\u7b97\u673a<\/span>\uff08mini-super<\/span><\/span><\/span>-computer\uff09<\/span>\u5411\u516c\u4f17\u63d0\u4f9b<\/span><\/span><\/span>ChatGPT<\/span><\/span><\/span>\u2014\u2014<\/span><\/span><\/span>\u8fd9\u5c31\u662f\u6211\u4eec\u6240\u8bf4\u7684<\/span><\/span><\/span>\u8ba1\u7b97\u673a<\/span><\/span><\/span>\u4e0e<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u7684\u7ed3\u5408<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u6211\u4eec\u501f\u52a9\u51e0\u5f20\u7167\u7247\u7b80\u5355\u4ecb\u7ecd<\/span><\/span><\/span>\u8ba1\u7b97\u673a\u548c\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u7684\u5173\u7cfb<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u8fd9\u5f20\u7167\u7247\u662f<\/span><\/span><\/span>2016 \u5e74 8 \u6708\u82f1\u4f1f\u8fbe<\/span> (Nvidia) <\/span>\u9ec4\u4ec1\u52cb\u4eb2\u624b\u8d60\u9001\u7ed9\u65f6\u5728OpenAI\u7684\u9a6c\u65af\u514b<\/span>\uff08Elon Reeve Musk\uff09<\/span>\u5168\u7403\u7b2c\u4e00\u90e8 AI \u8d85\u7ea7\u8ba1\u7b97\u673a DGX-1<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u8fd9\u90e8\u8ba1\u7b97\u673a<\/span><\/span><\/span>\u66f4\u9002\u5408<\/span><\/span><\/span>\u7528\u4e8eAI\u7684\u8ba1\u7b97<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>2023 <\/span><\/span><\/span>\u5e745\u670825 \u65e5\u82f1\u4f1f\u8fbe\u7684\u5e02\u503c\u589e\u52a02\u5343\u4ebf\u7f8e\u5143<\/span><\/span><\/span>\uff08<\/span><\/span><\/span>\u7834<\/span><\/span><\/span>\u7eaa<\/span><\/span><\/span>\u5f55<\/span><\/span><\/span>)<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u5230\u6708\u5e95<\/span><\/span><\/span>\u65f6<\/span><\/span><\/span>\u5168\u7403\u5e02\u503c\u6392\u7b2c\u516d<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u8d85\u8fc7Walmart \u52a0<\/span><\/span><\/span>\u4e0a<\/span><\/span><\/span>ExxonMobil<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>2025 \u5e74 1 \u6708 28 \u65e5<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span> DeepSeek \u53d1\u5e03 R1<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u82f1\u4f1f\u8fbe\u7684\u5e02\u503c\u66b4\u8dcc 5 \u5343\u4ebf\u7f8e\u5143<\/span> (\u7834<\/span><\/span><\/span>\u7eaa<\/span><\/span><\/span>\u5f55)<\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u4e0b\u56fe\u662f\u82f1\u4f1f\u8fbe<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd\u8ba1\u7b97\u673a<\/span><\/span><\/span>\u82af\u7247<\/span><\/span><\/span>GB2000<\/span><\/span><\/span>\u2014\u2014<\/span><\/span><\/span>\u8fd9\u662f\u4e0a\u9762\u7684\u4e24\u4e2aGPU\u4f5c\u5e73\u884c\u8ba1\u7b97\u5b9e\u65f6\u4e0e\u4e0b\u9762\u7684\u4e00\u4e2a\u4f5c\u7ebf\u6027\u8ba1\u7b97<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>CPU\u8f6c\u6362\u3002\u73b0\u65f6<\/span><\/span><\/span>\u82f1\u4f1f\u8fbe<\/span><\/span><\/span>\u7684H20\u82af\u7247\u662f\u7981\u8fd0\u6211\u56fd<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\uff0c\u6211\u4eec\u53ef\u4ee5<\/span><\/span><\/span>\u8fdb<\/span><\/span><\/span>\u53e3\u7684\u662f\u4f4e\u4e00\u6863\u7684H200<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u4e0b\u56fe\u662f<\/span><\/span><\/span>\u6211\u56fd\u751f\u4ea7\u7684\u534e\u4e3a<\/span><\/span><\/span>AI\u63a8\u7406\u82af\u7247<\/span><\/span><\/span>\uff08<\/span><\/span><\/span>\u5176\u4e2d<\/span><\/span><\/span>NPU<\/span><\/span><\/span>\u662f<\/span><\/span><\/span>\u642d\u8f7d\u4e868\u9897\u6607\u817e910B\uff0c<\/span><\/span><\/span>\u800c<\/span><\/span><\/span>CPU<\/span><\/span><\/span>\u662f<\/span><\/span><\/span>\u914d\u5907\u4e864\u9897\u9cb2\u9e4f920<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u5de6\u56fe<\/span><\/span><\/span>\uff09<\/span>\u548c<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u8d85\u7ea7<\/span><\/span><\/span>\u8ba1\u7b97\u673a<\/span><\/span><\/span>\u534e\u4e3a\u6607\u817e<\/span><\/span><\/span> Atlas900<\/span><\/span><\/span>\uff08\u53f3\u56fe<\/span><\/span><\/span>\uff09<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    2. <\/span>\u8ba1\u7b97\u673a\u4e0e\u8303\u7574<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u8ba1\u7b97\u673a\u7cfb\u7684\u8001\u5e08\u5f00\u7684\u6570\u5b66\u8bfe\u4ee5\u6570\u7406\u903b\u8f91\u4e3a\u4e3b\uff0c\u4ed6\u4eec\u4f1a\u8bb2\u6a21\u6001\u903b\u8f91<\/span><\/span><\/span>\uff08\u5982[<\/span><\/span><\/span>6<\/span><\/span><\/span>]<\/span><\/span><\/span>\uff09<\/span>\u6216\u4e0e<\/span><\/span><\/span>\u51fd\u6570\u7a0b\u5e8f\u8bbe\u8ba1<\/span><\/span><\/span>\u76f8\u5173\u7684<\/span><\/span><\/span>\u03bb\u6f14\u7b97<\/span><\/span><\/span>\uff08\u5982[<\/span><\/span><\/span>17<\/span><\/span><\/span>]<\/span><\/span><\/span>\uff09<\/span>\u3002\u81f3\u4e8e\u8ba1\u7b97\u673a\u4e0e<\/span><\/span><\/span>\u8303\u7574\u5b66\uff0c\u4e5f\u8bb8\u5927\u5bb6\u4e0d\u76f8\u4fe1\u4e24\u8005\u6709\u5173\u7cfb<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u6211\u4eec<\/span><\/span><\/span>\u4e0d\u6253\u7b97\u8bf4\u660e\uff0c\u4f46\u662f<\/span><\/span><\/span>\u6211\u4eec\u5e0c\u671b<\/span><\/span><\/span>\u7528\u51e0\u672c<\/span>\uff08\u4f8b\u5982<\/span><\/span><\/span>[<\/span><\/span><\/span>2][12][<\/span><\/span><\/span>24<\/span><\/span><\/span>][<\/span><\/span><\/span>25<\/span><\/span><\/span>][<\/span><\/span><\/span>32<\/span><\/span><\/span>][42]) <\/span><\/span><\/span>\u7ed9\u8ba1\u7b97\u673a\u7cfb\u7684\u5b66\u751f\u4e0a\u8303\u7574<\/span><\/span><\/span>\u5b66<\/span><\/span><\/span>\u7684\u8bfe\u672c<\/span><\/span><\/span>\u652f\u6301\u6211\u4eec\u89c2\u70b9<\/span><\/span><\/span>\u2014\u2014<\/span><\/span><\/span>\u8303\u7574\u5b66<\/span><\/span><\/span>\u662f\u53ef\u4ee5\u5e2e\u52a9\u660e\u767d\u548c\u89e3\u51b3\u8ba1\u7b97\u673a\u7406\u8bba\u95ee\u9898\u7684<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    3. <\/span>\u4eba\u5de5\u667a\u80fd\u4e0e\u8303\u7574\u5b66<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u4f59\u4e0b\u6211\u4eec\u4e13\u6ce8\u8ba8\u8bba\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u4e0e<\/span><\/span><\/span>\u8303\u7574\u5b66<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u4ece<\/span><\/span><\/span>\u4e0b<\/span><\/span><\/span>\u56fe\u53ef\u89c1<\/span><\/span><\/span>github\u6536<\/span><\/span><\/span>\u5f55<\/span><\/span><\/span>\u7684\u8303\u7574\u4e0e\u673a\u5668\u5b66\u4e60\u7684\u6587\u7ae0<\/span><\/span><\/span>\u51fa<\/span><\/span><\/span>\u73b0\u4e86<\/span><\/span><\/span>\u5feb\u901f\u589e<\/span><\/span><\/span>\u957f<\/span> <\/span><\/span><\/span>(<\/span><\/span><\/span>\u5728<\/span><\/span><\/span>arXiv<\/span><\/span><\/span>\u4e0a<\/span><\/span><\/span>\u4ea6\u7c7b\u4f3c<\/span><\/span><\/span>)<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u6700\u8fd1\u4eba\u4eec\u53d1\u73b0: \u8ba1\u7b97\u673a<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd\u548c\u8303\u7574\u5b66\u597d\u50cf\u6709\u5171\u540c\u7684\u7406\u8bba\u7ed3\u6784<\/span><\/span><\/span>\u3002\u6211\u4eec\u4e3e\u8303\u7574\u5b66\u8fdb\u5165\u795e\u7ecf\u7f51\u7edc\u7406\u8bba\u7684\u4e09\u4e2a\u4f8b\u5b50\u6765\u8bf4\u660e<\/span><\/span><\/span>\u8fd9\u4e2a\u73b0\u8c61<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span><\/section>\n
    (\u4e00) <\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edc\u5b66\u4e60\u7b97\u6cd5\u7684<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>\u7ed3\u6784<\/span><\/span><\/span>\uff1b<\/span><\/span><\/span><\/section>\n
    (\u4e8c) SNN<\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edc\u7684<\/span><\/span><\/span>\u4fe1\u606f<\/span><\/span><\/span>\u7ed3\u6784<\/span><\/span><\/span>\uff1b<\/span><\/span><\/span><\/section>\n
    (\u4e09) DNN<\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edc\u7684<\/span><\/span><\/span>\u62d3\u6251<\/span><\/span><\/span>\u7ed3\u6784<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u5728\u672a\u5f00\u59cb\u4e4b\u524d<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u6211\u4eec<\/span><\/span><\/span>\u5148<\/span><\/span><\/span>\u8bf4\u8bf4<\/span><\/span><\/span>\u8303\u7574\u662f\u4ec0\u4e48\uff1f<\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    4. <\/span>\u8303\u7574\u662f\u4ec0\u4e48\uff1f<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \uff081\uff09\u6570\u5b66\u662f\u4f7f\u5404\u4e0d\u540c\u7684\u7cfb\u7edf\u53ef\u4ee5\u5bf9\u8bdd\u7684\u8bed\u8a00\u3002<\/span><\/span><\/span><\/section>\n
    \uff082\uff09\u6570\u5b66\u662f\u7814\u7a76\u7ed3\u6784\u7684\u79d1\u5b66\u3002\u5f53\u4e24\u4e2a\u7edd\u4e0d\u76f8\u540c\u7684\u7cfb\u7edf\u6709\u5171\u540c\u7ed3\u6784\u7684\u65f6\u5019\uff0c\u8fd9\u4e24\u4e2a\u7cfb\u7edf\u4fbf\u53ef\u4ee5\u7ecf\u8fc7\u5171\u540c\u7ed3\u6784\u4f7f\u7528\u6570\u5b66\u5bf9\u8bdd\u3002<\/span><\/span><\/span><\/section>\n
    \uff083\uff09\u4ec0\u4e48<\/span><\/span><\/span>\u662f<\/span><\/span><\/span>\u4e00\u4e2a<\/span><\/span><\/span>\u6570\u5b66\u7ed3\u6784<\/span><\/span><\/span>\uff1f<\/span><\/span><\/span>\u6bcf\u5f53\u4e00\u4e2a\u7cfb\u7edf\u6709\u4e00\u4e2a\u8fc7\u7a0b\u628a\u4e24\u4ef6\u4e8b<\/span>\uff08\u6216<\/span><\/span><\/span>\u8bf4<\/span><\/span><\/span>\u4e24\u4e2a\u5143\u7d20\uff09<\/span><\/span><\/span>\u5408\u6210<\/span><\/span><\/span>\u4e3a\u65b0\u7684\u4e00\u4ef6\u4e8b\uff0c\u4fbf\u6709\u4e00\u4e2a\u6570\u5b66\u7ed3\u6784\u3002\u4f8b\u5982\u5728\u4ee3\u6570\u5b66<\/span><\/span><\/span>\u6211\u4eec\u8bf4\u2018<\/span><\/span><\/span>\u7fa4<\/span><\/span><\/span>\u2019G \u662f\u4e00\u79cd<\/span><\/span><\/span>\u6570\u5b66\u7ed3\u6784<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>G\u7684\u4efb\u4f55\u4e24\u4e2a\u5143x,y\u53ef\u4ee5\u5408\u6210\u4e3a\u4e00\u4e2a\u5c5e\u4e8eG\u7684\u5143x<\/span><\/span><\/span>\u25e6<\/span><\/span><\/span>y<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u6574\u6570Z\u7684\u52a0\u6cd5\u662f<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u7fa4<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\u7684\u4f8b\u5b50<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u4e24\u4e2a\u6574\u6570 x, y\u53ef\u4ee5<\/span><\/span><\/span>\u7528<\/span><\/span><\/span>\u6574\u6570<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u52a0\u6cd5\u5408\u6210\u4e3a\u4e00\u4e2a\u6574\u6570x+y<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/span><\/section>\n
    \uff08<\/span><\/span><\/span>4\uff09<\/span><\/span><\/span>\u4e00\u4e2a<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\u4e5f\u662f\u4e00\u79cd\u6570\u5b66\u7ed3\u6784<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u4e0d\u8fc7, <\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\u662f\u53ef\u4ee5\u7528\u6765\u7814\u7a76\u6570\u5b66\u7ed3\u6784\u7684\u6570\u5b66\u7ed3\u6784<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u4f8b\u5982<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>\u6211\u4eec\u53ef\u4ee5\u8003\u8651\u7531\u6240\u6709<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u7fa4<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\u7ec4\u6210\u7684\u8303\u7574<\/span><\/span><\/span>Grp<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/span><\/section>\n
    \u4e00\u4e2a<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>C\u6709\u4e24\u4e2a\u90e8\u5206<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>(1)\u662fC\u7684\u5bf9\u8c61<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>(2)\u662fC\u7684\u6001\u5c04<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u4f8b\u5982: \u6240\u6709<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u96c6<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\u7ec4\u6210\u7684\u8303\u7574Set\u7684\u5bf9\u8c61\u5c31\u662f\u4efb\u4f55\u4e00\u4e2a<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u96c6<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\uff1b<\/span><\/span><\/span>Set\u7684\u6001\u5c04\u5c31\u662f\u4efb\u4f55\u4e00\u4e2a<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u96c6\u7684\u6620\u5c04<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>f<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>X\u2192Y<\/span><\/span><\/span>\u3002<\/span><\/span><\/span> \u6211\u4eec\u7528\u4e00\u4e2a<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u96c6\u7684\u6620\u5c04<\/span><\/span><\/span>\u201d<\/span><\/span><\/span> f:X\u2192Y\u6765\u6bd4\u8f83\u4e24\u4e2a\u96c6X,Y<\/span><\/span><\/span>\uff1b<\/span><\/span><\/span>\u6211\u4eec\u7528\u4e00\u4e2a<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u8303\u7574\u7684\u51fd<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\u2019F:C\u2192D \u6765\u6bd4\u8f83\u4e24\u4e2a\u8303\u7574C,D<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \uff085\uff09\u62d3\u6251\u7a7a\u95f4\u662f<\/span>\u523b\u5212<\/span><\/span><\/span>\u8fde\u7eed\u6027<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u6570\u5b66\u7ed3\u6784<\/span><\/span><\/span>[40]<\/span><\/sup><\/strong><\/span>\u3002<\/span><\/span><\/span>\u7531\u6240\u6709\u2018<\/span><\/span><\/span>\u62d3\u6251\u7a7a\u95f4<\/span><\/span><\/span>\u2019\u7ec4\u6210\u7684\u8303\u7574<\/span><\/span><\/span>\u8bb0\u4e3a<\/span><\/span><\/span>Top<\/span>\u3002<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>\u5b66\u7684\u7b2c\u4e00\u4e2a\u91cd\u8981\u7684\u51fd\u5b50\u662f\u57fa\u672c\u7fa4\u51fd\u5b50<\/span><\/span><\/span>\u03c0<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>Top <\/span><\/span><\/span>\u2192<\/span><\/span><\/span>Grp<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u4f7f\u7528\u51fd\u5b50<\/span><\/span><\/span>\u03c0<\/span><\/span><\/span>\u53ef\u4ee5\u7b80\u5355\u7684\u89e3\u51b3\u5173\u4e8e\u62d3\u6251\u7a7a\u95f4\u7684\u6027\u8d28\u7684\u95ee\u9898<\/span><\/span><\/span>[41]<\/span><\/sup><\/strong><\/span>\u3002\u6211\u4eec\u5f53\u7559\u610f\u8fd9\u662f\u4e00\u4ef6\u5f88\u795e\u5947\u7684\u4e8b, <\/span><\/span><\/span>Top<\/span><\/span><\/span>\u662f\u8bb2\u8fde\u7eed, <\/span><\/span><\/span>Grp<\/span><\/span><\/span>\u662f\u8bb2\u52a0\u6cd5, \u5de6\u53f3\u4e24\u8fb9\u770b\u8d77\u6765\u6c92\u6709\u534a\u70b9\u5173\u7cfb, \u4f46\u662f\u6570\u5b66, \u5373\u662f\u51fd\u5b50<\/span><\/span><\/span>\u03c0<\/span><\/span><\/span>, \u628a<\/span><\/span><\/span>\u4e24<\/span><\/span><\/span>\u8fb9\u8054\u7edc\u8d77\u6765<\/span><\/span><\/span>\uff01<\/span><\/span><\/span>\u8fd9\u4e2a\u628a\u4e24\u79cd\u4e8b\u7269\u8054\u7edc\u8d77\u6765\u7684\u60f3\u6cd5\u5e94\u4f1a<\/span><\/span><\/span>\u5bf9<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u7406\u8bba\u6709<\/span><\/span><\/span>\u6240<\/span><\/span><\/span>\u8d21\u732e\u3002<\/span><\/span><\/span><\/section>\n
    \uff08<\/span><\/span><\/span>6\uff09\u76ee\u524d\u6211\u56fd\u6bd4\u8f83\u6210\u719f\u7684\u6570\u5b66\u7cfb\u7684\u672c\u79d1\u751f<\/span><\/span><\/span>\u80fd<\/span><\/span><\/span>\u5b66<\/span><\/span><\/span>\u5230<\/span><\/span><\/span>\u4e00\u70b9<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>\u5b66\u3002\u8303\u7574\u5b66\u5927\u5e08<\/span><\/span><\/span>\u3001\u7f8e\u56fd\u6570\u5b66\u5bb6<\/span><\/span><\/span>Francis William <\/span><\/span><\/span>Lawvere<\/span><\/span><\/span>\u4e3a\u4e2d\u5b66\u751f\u5199\u4e86\u4e00\u672c\u5b66\u4e60\u8303\u7574\u5b66\u7684<\/span><\/span><\/span>\u6559\u6750<\/span><\/span><\/span>[22](\u4e0d\u5355\u8303\u7574\u5b66<\/span><\/span><\/sup><\/strong><\/span>\uff0c<\/span><\/span><\/span>\u7f8e\u56fd\u4eba\u4ea6\u9f13\u52b1<\/span><\/span><\/span>\u9752<\/span><\/span><\/span>\u5c11\u5e74\u5b66\u4e60\u4eba\u5de5<\/span><\/span><\/span>\u667a\u80fd<\/span><\/span><\/span>\u7f16\u7a0b\u8bed\u8a00Python<\/span><\/span><\/span>[<\/span><\/span><\/span>8]<\/span><\/span><\/span>)<\/span>\u3002\u672c\u6587\u4ee5\u4e0b\u51fa\u73b0\u7684\u8303\u7574\u90fd\u662f<\/span><\/span><\/span>\u5e7a<\/span><\/span><\/span>\u534a\u8303\u7574<\/span> <\/span><\/span><\/span>(symmetric monoidal category)<\/span>\uff0c<\/span><\/span><\/span>\u5b9a\u4e49\u89c1<\/span><\/span><\/span>[21]<\/span><\/span><\/span>\u674e\u6587\u5a01\u7684<\/span><\/span><\/span>\u4ee3\u6570\u5b66\u65b9\u6cd5\u3002<\/span><\/span><\/span>\u5b9a\u4e49<\/span><\/span><\/span>\u6570\u5b66\u7ed3\u6784<\/span><\/span><\/span>\u662f\u9700\u8981\u4f7f\u7528\u6570\u7406\u903b\u8f91\u7684\uff0c\u4e0d\u8fc7\u770b\u770b[<\/span><\/span><\/span>19][20]<\/span><\/span><\/span>\u4ea6<\/span><\/span><\/span>\u53ef\u4ee5<\/span><\/span><\/span>\u589e\u52a0<\/span><\/span><\/span>\u7406\u89e3\u3002<\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    5. <\/span>\u7528\u53ef\u5fae\u51fd\u6570\u9020\u4e00\u4e2a\u8303\u7574\u7684\u4f8b\u5b50<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u4ee5<\/span><\/span><\/span><\/span><\/sub>\u8bb0\u5b9e\u6570, <\/span><\/span><\/span><\/span><\/sub>\u8bb0n\u7ef4\u5b9e\u6570\u5411\u91cf\u7a7a\u95f4<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u4ee5P\u8bb0<\/span><\/span><\/span>\u53c2\u6570<\/span><\/span><\/span>\u7a7a\u95f4<\/span><\/span><\/span><\/span><\/sub>\uff0c<\/span><\/span><\/span>\u79f0\u53ef\u5fae\u51fd\u6570I:<\/span><\/span><\/span>P<\/span><\/span><\/span>\u00d7<\/span><\/span><\/span><\/span><\/sub>\u2192<\/span><\/span><\/span><\/span><\/sub>\u4e3a\u4ece<\/span><\/span><\/span><\/span><\/sub>\u81f3<\/span><\/span><\/span><\/span><\/sub>\u7684\u5e26\u53c2\u6570\u53ef\u5fae\u51fd\u6570<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u7ed9\u51fa\u4ece<\/span><\/span><\/span><\/span><\/sub>\u81f3<\/span><\/span><\/span><\/span><\/sub>\u7684\u5e26\u53c2\u6570\u53ef\u5fae\u51fd\u6570(P,I)<\/span><\/span><\/span> \u548c<\/span><\/span><\/span>(P\u2032,I\u2032)<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u82e5\u6709\u53ef\u5fae\u51fd\u6570 f : P \u2192P\u2032 \u4f7f\u5f97 <\/span><\/span><\/span>\u2200<\/span><\/span><\/span>p<\/span><\/sub>\u2208P, a\u2208<\/span><\/span><\/span><\/span><\/sub>\u6709I\u2032(f(p),a) =I(p,a) <\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u4fbf\u8bb0 (P,I) <\/span><\/span><\/span>\u223c<\/span><\/span><\/span> (P\u2032,I\u2032)\u3002<\/span><\/span><\/span>\u223c<\/span><\/span><\/span>\u662f\u7b49\u4ef7\u5173\u7cfb\u3002<\/span><\/span><\/span><\/section>\n
    \u5b9a\u4e49<\/span><\/span><\/span>\u8303<\/span><\/span><\/span>Para<\/span><\/span><\/span>\u7684\u5bf9\u8c61\u662f<\/span><\/span><\/span><\/span><\/sub>, \u4ece<\/span><\/span><\/span><\/span><\/sub>\u81f3<\/span><\/span><\/span><\/span><\/sub>\u7684\u6001\u5c04\u662f\u4ece<\/span><\/span><\/span><\/span><\/sub>\u81f3<\/span><\/span><\/span><\/span><\/sub>\u7684\u5e36\u53c2\u6570\u53ef\u5fae\u51fd\u6570\u7684\u7b49\u4ef7\u7c7b<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u53d6<\/span><\/span><\/span><\/span><\/sub>\u3002<\/span><\/span><\/span><\/section>\n
    \u8bbek=m\u5219\u5408\u6210(Q,J)<\/span><\/span><\/span>\u25e6<\/span><\/span><\/span>(P,I) <\/span><\/span><\/span>\u662f (P\u00d7Q,I<\/span><\/span><\/span>\u2217<\/span><\/span><\/span>J) \uff0c<\/span><\/span><\/span>\u5176\u4e2d<\/span><\/span><\/span>(I<\/span><\/span><\/span>\u2217<\/span><\/span><\/span>J)(p,q,a) = J(q,I(p,a))\u3002<\/span><\/span><\/span><\/section>\n
    \u5b9a\u4e49(P,I)<\/span><\/span><\/span>\u2297<\/span><\/span><\/span>(Q,J) \u4e3a (P\u00d7Q,I\u2225J) <\/span><\/span><\/span>,<\/span><\/span><\/span>\u5176\u4e2d (I \u2225 J)(p,q,a,c) = (I(p,a),J(q,c)). <\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    6. <\/span>\u4eba\u5de5\u667a\u80fd\u4e0e\u8303\u7574\u5b66\u7684\u7b2c\u4e00\u4e2a\u4f8b\u5b50\uff1a<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n

    \u795e\u7ecf\u7f51\u7edc<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u63a5\u4e0b\u6765\u6211\u4eec<\/span><\/span><\/span>\u8bf4\u660e\u600e\u6837\u628a\u795e\u7ecf\u7f51\u7edc\u653e\u5728\u4e00\u8d77\u6210\u4e3a\u4e00\u4e2a<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>\uff0c\u8fd9<\/span><\/span><\/span>\u662f\u6211\u4eec<\/span><\/span><\/span>\u8ba8\u8bba\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u4e0e<\/span><\/span><\/span>\u8303\u7574\u5b66<\/span><\/span><\/span>\u7684\u7b2c\u4e00\u4e2a\u4f8b\u5b50\u3002<\/span><\/span><\/span><\/section>\n
    \u4eba\u8111\u7531<\/span><\/span><\/span>860\u4ebf\u4e2a\u795e\u7ecf\u5143<\/span> (neuron) <\/span>\u7ec4\u6210, \u8fd9\u4e9b\u795e\u7ecf\u5143\u7531100\u4e07\u4ebf\u4e2a\u7a81\u89e6 <\/span>(synapse) <\/span><\/span><\/span><\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u8054\u7edc\u8d77\u6765\u3002<\/span><\/span><\/span>\u4ee5\u4e0b\u662f\u4e00\u5f20\u7f8e\u4e3d\u7684\u663e\u5fae\u955c\u4e0b\u7684\u4eba\u8111\u795e\u7ecf\u7f51<\/span><\/span><\/span>\u7edc<\/span><\/span><\/span>\u56fe<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u4e0b\u56fe\u5c55\u793a\u4e24\u4e2a<\/span><\/span><\/span>\u795e\u7ecf\u5143<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u7a81\u89e6<\/span><\/span><\/span><\/span>\u8054\u7cfb<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u6211\u4eec\u5047\u8bbe<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd<\/span><\/span><\/span>\u662f\u7528\u8ba1\u7b97\u673a\u5efa\u6a21\u6765\u6a21\u4eff\u4eba\u8111\u3002<\/span><\/span><\/span>\u6211\u4eec\u7b80\u5316\u5b9e\u51b5<\/span><\/span><\/span>\u2014\u2014<\/span><\/span><\/span>\u628a\u795e\u7ecf\u5143\u5b89\u6392\u5728\u4e00\u7cfb\u5217\u7684<\/span><\/span><\/span>\u9762\u677f <\/span><\/span><\/span>(panel)<\/span>\uff0c<\/span><\/span><\/span>\u6211\u4eec\u7528\u7ebf\u6761\u4ee3\u8868<\/span><\/span><\/span>\u7a81\u8d77<\/span>\uff08\u5305\u62ec\u8f74\u7a81\u548c\u6811\u7a81\uff09<\/span><\/span><\/span><\/span>\u4ee5\u663e\u793a\u4e24\u4e2a\u76f8\u7ee7\u7684\u9762\u677f\u7684\u795e\u7ecf\u5143\u7684\u8054\u7edc\u3002\u6211\u4eec\u79f0\u4e24\u4e2a\u9762\u677f\u4e4b\u95f4\u7684\u6240\u6709\u8054\u7edc\u7ebf\u6761\u4e3a\u795e\u7ecf\u7f51\u7edc\u7684<\/span><\/span><\/span>\u8054\u7edc\u5c42<\/span> <\/span><\/span><\/span>(layer)<\/span>\uff0c<\/span><\/span><\/span> \u5c31\u662f\u8bf4\u4ece\u4e00\u4e2a\u9762\u677f\u4e0a\u7684\u795e\u7ecf\u5143\u5230\u53e6\u4e00\u9762\u677f\u4e0a\u6240\u6709\u7684\u795e\u7ecf\u5143\u7684\u7a81\u8d77<\/span><\/span><\/span><\/span>\u7ec4\u6210\u4e24\u4e2a\u9762\u677f\u4e4b\u95f4\u7684\u4e00\u4e2a\u8054\u7edc\u5c42<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u6211\u4eec\u7528\u4e00\u4e2a\u96c6C={\uff08i\uff0cj<\/span><\/span><\/span>\uff09<\/span><\/span><\/span>: <\/span><\/span><\/span>1 \u2264i \u2264m<\/span><\/span><\/span>, <\/span><\/span><\/span>1 \u2264j \u2264n<\/span><\/span><\/span>}\u6765\u8bb0\u5f55\u4e00\u4e2a(m,n)-<\/span><\/span><\/span>\u578b<\/span><\/span><\/span>\u8054\u7edc\u5c42<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>\u8fd9\u662f\u8bf4\u6709m\u4e2a\u795e\u7ecf\u5143<\/span><\/span><\/span>\u5728\u7b2c\u4e00\u4e2a<\/span><\/span><\/span>\u9762\u677f<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>n<\/span><\/span><\/span>\u4e2a\u795e\u7ecf\u5143<\/span><\/span><\/span>\u5728\u7b2c\u4e8c\u4e2a<\/span><\/span><\/span>\u9762\u677f<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u82e5\u4e00\u5bf9\u6574\u6570\uff08i\uff0cj\uff09\u5728\u96c6C\u51fa\u73b0<\/span><\/span><\/span>\u662f\u6307\u5de6\u8fb9\u9762\u677f\u7b2c<\/span><\/span><\/span>i<\/span><\/span><\/span>\u4e2a\u795e\u7ecf\u5143\u662f<\/span><\/span><\/span>\u8054\u7edc\u81f3\u53f3\u8fb9\u9762\u677f\u7b2cj<\/span><\/span><\/span>\u4e2a\u795e\u7ecf\u5143\u3002<\/span><\/span><\/span>\u4ee5\u4e0b\u662f\u4e00\u4e2a\u7b80\u5358\u7684\u4f8b\u5b50<\/span><\/span><\/span>\u2014\u2014<\/span><\/span><\/span>\u4e00\u4e2a(4,3)-<\/span><\/span><\/span>\u578b<\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edc\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u6211\u4eec\u63a8\u5e7f\u4e00\u70b9<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u5b9a\u4e49<\/span><\/span><\/span>(<\/span><\/span><\/span>n<\/span><\/span><\/span>0<\/span><\/span><\/sub>,<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span>k<\/span><\/span><\/sub>)-\u578b<\/span>(\u524d\u9988)<\/span>\u795e\u7ecf\u7f51\u7edcN<\/span><\/span><\/span>\u662f\u6709k+1\u4e2a<\/span><\/span><\/span>\u9762\u677f<\/span><\/span><\/span>\u6392\u8d77\u6765, \u7b2ci\u4e2a<\/span><\/span><\/span>\u9762\u677f<\/span>\u5230<\/span><\/span><\/span>\u7b2ci+1\u4e2a<\/span><\/span><\/span>\u9762\u677f<\/span>\u662f\u4e00\u4e2a<\/span><\/span><\/span>(<\/span><\/span><\/span>n<\/span><\/span><\/span>i<\/span><\/span><\/sub>,<\/span><\/span><\/span><\/span>n<\/span><\/span><\/span>i+1<\/span><\/span><\/sub>)<\/span><\/span><\/span>–<\/span><\/span><\/span>\u578b<\/span><\/span><\/span>\u8054\u7edc\u5c42<\/span><\/span><\/span>C<\/span><\/span><\/span>i<\/span><\/span><\/sub>\u3002<\/span><\/span><\/span>\u6211\u4eec\u4ea6\u53ef\u4ee5\u8bf4<\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edcN<\/span><\/span><\/span>\u662f<\/span><\/span><\/span>C<\/span><\/span><\/span>1<\/span><\/span><\/sub>\uff0c <\/span><\/span><\/span>C<\/span><\/span><\/span>2<\/span><\/span><\/sub>\uff0c\u2026<\/span><\/span><\/span>, <\/span><\/span><\/span>C<\/span><\/span><\/span>k<\/span><\/span><\/sub>-\u578b\u7684<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u5b9a\u4e49<\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edc\u8303\u7574<\/span><\/span><\/span>NNe<\/span><\/span><\/span>t<\/span><\/span><\/span>\u7684\u5bf9\u8c61\u4e3a\u81ea\u7136\u6570,<\/span><\/span><\/span> \u6001\u5c04<\/span><\/span><\/span>$Hom (m,n)<\/span><\/span><\/span>\u4e3a\u6240\u6709<\/span><\/span><\/span>(m,n)-\u578b\u795e\u7ecf\u7f51\u7edc\u3002<\/span><\/span><\/span>\u7528\u62fc\u63a5<\/span> <\/span><\/span><\/span>(concatenation) <\/span>\u5b9a\u4e49\u7f51\u7edc\u7684\u5408\u6210<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u6052\u7b49\u6001\u5c04\u662f<\/span><\/span><\/span>0<\/span><\/span><\/span>\u5c42\u795e\u7ecf\u7f51\u7edc<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u8fd9\u6837\u6211\u4eec\u8fbe\u5230\u4e86\u7b2c\u4e00\u4e2a\u76ee\u6807<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edc\u653e\u5728\u4e00\u8d77\u662f\u4e00\u4e2a<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    7. <\/span>\u600e\u6837\u5b66\u4e60\u7684\uff1f <\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u4e00\u4e2a<\/span><\/span><\/span>\u9762\u677f<\/span><\/span><\/span>\u7684\u795e\u7ecf\u5143\u600e\u6837\u8f93\u51fa\u4fe1\u606f<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>\u6211<\/span><\/span><\/span>\u4eec<\/span><\/span><\/span>\u5047\u8bbe\u7b2ci\u4e2a\u795e\u7ecf\u5143\u7684<\/span><\/span><\/span>\u7535\u52bf<\/span><\/span><\/span>\u662f<\/span><\/span><\/span>x<\/span><\/span><\/span>i<\/span><\/span><\/sub>\uff0c<\/span><\/span><\/span>\u800c<\/span><\/span><\/span>\u8f93\u51fa\u5f3a\u5ea6\u662f<\/span><\/span><\/span>x<\/span><\/span><\/span>i<\/span><\/span><\/sub>\u7684<\/span><\/span><\/span>w<\/span><\/span><\/span>i<\/span><\/span><\/sub>\u500d<\/span> (\u79f0<\/span><\/span><\/span>w<\/span><\/span><\/span>i<\/span><\/span><\/span>\u4e3a\u6743<\/span><\/span><\/span>)<\/span>, \u4e8e\u662f\u603b<\/span><\/span><\/span>\u8f93\u51fa<\/span><\/span><\/span>\u7535\u52bf<\/span><\/span><\/span>\u662fnet=<\/span><\/span><\/span>\u2211<\/span><\/span><\/span>w<\/span><\/span><\/span>i<\/span><\/span><\/sub>x<\/span><\/span><\/span>i<\/span><\/span><\/sub><\/span>\uff0c<\/span><\/span><\/span>\u7136\u540e<\/span><\/span><\/span>\u8bbe<\/span><\/span><\/span>\u03c3<\/span><\/span><\/span>\u662f\u4e2a\u7ba1\u7ebf\u8def\u5f00\u5173\u7684\u51fd\u6570\uff0c\u5373\u5f53<\/span><\/span><\/span>\u7535\u52bf<\/span><\/span><\/span>\u8fbe\u5230\u67d0\u4e2a\u6570\u5b57\u4fbf\u6253\u5f00<\/span><\/span><\/span>\u7ebf\u8def\uff0c\u628a\u4fe1\u53f7<\/span><\/span><\/span>\u8f93\u51fa\u5230\u4e0b\u4e00\u4e2a<\/span><\/span><\/span>\u9762\u677f<\/span><\/span><\/span>\u7684\u67d0\u4e00\u4e2a\u795e\u7ecf\u5143\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u628a\u8fd9\u4e2a\u60f3\u6cd5\u63a8\u5e7f<\/span><\/span><\/span>\u4e00\u4e0b\uff0c<\/span><\/span><\/span>\u4e0b\u56fe\u662f\u4e09<\/span><\/span><\/span>\u5757<\/span><\/span><\/span>\u9762\u677f\u4e24\u5c42\u7684\u7f51\u7edc, \u8fd9\u65f6\u6bcf\u4e00\u5c42\u7684\u6743\u4fbf\u5199\u4e3a\u77e9\u9635(<\/span><\/span><\/span>w<\/span><\/span><\/span>ij<\/span><\/span><\/sub>)\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n
    \u53d6\u53ef\u5fae\u51fd\u6570<\/span><\/span><\/span><\/span><\/sub>\u3002\u5b9a\u4e49<\/span><\/span><\/span>I<\/span><\/span><\/span>\u03c3<\/span><\/span><\/sup>\u628a<\/span><\/span><\/span>1<\/span><\/span><\/span>-\u5c42\uff08<\/span><\/span><\/span>m<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>n<\/span><\/span><\/span>\uff09<\/span><\/span><\/span>-\u578b<\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edcC\u6620\u4e3a\u5e26\u53c2\u6570\u53ef\u5fae\u51fd\u6570<\/span><\/span><\/span> : <\/span><\/span><\/span><\/section>\n
    <\/section>\n
    \u5176\u4e2d<\/span><\/span><\/span>W\u662fn<\/span><\/span><\/span>\u00d7<\/span><\/span><\/span>m\u77e9\u9635, <\/span><\/span><\/span><\/span><\/sub>\uff0c\u5411\u91cfa\u662f\u5728<\/span><\/span><\/span><\/span><\/sub>,  \u5411\u91cfx\u662f\u5728<\/span><\/span><\/span><\/span><\/sub>\u3002\u82e5\u795e\u7ecf\u7f51\u7edc<\/span><\/span><\/span>\u662f<\/span><\/span><\/span>C<\/span><\/span><\/span>1<\/span><\/span><\/sub>, <\/span><\/span><\/span>\u22ef,<\/span><\/span><\/span>C<\/span><\/span><\/span>k<\/span><\/span><\/sub>–<\/span><\/span><\/span>\u578b\u7684<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u5219\u8bbe<\/span><\/span><\/span><\/section>\n
    <\/section>\n
    \u5b9a\u7406<\/span><\/span><\/span>.<\/span><\/span><\/span><\/span>I<\/span><\/span><\/span>\u03c3<\/span><\/span><\/sup> : <\/span><\/span><\/span>NNet<\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span><\/span>Para<\/span><\/span><\/span>\u662f\u51fd\u5b50\u3002<\/span><\/span><\/span><\/span><\/section>\n
    \u6211\u4eec\u53ef\u4ee5\u6784\u9020\u4e00\u4e2a<\/span><\/span><\/span>\u8303\u7574<\/span><\/span><\/span>Learn<\/span><\/span><\/span>\u4f7f\u5f97<\/span><\/span><\/span>Learn<\/span><\/span><\/span> \u7684\u5bf9\u8c61<\/span><\/span><\/span>\u662f\u96c6<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u6001\u5c04<\/span>\u662f<\/span><\/span><\/span>\u5b66\u4e60\u7b97\u6cd5<\/span><\/span><\/span>\u7b49\u4ef7\u7c7b<\/span><\/span><\/span>; \u5e76\u4e14\u6784\u9020\u4e00\u4e2a<\/span><\/span><\/span>\u51fd\u5b50L:<\/span><\/span><\/span><\/span>Para<\/span><\/span><\/span>\u2192<\/span><\/span><\/span>Learn<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u4e8e\u662f<\/span><\/span><\/span>\u5f97<\/span><\/span><\/span>\u51fd\u5b50<\/span><\/span><\/span>L\u2218<\/span><\/span><\/span>I<\/span><\/span><\/span>\u03c3<\/span><\/span><\/sup> : <\/span><\/span><\/span>NNet<\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span>Learn<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u53ef\u4ee5\u5229\u7528\u8fd9\u4e2a\u51fd\u5b50\u7684\u6027\u8d28\u53bb\u4e86\u89e3<\/span><\/span><\/span>\u5b66\u4e60\u7b97\u6cd5<\/span><\/span><\/span>\u7684\u4ee3\u6570\u7ed3\u6784\u3002<\/span><\/span><\/span>\u4ee5\u4e0a\u6765\u81eaMIT<\/span><\/span><\/span>\u8ba1\u7b97\u673a\u7cfb<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u6587\u7ae0<\/span><\/span><\/span>[13]<\/span><\/sup><\/strong><\/span>\u3002 <\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    8. <\/span>\u4eba\u5de5\u667a\u80fd\u4e0e\u8303\u7574\u5b66\u7684\u7b2c\u4e8c\u4e2a\u4f8b\u5b50\uff1a\u4fe1\u606f<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u4ee5<\/span><\/span><\/span>T\u8bb0\u7edd\u5bf9\u6e29\u5ea6<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>Q\u8bb0\u70ed\u91cf<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u70ed\u52a8\u529b\u5b66<\/span><\/span><\/span>\u7528\u4ee5\u4e0b\u516c\u5f0f<\/span><\/span><\/span>\u5f15\u5165\u51fd\u6570\u71b5(entropy)S<\/span><\/span><\/span>:<\/span><\/span><\/span>dS =<\/span><\/span><\/span>dQ\/<\/span><\/span><\/span>T<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u501f\u7528\u5230\u4fe1\u606f\u8bba<\/span><\/span><\/span>: \u4fe1\u606f\u7684\u589e\u91cf=\u4e0d\u786e\u5b9a\u6027\u7684<\/span><\/span><\/span>\u51cf<\/span><\/span><\/span>\u91cf<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    (1) \u5b9a\u4e49\u4e8b\u4ef6A\u7684\u4fe1\u606f\u5167\u5bb9\u4e3aI(A) :=-<\/span><\/span><\/span>l<\/span><\/span><\/span>og P(A), <\/span><\/span><\/span>\u5176\u4e2d<\/span><\/span><\/span>P\u4e3a\u6982\u7387<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    (2) \u6982\u7387\u7a7a\u95f4\u7684\u6709\u9650\u53ef\u6d4b\u5206\u5272 <\/span><\/span><\/span>\u03b1<\/span><\/span><\/span>= {<\/span><\/span><\/span>A<\/span><\/span><\/span>1<\/span><\/span><\/sub>, <\/span><\/span><\/span>\u22ef<\/span><\/span><\/span>, <\/span><\/span><\/span>A<\/span><\/span><\/span>N<\/span><\/span><\/sub> }<\/span><\/span><\/span><\/span>\u7684\u4fe1\u606f\u5167\u5bb9\u662f<\/span><\/span><\/span><\/span><\/sub>\uff0c<\/span><\/span><\/span>\u5176\u4e2d<\/span><\/span><\/span>1<\/span><\/span><\/span>A<\/span><\/span><\/sub>\u662fA\u7684\u7279\u5f81\u51fd\u6570<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    (3) \u6709\u9650\u53ef\u6d4b\u5206\u5272<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span>\u7684<\/span><\/span><\/span>\u71b5<\/span><\/span><\/span>\u5b9a\u4e49\u4e3a<\/span><\/span><\/span>\u03b1<\/span><\/span><\/span>\u7684\u4fe1\u606f\u5167\u5bb9\u7684\u5e73\u5747\u503c<\/span><\/span><\/span>(<\/span><\/span><\/span>[10]<\/span><\/span><\/span> 2.1\u8282)<\/span><\/span><\/span><\/section>\n
    <\/section>\n
    \u73b0\u4ee3\u4fe1\u606f\u8bba\u662f\u4ece<\/span><\/span><\/span>Shannon 1948\u5e74Bell Journal<\/span><\/span><\/span>[28]<\/span><\/sup><\/strong><\/span>\u53caCarnap\u548cBar-Hillel 1952\u5e74MIT Electronics Laboratory\u7684\u6587\u7ae0<\/span><\/span><\/span>[9]<\/span><\/sup><\/strong><\/span>\u5f00\u59cb<\/span><\/span><\/span>\u3002\u4e4b\u540e\u7684\u53d1\u5c55\u4e3b\u8981\u662f\u5728\u5e94\u7528<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u52302015\u5e74\u62d3\u6251\u8fdb\u5165\u4fe1\u606f\u8bba, Baudot\u548cBennequin<\/span><\/span><\/span>[5]<\/span><\/sup><\/strong><\/span>\u6307\u51fa\u4fe1\u606f\u8bba\u7684\u71b5\u662f\u4e0a\u540c\u8c03\u7c7b<\/span><\/span><\/span>, \u8fd9\u662f\u4fe1\u606f\u8bba\u7684\u4e00\u4e2a\u5212\u65f6\u4ee3\u7684\u66f4\u65b0\u3002\u5230\u4e86<\/span><\/span><\/span>2020\u5e74Vigneaux<\/span><\/span><\/span>[30]<\/span><\/sup><\/strong><\/span>\u63d0\u51fa\u4fe1\u606f\u7ed3\u6784<\/span> (information structures) <\/span>\u7edf\u4e00\u5404\u4e2a\u89c2\u70b9<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u6211\u4eec\u8303\u7574<\/span><\/span><\/span>S\u7684\u5bf9\u8c61<\/span><\/span><\/span>\u4e3a\u5b9a\u4e49\u5728<\/span><\/span><\/span>\u6709\u9650\u6982\u7387\u7a7a\u95f4<\/span><\/span><\/span>\u4e0a\u7684\u5b9e\u6570\u503c<\/span><\/span><\/span>\u968f\u673a\u53d8\u5143, <\/span><\/span><\/span>\u4ee5<\/span><\/span><\/span>M<\/span><\/span><\/span>X<\/span><\/span><\/sub>\u8bb0X<\/span><\/span><\/span>\u7684\u8c61<\/span><\/span><\/span>\u3002\u8981\u6c42\u8303\u7574<\/span><\/span><\/span>S\u7684\u6001\u5c04<\/span><\/span><\/span>\u03c0<\/span><\/span><\/span>: X<\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span> Y\u6ee1\u8db3\u6761\u4ef6:<\/span><\/span><\/span><\/span>M<\/span><\/span><\/span>X<\/span><\/span><\/sub>\u662f<\/span><\/span><\/span>M<\/span><\/span><\/span>Y<\/span><\/span><\/sub>\u7684\u5b50\u96c6<\/span><\/span><\/span>\u3002\u4ee5<\/span><\/span><\/span>F\u8bb0\u6709\u9650\u96c6\u8303\u7574<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u5219\u6709<\/span><\/span><\/span>\u51fd\u5b50M : S<\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span>F\u628aX\u6620\u4e3a<\/span><\/span><\/span>M<\/span><\/span><\/span>X<\/span><\/span><\/sub>; \u628a\u6001\u5c04<\/span><\/span><\/span>\u03c0<\/span><\/span><\/span>: X<\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span> Y<\/span><\/span><\/span>\u6620\u4e3a\u5355\u5c04<\/span><\/span><\/span>M(<\/span><\/span><\/span>\u03c0<\/span><\/span><\/span>) : <\/span><\/span><\/span>M<\/span><\/span><\/span>X<\/span><\/span><\/sub>\u2192<\/span><\/span><\/span>M<\/span><\/span><\/span>Y<\/span><\/span><\/sub>\u3002<\/span><\/span><\/span>\u79f0<\/span><\/span><\/span>(S, M)\u4e3a<\/span><\/span><\/span>\u4e00\u4e2a<\/span><\/span><\/span>\u4fe1\u606f\u7ed3\u6784<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u5b9a\u4e49\u8303\u7574<\/span><\/span><\/span>InfoStr<\/span><\/span><\/span>\u7684\u5bf9\u8c61\u4e3a\u4fe1\u606f\u7ed3\u6784(S, M);<\/span><\/span><\/span><\/span>\u6001\u5c04\u4e3a<\/span><\/span><\/span>\u03c6=(<\/span><\/span><\/span>\u03c6<\/span><\/span><\/span>0<\/span><\/span><\/sub>,<\/span><\/span><\/span>\u03c6<\/span><\/span><\/span>1<\/span><\/span><\/sub>)<\/span><\/span><\/span>: (S, M) <\/span><\/span><\/span>\u2192<\/span><\/span><\/span> (S’, M’)\u5176\u4e2d<\/span><\/span><\/span>\u03c6<\/span><\/span><\/span>0<\/span><\/span><\/sub> : S <\/span><\/span><\/span>\u2192<\/span><\/span><\/span>S’,<\/span><\/span><\/span><\/span>\u03c6<\/span><\/span><\/span>1<\/span><\/span><\/sub> : <\/span><\/span><\/span>M <\/span><\/span><\/span>\u2192<\/span><\/span><\/span> M’ <\/span><\/span><\/span>\u2218<\/span><\/span><\/span>\u03c6<\/span><\/span><\/span>0<\/span><\/span><\/sub>\u3002<\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    9. <\/span>\u56de\u5230\u795e\u7ecf\u7f51\u7edc<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    (1) \u751f\u7406\u795e\u7ecf\u5143\u5728\u63a5\u6536\u8f93\u5165<\/span>\uff08\u5149\u6216\u5916\u90e8\u7535\u8f93\u5165\u7b49\u611f\u5b98\u523a\u6fc0\uff09<\/span>\u6216\u88ab\u7ec6\u83cc\u653b\u51fb\u65f6\u4f1a\u88ab\u6fc0\u6d3b\u800c\u4ea7\u751f\u7535<\/span><\/span><\/span>\u52bf\u3002<\/span><\/span><\/span><\/section>\n
    (2)<\/span><\/span><\/span><\/span>\u795e\u7ecf\u901a\u8baf<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>\u795e\u7ecf\u5143\u4e4b\u95f4\u7684\u901a\u8baf\u901a\u8fc7\u7535\u4fe1\u53f7\u548c\u5316\u5b66\u4fe1\u53f7\u8fdb\u884c\u3002\u5f53\u4e00\u4e2a\u795e\u7ecf\u5143\u63a5\u6536\u5230\u6765\u81ea\u5176\u4ed6\u795e\u7ecf\u5143\u7684\u4fe1\u53f7\u65f6\uff0c\u5b83\u4f1a\u5728\u7ec6\u80de\u4f53\u5185\u5904\u7406\u8fd9\u4e9b\u4fe1\u606f\uff0c\u5982\u679c\u7d2f\u79ef\u7684\u8f93\u5165\u8db3\u591f\u5f3a\uff0c\u5c31\u4f1a\u4ea7\u751f\u4e00\u79cd\u79f0\u4e3a<\/span><\/span><\/span>\u52a8\u4f5c\u7535\u52bf\u7684\u7535\u8109\u51b2\u3002<\/span><\/span><\/span><\/section>\n
    (3) \u5982\u679c\u6211\u4eec\u5728\u795e\u7ecf\u7f51\u7edc\u52a0\u5165\u795e\u7ecf\u5143<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u7528\u53d1\u5c04\u8109\u51b2<\/span> <\/span><\/span><\/span>(spike) <\/span><\/span><\/span>\u5e8f\u5217\u4f20\u8f93\u4fe1\u606f<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u6211\u4eec\u5f97\u5230SN<\/span><\/span><\/span>N <\/span>(spiking neural network) <\/span><\/span><\/span>[26]<\/span><\/sup><\/strong><\/span>\uff08\u5982<\/span><\/span><\/span>\u4e0b\u56fe<\/span><\/span><\/span>\uff09<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n<\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    10. \u7535\u7801<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u795e\u7ecf\u5143\u5728\u7279\u5b9a\u65f6\u95f4\u95f4\u9694\u5185\u8981\u4e48\u6d3b\u8dc3, \u5373<\/span><\/span><\/span>\u6709\u4e00\u4e2a\u8109\u51b2<\/span><\/span><\/span>\uff1b<\/span><\/span><\/span>\u8981\u4e48\u4e0d\u6d3b\u8dc3, \u5373\u6ca1<\/span><\/span><\/span>\u6709\u8109\u51b2<\/span><\/span><\/span>\u3002\u5047\u8bbe\u6211\u4eec\u628a\u89c2\u5bdf\u65f6\u6bb5<\/span><\/span><\/span>\u5206\u4e3an\u6bb5,<\/span><\/span><\/span><\/span>\u4f7f\u5f97\u6bcf\u4e2a\u5c0f\u65f6\u6bb5\u6700\u591a\u53ea\u6709\u4e00\u4e2a\u8109\u51b2<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u82e5\u7b2ci\u4e2a\u5c0f\u65f6\u6bb5\u6709\u8109\u51b2\u6211\u4eec\u8bb01<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u6ca1\u6709\u5219\u8bb0<\/span><\/span><\/span>0, \u4e8e\u662f<\/span><\/span><\/span>\u6211\u4eec\u5f97\u4e00\u4e2a<\/span><\/span><\/span>n\u4f4d\u4e8c\u8fdb\u6570, \u4f8b\u59821011<\/span><\/span><\/span>\u3002\u5047\u5982\u6211\u4eec\u505a<\/span><\/span><\/span>k\u4e2a\u89c2\u5bdf\u4fbf\u751f\u6210\u4e00\u4e2a\u4e8c\u8fdb<\/span><\/span><\/span>\u7801<\/span><\/span><\/span>C<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u4eca\u5e74\u534e\u4e3a\u5df2\u5f00\u59cb\u751f\u4ea73\u8fdb\u5236AI\u82af\u7247<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u800c\u91cf\u5b50\u7801\u662f<\/span><\/span><\/span>p<\/span><\/span><\/span>m<\/span><\/span><\/sup>\u8fdb\u5236(p\u662f\u7d20\u6570<\/span><\/span><\/span>) (<\/span><\/span><\/span>[33][34]<\/span><\/span><\/span>)\u3002\u5230\u4eca\u5929\u8fd8\u672a\u89c1\u533b\u5b66\u6216\u751f\u7269\u5b66\u7684\u7406\u8bba\u8bf4\u4eba\u8111\u4e0d\u53ef\u80fd\u505a\u91cf\u5b50\u8ba1\u7b97\u3002<\/span><\/span><\/span><\/section>\n
    \u8bbeF\u662f\u6709<\/span><\/span><\/span>p<\/span><\/span><\/span>m<\/span><\/span><\/sup>\u4e2a\u5143\u7d20\u7684\u57df<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u4e00\u4e2a<\/span><\/span><\/span>[n,k]<\/span><\/span><\/span>p<\/span><\/span><\/span>m<\/span><\/span><\/sup>\u7801\u662f\u6307<\/span><\/span><\/span>F<\/span><\/span><\/span>n<\/span><\/span><\/sup>\u7684\u6709<\/span><\/span><\/span>p<\/span><\/span><\/span>mk<\/span><\/span><\/sup>\u4e2a\u5143\u7d20\u7684<\/span><\/span><\/span>\u5b50\u96c6C\u3002\u4ee5<\/span><\/span><\/span>d<\/span><\/span><\/span>C<\/span><\/span><\/sub>(c, c<\/span><\/span><\/span>\u2019<\/span><\/span><\/span>)\u8bb0\u96c6{1 <\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>i\u2264<\/span><\/span><\/span> n : <\/span><\/span><\/span>c<\/span><\/span><\/span>i<\/span><\/span><\/sub>\u4e0d\u7b49\u4e8e<\/span><\/span><\/span>c’<\/span><\/span><\/span>i<\/span><\/span><\/sub>} \u7684\u5143\u7d20\u4e2a\u6570\u3002<\/span><\/span><\/span><\/section>\n
    \u5b9a\u4e49\u8303\u7574<\/span><\/span><\/span>Codes<\/span><\/span><\/span>(n)\u7684\u5bf9\u8c61\u4e3a<\/span><\/span><\/span><\/span><\/sub><\/span>\u7801<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u6001\u5c04\u662f\u6620\u5c04<\/span><\/span><\/span>\u03c6<\/span><\/span><\/span> : C<\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span> C’\u4f7f\u5f97<\/span><\/span><\/span>\u03c6(<\/span><\/span><\/span>C) <\/span><\/span><\/span>\u662f<\/span><\/span><\/span> C’<\/span><\/span><\/span>\u7684\u5b50\u96c6<\/span><\/span><\/span>\u548c<\/span><\/span><\/span>d<\/span><\/span><\/span>C’<\/span><\/span><\/sub>( <\/span><\/span><\/span>\u03c6<\/span><\/span><\/span>c<\/span><\/sub>, <\/span><\/span><\/span>\u03c6<\/span><\/span><\/span>c<\/span><\/span><\/span>\u2019<\/span><\/span><\/span><\/sub>) <\/span><\/span><\/span>\u2264<\/span><\/span><\/span><\/span>d<\/span><\/span><\/span>C<\/span><\/span><\/sub>(c, c<\/span><\/span><\/span>\u2019<\/span><\/span><\/span>)\u3002<\/span><\/span><\/span><\/section>\n
    \u8bbeC\u662f<\/span><\/span><\/span><\/span><\/sub><\/span>\u7801<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u53d6<\/span><\/span><\/span>I<\/span><\/span><\/span>(C )\u4e3a\u968f\u673a\u53d8\u5143X : C <\/span><\/span><\/span>\u2192<\/span><\/span><\/span>R<\/span><\/span><\/span>\u7ec4\u6210\u7684\u96c6\u3002<\/span><\/span><\/span>\u53d6<\/span><\/span><\/span>Codes<\/span><\/span><\/span>(n)<\/span><\/span><\/span>\u7684\u6001\u5c04<\/span><\/span><\/span>\u03c6<\/span><\/span><\/span> : C<\/span><\/span><\/span><\/span>\u2192<\/span><\/span><\/span> C’<\/span><\/span><\/span>, \u82e5X<\/span><\/span><\/span>\u2019<\/span><\/span><\/span> \u5c5e\u4e8e<\/span><\/span><\/span>I<\/span><\/span><\/span>(C <\/span><\/span><\/span>\u2018<\/span><\/span><\/span>)\u5219\u53d6<\/span><\/span><\/span>I<\/span><\/span><\/span>(<\/span><\/span><\/span>\u03c6<\/span><\/span><\/span> )(X<\/span><\/span><\/span>\u2019<\/span><\/span><\/span>) = X<\/span><\/span><\/span>\u2019<\/span><\/span><\/span><\/span>\u00b0 \u03c6<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u5b9a\u7406. <\/span><\/span><\/span><\/span>: <\/span><\/span><\/span>Codes<\/span><\/span><\/span>(<\/span><\/span><\/span>n<\/span><\/span><\/span>)<\/span><\/span><\/span>\u2192<\/span><\/span><\/span>InfoStr<\/span><\/span><\/span><\/span>\u662f\u51fd\u5b50<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u8fd9\u662f\u5fb7\u56fdBonn Max Planck Institute\u7684\u5de5\u4f5c<\/span><\/span><\/span>[23]<\/span><\/sup><\/strong><\/span>\u3002<\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    11. <\/span>\u4eba\u5de5\u667a\u80fd\u4e0e\u8303\u7574\u5b66\u7684\u7b2c\u4e09\u4e2a\u4f8b\u5b50\uff1aTopos<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u4e00\u4e2a<\/span><\/span><\/span>Topos<\/span><\/span><\/span>\u662f\u4e00\u4e2a\u6709\u4e30\u5bcc\u7ed3\u6784\u7684\u8303\u7574\u3002\u4e00\u76f4\u4ee5\u6765<\/span><\/span><\/span>Topos<\/span><\/span><\/span>\u7406\u8bba\u53ea\u662f\u5c0f\u90e8\u4efd\u7684\u4ee3\u6570\u51e0\u4f55\u5b66<\/span>\u5bb6<\/span><\/span><\/span>(<\/span><\/span><\/span>[35][37]<\/span><\/span><\/span>)<\/span>\u548c\u51e0\u4f55\u903b\u8f91\u5b66\u5bb6\u7684\u5de5\u5177<\/span><\/span><\/span>(<\/span><\/span><\/span>[36]<\/span><\/span><\/span>)<\/span>\u3002\u4f46\u662f\u8fdb\u5165\u4e86<\/span><\/span><\/span>21\u4e16\u7eaa\u4fbf\u770b\u89c1<\/span><\/span><\/span>Topos<\/span><\/span><\/span>\u5728\u4fe1\u606f\u7cfb\u7edf\u63a7\u5236<\/span><\/span><\/span> (<\/span><\/span><\/span>[1]<\/span><\/span><\/span>)<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u7f51\u8def\u7ba1\u7406<\/span><\/span><\/span>(<\/span><\/span><\/span>[7]<\/span><\/span><\/span>)<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u795e\u7ecf\u7f51\u7edc<\/span><\/span><\/span>(<\/span><\/span><\/span>[4][5][19]<\/span><\/span><\/span>)<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>6G\u901a\u8baf<\/span><\/span><\/span>(<\/span><\/span><\/span>[3]<\/span><\/span><\/span>)<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u4eba<\/span><\/span><\/span>\u5de5\u667a\u80fd<\/span><\/span><\/span> (<\/span><\/span><\/span>[18]<\/span><\/span><\/span>)<\/span><\/span><\/span>\u548c\u91cf\u5b50<\/span><\/span><\/span>\u4eba<\/span><\/span><\/span>\u5de5\u667a\u80fd<\/span><\/span><\/span>(<\/span><\/span><\/span>[31]<\/span><\/span><\/span>)<\/span><\/span><\/span>\u7684\u5e94\u7528<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u6211\u4eec\u5047\u8bbe\u9762\u677f\u662f\u6309\u504f\u5e8f<\/span><\/span><\/span>\u2264<\/span><\/span><\/span>\u6392\u5217<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u6211\u4eec\u5f15\u5165\u56fe<\/span><\/span><\/span>\u0393<\/span><\/span><\/span>\u3002\u53d6<\/span><\/span><\/span>\u7f51\u7edc<\/span><\/span><\/span>N<\/span><\/span><\/span>\u7684\u9762\u677f\u4e3a<\/span><\/span><\/span>\u0393<\/span><\/span><\/span>\u7684\u9876\u70b9; \u82e5\u6709<\/span><\/span><\/span>a\u2264b<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u5219\u4ee5a<\/span><\/span><\/span>\u2192<\/span><\/span><\/span>b\u4e3a<\/span><\/span><\/span>\u0393<\/span><\/span><\/span>\u7684\u7bad<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u4ee5C(<\/span><\/span><\/span>\u0393<\/span><\/span><\/span>)\u8bb0<\/span><\/span><\/span>\u0393<\/span><\/span><\/span>\u7684\u8def\u8303\u7574; \u5728<\/span><\/span><\/span>C(\u0393)<\/span><\/span><\/span>op<\/span><\/span><\/sup>\u53d6Grothendieck\u62d3\u6251\u5f97\u4f4d\u5f62<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u5bf9b\u2264c, \u8bbe\u96c6<\/span><\/span><\/span>W<\/span><\/span><\/span>bc<\/span><\/span><\/sub>\u7684\u5143\u7d20\u4e3a\u4ece\u9762\u677fb\u4e0a\u7684\u795e\u7ecf\u5143\u81f3\u9762\u677fc\u4e0a\u7684\u795e\u7ecf\u5143\u7684\u7a81\u89e6<\/span><\/span><\/span><\/span>\u7684\u6743<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u53d6\u9876\u70b9a\uff0c\u8bbe<\/span><\/span><\/span>
    <\/span><\/section>\n
    <\/section>\n
    \u82e5a\u2192b\u662f<\/span><\/span><\/span>\u0393<\/span><\/span><\/span>\u7684\u7bad\uff0c\u53d6\u6295\u5c04<\/span><\/span><\/span>
    <\/span><\/section>\n
    <\/section>\n
    \u5219{<\/span><\/span><\/span>W<\/span><\/span>a<\/span><\/sub>,<\/span><\/span><\/span>W<\/span><\/span><\/span>a\u2192b<\/span><\/span><\/sub>} \u51b3\u5b9a\u51fd\u5b50 W: C(<\/span><\/span><\/span>\u0393<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>\u2192Set<\/span><\/span><\/span>\u3002<\/span><\/span><\/span>\u5e76\u4e14 W \u662f\u5c42, \u662f\u5c5e\u4e8e<\/span><\/span><\/span>C(\u0393)<\/span><\/span><\/span>op<\/span><\/span><\/sup>\u7684<\/span><\/span><\/span>Topos<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u5b9a\u7406<\/span><\/span><\/span>.<\/span><\/span><\/span><\/span>\u6bcf\u4e00\u4e2a<\/span><\/span><\/span>\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc(<\/span><\/span><\/span>DNN<\/span><\/span><\/span>)<\/span><\/span><\/span><\/span>N<\/span><\/span><\/span> \u6709\u4e00\u4e2a<\/span><\/span><\/span>Topos<\/span><\/span><\/span>T(N) \u4f7f\u5f97 N\u7684<\/span><\/span><\/span>\u52a8\u529b\u7cfb\u7edf\u662f\u5c5e\u4e8e<\/span><\/span><\/span>T(N)\u7684\u5c42(sheaf)<\/span><\/span><\/span>\u3002<\/span><\/span><\/span><\/section>\n
    \u4ee5\u4e0a\u6765\u81ea<\/span><\/span><\/span>\u534e\u4e3a<\/span><\/span><\/span>\u5df4\u9ece\u5b9e\u9a8c\u5ba4<\/span><\/span><\/span>Bennequin, Belfiore 2022<\/span><\/span><\/span>\u5e74\u7684\u5de5\u4f5c<\/span><\/span><\/span>[4]<\/span><\/sup><\/strong><\/span>\u3002<\/span><\/span><\/span><\/section>\n

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

    \n
    \n
    \n
    \n
    \n

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

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

    12. \u7ed3\u8bed<\/span><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

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

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


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

    \u4e3a\u4e86<\/span><\/span><\/span>\u9f13\u52b1<\/span><\/span><\/span>\u5b66\u751f\u5b66\u4e60<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u8001\u5e08\u5e38\u8bf4<\/span><\/span><\/span>\u201c<\/span><\/span><\/span>\u6570\u5b66\u5f88\u6709\u7528<\/span><\/span><\/span>\u201d<\/span><\/span><\/span>\u3002\u8001\u5e08\u8bf4<\/span><\/span><\/span>\uff1a<\/span><\/span><\/span>\u4eca\u5929\u4f60\u638c\u63e1\u591a\u5c11<\/span><\/span><\/span>\u6570\u5b66<\/span><\/span><\/span>\u5c31\u662f\u4f60\u53ef\u4ee5\u638c\u63e1\u591a\u5c11\u4f60\u660e\u5929\u7684<\/span><\/span><\/span>\u524d\u7a0b<\/span><\/span><\/span> ! <\/span><\/span><\/span>\u5728\u6b64\u6211\u4eec\u8ba9<\/span><\/span><\/span>\u4f01\u4e1a\u8bf4\u540c\u6837\u7684\u8bdd\u3002<\/span><\/span><\/span>2020\u5e749\u670812\u65e5<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u534e\u4e3a\u5f90\u6587\u4f1f\u5728\u7531\u4e2d\u56fd\u5de5\u4e1a\u4e0e\u5e94\u7528\u6570\u5b66\u5b66\u4f1a\u4e3e\u529e\u7684<\/span><\/span><\/span>\u6570\u5b66\u4fc3\u8fdb\u4f01\u4e1a\u521b\u65b0\u53d1\u5c55\u8bba\u575b<\/span><\/span><\/span>\u63d0\u51fa\u901a\u8baf\u6570\u5b66\u7684<\/span><\/span><\/span>\u534e\u4e3a10\u5927\u95ee\u9898<\/span><\/span><\/span>\u3002\u8fd9\u662f\u503c\u5f97\u6211\u4eec\u4e0b\u8f7d\u770b\u770b\u751f\u4ea7\u524d\u7ebf\u5bf9\u6570\u5b66\u7684\u6c42\u52a9\u3002\u6b64\u5916<\/span><\/span><\/span>2021\u5e74\u5251\u6865\u5927\u5b66\u51fa\u7248\u793e\u4e3a\u534e\u4e3a\u51fa\u7248\u6587\u96c6<\/span><\/span><\/span>[16]<\/span><\/sup><\/strong><\/span>, \u4e66\u4e2d\u534e\u4e3a\u7684\u4e0d\u540c\u90e8\u95e8\u7684\u4e3b\u7ba1\u8c08\u4ed6\u4eec\u5de5\u4f5c\u4e0a\u9047\u5230\u7684\u6570\u5b66\u3002\u6211\u4eec\u76f8\u4fe1\u534e\u4e3a\u4e4b\u5916\u50cf<\/span><\/span><\/span>\u963f<\/span><\/span><\/span>\u91cc\u5df4\u5df4<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u817e\u8baf\u7b49\u4e5f\u5728\u4f7f\u7528\u5927\u91cf\u6570\u5b66\u3002\u4e0d\u8fc7\u4ece\u6587\u732e<\/span><\/span><\/span>[14][38][39]<\/span><\/span><\/span>\u6240\u89c1<\/span><\/span><\/span>\uff0c<\/span><\/span><\/span>\u4eba\u5de5\u667a\u80fd\u7528\u7684\u6570\u5b66\u53ea\u662f\u5fae\u79ef\u5206<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u7ebf\u6027\u4ee3\u6570<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>\u6570\u503c\u7b97\u6cd5<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>Bayes\u7edf\u8ba1<\/span><\/span><\/span>\u3001<\/span><\/span><\/span>Monte Carlo\u65b9\u6cd5<\/span><\/span><\/span>\u548c<\/span><\/span><\/span>\u968f\u673a\u68af\u5ea6\u4e0b\u964d\u3002\u6211\u4eec\u53ea\u80fd\u5e0c\u671b\u6709\u4e00\u5929\u4eba\u5de5\u667a\u80fd\u53ef\u4ee5\u7528\u4e0a\u66f4\u6df1\u523b\u7684\u6570\u5b66\u3002<\/span><\/span><\/span><\/section>\n
    \n
    <\/section>\n

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

    <\/span><\/section>\n
    \u4f5c\u8005\u7b80\u4ecb<\/span><\/span><\/section>\n

    \u9ece\u666f\u8f89<\/span>\uff0c<\/span><\/span>1974\u5e74\u6570\u5b66\u535a\u58eb\uff0c<\/span><\/span>\u8bba\u6587<\/span><\/span>\u5bfc\u5e08<\/span><\/span>\u662f<\/span><\/span>\u6717\u5170<\/span><\/span>\u5179<\/span><\/span>\u3002\u6bd5\u4e1a\u540e\u5728<\/span><\/span>\u4e2d\u56fd<\/span><\/span>\u9999\u6e2f<\/span><\/span>\u3001\u4e2d\u56fd<\/span><\/span>\u53f0\u6e7e<\/span><\/span>\u548c\u56fd\u5916<\/span><\/span>\u7684\u5927\u5b66\u5f53\u8001\u5e08\u3002\u81ea1978\u5e74<\/span><\/span>\u66fe<\/span><\/span>\u5728<\/span><\/span>\u4e2d\u5c71\u5927\u5b66<\/span><\/span>\u3001<\/span><\/span>\u534e\u4e1c\u5e08\u8303\u5927\u5b66<\/span><\/span>\u3001<\/span><\/span>\u5317\u4eac\u5927\u5b66<\/span><\/span>\u3001<\/span><\/span>\u9996\u90fd\u5e08\u8303\u5927\u5b66<\/span><\/span>\u3001<\/span><\/span>\u5c71\u4e1c\u5927\u5b66<\/span><\/span>\u3001<\/span><\/span>\u6cb3\u5357\u5927\u5b66<\/span><\/span>\u3001<\/span><\/span>\u4e1c\u5317\u5e08\u8303\u5927\u5b66<\/span><\/span>\u3001<\/span><\/span>\u6cb3\u5357\u5e08\u8303\u5927\u5b66\u7b49<\/span><\/span>\u5927\u5b66\u8bb2\u8bfe\u3002\u7814\u7a76\u65b9\u5411\u662f\u4ee3\u6570\u6570\u8bba\u548c\u6570\u5b66\u6559\u80b2\u3002\u5728\u79d1\u5b66\u51fa\u7248\u793e<\/span><\/span>\u3001<\/span><\/span>\u9ad8\u7b49\u6559\u80b2\u51fa\u7248\u793e<\/span><\/span>\u3001<\/span><\/span>\u5317\u4eac\u5927\u5b66\u51fa\u7248\u793e<\/span><\/span>\u548c<\/span><\/span>\u6cb3\u5357\u5927\u5b66<\/span><\/span>\u51fa\u7248\u793e\u51fa\u7248\u591a\u90e8\u6570\u5b66<\/span><\/span>\u4e13\u8457<\/span><\/span>\u3002<\/span><\/span><\/span><\/p>\n


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

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

    \n
    \n
    \n
    \n


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

    [1] <\/span><\/span>Avsykevich<\/span><\/span> D<\/span><\/span>,<\/span><\/span><\/span>Formalization of<\/span><\/span><\/span>control systems based on the topos theory, Military Space Academy, St. Petersburg,<\/span><\/span><\/span>(2021).<\/span><\/span><\/section>\n
    [<\/span><\/span>2<\/span><\/span>] <\/span><\/span>Barr <\/span><\/span>M<\/span><\/span>, Wells<\/span><\/span> C<\/span><\/span> \uff0cCategory Theory for Computing Science\uff0cGrundlehren der<\/span><\/span><\/span>math. Wiss. 278, Springer-Verlag, Berlin, (1985).<\/span><\/span><\/section>\n
    [<\/span><\/span>3<\/span><\/span>] <\/span><\/span>Belfiore J, Debbah M, Towards an intelligent 6G – a Topos perspective, Huawei Algo. Lab. Paris (2019).<\/span><\/span><\/section>\n
    [4]<\/span><\/span> Belfiore J, Bennequin D, Topos and stacks of deep neural networks, Huawei Adv. Wireless <\/span><\/span>Lab. Paris arXiv, 2106.14587, (2022).<\/span><\/span><\/section>\n
    [5]<\/span><\/span><\/span>Baudot <\/span><\/span>P, <\/span><\/span>Bennequin<\/span><\/span> D, <\/span><\/span>The homological nature of entropy<\/span><\/span>, <\/span><\/span>Entropy, 17(5):3253\u20133318, <\/span><\/span>(<\/span><\/span>2015<\/span><\/span>)<\/span><\/span>.<\/span><\/span><\/section>\n
    [6]<\/span><\/span><\/span>Blackburn<\/span><\/span> P<\/span><\/span>, de Rijke<\/span><\/span> M<\/span><\/span>, Venema<\/span><\/span> Y,<\/span><\/span> Modal Logic. Cambridge University Press (2002).<\/span><\/span><\/section>\n
    [7]<\/span><\/span><\/span>Bonick<\/span><\/span> J<\/span><\/span>,<\/span><\/span><\/span>Higher category theory as a paradigm for network applications,<\/span><\/span><\/span>Signal Processing, Sensor Fusion, and Target Recognition,<\/span><\/span><\/span>United States Defense and Security Symposium,   (2006).<\/span><\/span><\/section>\n
    [8]<\/span><\/span> Briggs J, P<\/span><\/span>ython for Kids<\/span><\/span><\/span>: A Playful Introduction to Programming<\/span><\/span><\/span>, <\/span><\/span>No Starch Press; <\/span><\/span>( <\/span><\/span>2022<\/span><\/span>)<\/span><\/span><\/section>\n
    [9]<\/span><\/span> Carnap R, Bar-Hillel Y, An outline of a theory of semantic information, MIT <\/span><\/span><\/section>\n
    Lab. <\/span><\/span>O<\/span><\/span>f Electronics, Technical Report 247 (1952).<\/span><\/span><\/section>\n
    [10]<\/span><\/span> Cover T, Thomas J, Elements of Information Theory, Wiley, (2006)<\/span><\/span><\/section>\n
    [11]<\/span><\/span><\/span>Fausett L<\/span><\/span>, <\/span><\/span>Fundamentals of Neural Networks_ Architectures, Algorithms, and Applications<\/span><\/span>, <\/span><\/span>Prentice Hall, Englewood Cliffs (1993).  <\/span><\/span><\/section>\n
    [12]<\/span><\/span><\/span>Fiadero<\/span><\/span> J<\/span><\/span>, Categories for Software Engineering<\/span><\/span>, Springer. (2005).<\/span><\/span><\/section>\n
    [13]<\/span><\/span><\/span>Fong<\/span><\/span> B<\/span><\/span>, Spivak<\/span><\/span> D<\/span><\/span>, Tuyeras<\/span><\/span> R<\/span><\/span>, Backprop as functor, ArXiv<\/span><\/span> 1171.10455<\/span><\/span>,<\/span><\/span> (2019)<\/span><\/span><\/section>\n
    [14]<\/span><\/span><\/span>Goodfellow<\/span><\/span> I<\/span><\/span>, Bengio<\/span><\/span> Y<\/span><\/span>, Courville<\/span><\/span> A, <\/span><\/span>Deep Learning<\/span><\/span>, MIT Press\uff0c(2016).<\/span><\/span><\/section>\n
    [15]<\/span><\/span><\/span>Gust<\/span><\/span> H<\/span><\/span>,<\/span><\/span><\/span>Learning Models of Neural Networks Based on<\/span><\/span><\/span>Topos Theory, (2007).<\/span><\/span><\/section>\n
    [16]<\/span><\/span><\/span>Heng<\/span><\/span> L<\/span><\/span>, McColl B<\/span><\/span><\/span>(Editors), Mathematics for Future Computing and Communications-Huawei volume, Cambridge University Press, (2021).<\/span><\/span><\/section>\n
    [17]<\/span><\/span><\/span>Hindley<\/span><\/span> J<\/span><\/span>, Seldin<\/span><\/span> J<\/span><\/span>, Lambda Calculus and Combinatorics, Cambridge University Press, (2008).<\/span><\/span><\/section>\n
    [18]<\/span><\/span><\/span>Lafforgue<\/span><\/span> L<\/span><\/span>, Some possible roles for AI<\/span><\/span><\/span>of Grothendieck topos theory, Huawei Research Center, France, (2022).<\/span><\/span><\/section>\n
    [19]<\/span><\/span><\/span>\u9ece\u666f\u8f89,<\/span><\/span> \u51af\u6dd1\u971e<\/span><\/span>\uff0c<\/span><\/span>\u6881\u5fd7\u658c,<\/span><\/span><\/span>\u4fde\u5c0f\u7965\uff0c\u6731\u4e00\u5fc3, \u4e2d\u5b66\u6570\u5b66\u548c\u5927\u5b66\u6570\u5b66\u7684\u672c\u8d28\u533a\u522b\u5bf9\u5b66\u4e60\u548c\u6559\u5b66\u7684\u5f71\u54cd,<\/span><\/span> \u6570\u5b66\u901a\u62a5<\/span><\/span>, 59(3), 1-6; 59(4), 7-10,17; (2020).<\/span><\/span><\/section>\n
    [<\/span><\/span>20]<\/span><\/span> \u9ece\u666f<\/span><\/span>\u8f89<\/span><\/span>, \u5f62\u5f0f\u7cfb\u7edf\u548c\u6570\u5b66, \u6570\u5b66\u6587\u5316, 11-3 (2020) 37-44.<\/span><\/span><\/section>\n
    [21]<\/span><\/span> \u674e\u6587\u5a01<\/span><\/span>,<\/span><\/span><\/span>\u4ee3\u6570\u5b66\u65b9\u6cd5<\/span><\/span>, <\/span><\/span>\u5377\u4e00,\u4e8c, \u5317\u4eac, \u9ad8\u7b49\u6559\u80b2\u51fa\u7248\u793e (2016).<\/span><\/span><\/section>\n
    [22]<\/span><\/span> Lawvere W, Schanuel S, <\/span><\/span>C<\/span><\/span>onceptual Mathematics: A First Introduction to Categories Cambridge University Press; ( 2009)<\/span><\/span><\/section>\n
    [23]<\/span><\/span><\/span>Manin<\/span><\/span> Y<\/span><\/span>, Marcolli<\/span><\/span> M<\/span><\/span>, Homotopy-theoretic and categorical models of neural information<\/span><\/span><\/span>networks, Compositionality, 6 (2024).<\/span><\/span><\/section>\n
    [24]<\/span><\/span><\/span>Milewski<\/span><\/span> B\uff0c <\/span><\/span>Category Theory for Programmers\uff0c<\/span><\/span><\/span>Creative Commons<\/span><\/span><\/span>Attribution<\/span><\/span> \uff082017\uff09<\/span><\/span>.<\/span><\/span><\/section>\n
    [25]<\/span><\/span><\/span>Pierce<\/span><\/span> B<\/span><\/span>, Basic Category Theory for Computer Scientists\uff0c<\/span><\/span>MITP ress\uff0c1991<\/span><\/span>.<\/span><\/span><\/section>\n
    [26]<\/span><\/span> Quiroga<\/span><\/span> R<\/span><\/span>, Panzeri <\/span><\/span>S<\/span><\/span>(Ed)<\/span><\/span>\uff0c<\/span><\/span>Principles of Neural Coding, CRC Press, (2013).<\/span><\/span><\/section>\n
    [27]<\/span><\/span> \u90b1\u9521\u9e4f, <\/span><\/span>\u795e\u7ecf\u7f51\u7edc\u4e0e\u6df1\u5ea6\u5b66\u4e60\uff0c\u590d\u65e6\u5927\u5b66\uff08<\/span><\/span>2021\uff09<\/span><\/span>https:\/\/nndl.github.io<\/span><\/span><\/section>\n
    [28]<\/span><\/span><\/span>Shannon<\/span><\/span> C, <\/span><\/span>A mathematical theory of communication. Bell System Technical Journal,<\/span><\/span><\/span>27:379\u2013423, 623\u2013656,<\/span><\/span>(<\/span><\/span>1948.<\/span><\/span>)<\/span><\/span><\/section>\n
    [29]<\/span><\/span><\/span>Vallani<\/span><\/span> M<\/span><\/span>, Topos of Google Transformer Networks, <\/span><\/span>arXiv, 2403.18415<\/span><\/span>(2024).<\/span><\/span><\/section>\n
    [30]<\/span><\/span> Vigneaux J, Information structures and their cohomology, <\/span><\/span>Theory and Applications of Categories, Vol. 35, No. 38, <\/span><\/span>(<\/span><\/span>2020<\/span><\/span>)<\/span><\/span>, 1476\u20131529.<\/span><\/span><\/section>\n
    [31]<\/span><\/span><\/span>D. Youvan, Topos Theory and the Evolution of Quantum AI, (2024)<\/span><\/span><\/section>\n
    [32]<\/span><\/span><\/span>Walters<\/span><\/span> R<\/span><\/span><\/span>, <\/span><\/span>Categories and Computer Science<\/span><\/span>, Cambridge University Press (2009).<\/span><\/span><\/section>\n
    [33]<\/span><\/span> \u51af\u514b\u52e4, \u9648\u8c6a, \u91cf\u5b50\u7ea0\u9519\u7801, \u79d1\u5b66\u51fa\u7248\u793e (2010). <\/span><\/span><\/section>\n
    [34]<\/span><\/span><\/span>W.F. Ke, <\/span><\/span>\u9ece\u666f<\/span><\/span>\u8f89<\/span><\/span>,<\/span><\/span><\/span>R. B. Zhang<\/span><\/span>, <\/span><\/span>Quantum codes from Hadamard matrices, Linear and Multilinear Alg<\/span><\/span>ebra<\/span><\/span>, 58 (2010) 847<\/span><\/span>–<\/span><\/span>854.<\/span><\/span><\/section>\n
    [35]<\/span><\/span><\/span>Grothendieck<\/span><\/span> A<\/span><\/span>, et. al.,Seminaire de Geometric Algebrique<\/span><\/span>, <\/span><\/span>[SGA 4]<\/span><\/span>, <\/span><\/span>Springer Lect. Notes Math., 269<\/span><\/span> (1972).<\/span><\/span><\/section>\n
    [36]<\/span><\/span> J<\/span><\/span>ohnstone<\/span><\/span> P<\/span><\/span>, Topos Theory, Dover Books, (2014).<\/span><\/span><\/section>\n
    [37]<\/span><\/span><\/span>\u9ece\u666f\u8f89<\/span><\/span>\uff0c<\/span><\/span>TOPOS\u7406\u8bba, <\/span><\/span>\u9ad8\u7b49\u6559\u80b2\u51fa\u7248\u793e<\/span><\/span>, 2025\u5e74\u5c06\u51fa\u7248\u3002<\/span><\/span><\/section>\n
    [38]<\/span><\/span><\/span>Hardt M, Recht B, Singer Y, <\/span><\/span>Train faster, generalize better: Stability of stochastic gradient descent<\/span><\/span>, <\/span><\/span>Proceedings of The 33rd International Conference on Machine Learning<\/span><\/span>, <\/span><\/span>J. Machine Learing Research, <\/span><\/span>48<\/span><\/span> (2016) <\/span><\/span>1225<\/span><\/span>– <\/span><\/span>234<\/span><\/span>.<\/span><\/span><\/section>\n
    [39]<\/span><\/span> Murphy K, Machine Learning, MIT Press (2012).<\/span><\/span><\/section>\n
    [40] <\/span><\/span>\u6c5f\u8f89\u6709<\/span><\/span>, <\/span><\/span>\u62d3\u6251\u5b66\u57fa\u7840<\/span><\/span>, <\/span><\/span>\u5317\u4eac<\/span><\/span>\uff1a<\/span><\/span>\u79d1\u5b66\u51fa<\/span><\/span>\u7248\u793e(2001).<\/span><\/span><\/section>\n
    [41]<\/span><\/span> Massey<\/span><\/span> W<\/span><\/span>,<\/span><\/span><\/span>Alg<\/span><\/span>ebraic <\/span><\/span>Topology<\/span><\/span><\/span>–<\/span><\/span><\/span>an introduction, Springer<\/span><\/span> (1977)<\/span><\/span><\/section>\n
    [42]<\/span><\/span> Pitt D, Rydeheard D, Johnstone P (eds.), Category Theory and Computer Science, Springer Lecture Notes in Computer Science 953, (1995).<\/span><\/span><\/section>\n
    [43]<\/span><\/span><\/span>E. Glazer, et.al. , <\/span><\/span>FrontierMath: A Benchmark for Evaluating Advanced Mathematical Reasoning in AI<\/span><\/span>, (2024), <\/span><\/span>arXiv:2411.04872v5<\/span><\/span> <\/span><\/span><\/section>\n


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    \u53c2\u8003\u6587\u732e\u53ef<\/span>\u4e0a\u4e0b\u6ed1\u52a8<\/span><\/strong>\u67e5\u770b<\/span><\/p>\n<\/section>\n<\/section>\n


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    \u8303\u7574\u8bba\u7cfb\u5217\u8bfe\u7a0b<\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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    \u8303\u7574\u8bba\u88ab\u79f0\u4e3a\u662f\u201c\u6570\u5b66\u7684\u6570\u5b66\u201d\uff0c\u4e3a\u73b0\u4ee3\u6570\u5b66\u63d0\u4f9b\u4e86\u4e00\u5957\u57fa\u7840\u8bed\u8a00\u3002\u8303\u7574\u8bba\u662f\u4e00\u4e2a\u5173\u4e8e\u5173\u7cfb\u7684\u7406\u8bba\uff0c\u63cf\u8ff0\u5e76\u7814\u7a76\u5173\u7cfb\u7684\u6240\u6709\u53ef\u80fd\u6027\u8d28\u3002\u5b83\u63d0\u4f9b\u4e86\u4e00\u79cd\u7cfb\u7edf\u3001\u7cbe\u786e\u3001\u62bd\u8c61\u7684\u8de8\u9886\u57df\u79d1\u5b66\u65b9\u6cd5\u8bba\uff0c\u53ef\u76f4\u63a5\u4ed8\u8bf8\u4e8e\u5404\u9886\u57df\u8003\u5bdf\u7684\u95ee\u9898\uff0c\u5bfb\u6c42\u8de8\u9886\u57df\u7684\u89e3\u51b3\u4e4b\u9053\u3002\u8fd9\u79cd\u6570\u5b66\u8bed\u8a00\u4e0e\u590d\u6742\u6027\u79d1\u5b66\u6709\u4f17\u591a\u76f8\u4f3c\u4e4b\u5904\uff0c\u52a0\u4e4b\u5176\u672c\u8eab\u4f5c\u4e3a\u6570\u5b66\u5de5\u5177\u7684\u4e25\u5bc6\u6027\uff0c\u540e\u7eed\u53ef\u80fd\u80fd\u4e3a\u89e3\u51b3\u590d\u6742\u6027\u79d1\u5b66\u95ee\u9898\u63d0\u4f9b\u4e00\u628a\u94a5\u5319\u3002\u8303\u7574\u8bba\u7cfb\u5217\u8bfe\u7a0b\u5305\u542b\u4e24\u5b63\u7684\u201c\u8303\u7574\u8bba\u5165\u95e8\u7cfb\u5217\u8bfe\u7a0b\u201d\uff0c\u548c\u6700\u65b0\u7684\u300c\u8303\u7574\u8bba\u4e0e\u673a\u5668\u5b66\u4e60\u300d\u8bfe\u7a0b\uff0c\u8be5\u8bfe\u7a0b\u4ece\u8303\u7574\u89c2\u70b9\u5207\u5165\u673a\u5668\u5b66\u4e60\uff0c\u5305\u62ec\u5bf9\u673a\u5668\u5b66\u4e60\u7684\u67d0\u4e9b\u65b9\u6cd5\u8bba\u5efa\u7acb\u80cc\u666f\u7684\u5177\u4f53\u8303\u7574\u7684\u7814\u7a76\u65b9\u6cd5\uff0c\u4ee5\u53ca\u4ece\u795e\u7ecf\u7f51\u7edc\u67b6\u6784\u7b49\u51fa\u53d1\u7814\u7a76\u5728\u8303\u7574\u4e0a\u7684\u67d0\u4e9b\u7ed3\u6784\u3002<\/span><\/span><\/span>\u53c2\u770b\uff1a<\/span><\/span><\/span><\/section>\n