{"id":33452,"date":"2022-03-29T22:10:57","date_gmt":"2022-03-29T14:10:57","guid":{"rendered":"https:\/\/swarma.org\/?p=33452"},"modified":"2022-03-29T22:10:57","modified_gmt":"2022-03-29T14:10:57","slug":"%e5%a4%8d%e6%9d%82%e7%b3%bb%e7%bb%9f%e7%9a%84%e6%95%b0%e6%8d%ae%e9%a9%b1%e5%8a%a8%e5%bb%ba%e6%a8%a1%ef%bc%9a%e5%82%a8%e5%a4%87%e6%b1%a0%e8%ae%a1%e7%ae%97%e6%95%99%e7%a8%8b","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=33452","title":{"rendered":"\u590d\u6742\u7cfb\u7edf\u7684\u6570\u636e\u9a71\u52a8\u5efa\u6a21\uff1a\u50a8\u5907\u6c60\u8ba1\u7b97\u6559\u7a0b"},"content":{"rendered":"
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<\/strong>\u5bf9\u5929\u6c14\u3001\u6e4d\u6d41\u548c\u80a1\u7968\u5e02\u573a\u7b49\u6df7\u6c8c\u7cfb\u7edf\uff0c\u5373\u4f7f\u521d\u59cb\u6761\u4ef6\u4e0b\u6709\u5fae\u5c0f\u7684\u4e0d\u786e\u5b9a\u6027\uff0c\u4e5f\u4f1a\u5bfc\u81f4\u8bef\u5dee\u6307\u6570\u589e\u957f\uff0c\u4ece\u800c\u9650\u5236\u4e86\u5bf9\u6b64\u7c7b\u7cfb\u7edf\u7684\u4efb\u4f55\u9884\u6d4b\u3002\u867d\u7136\u5f53\u524d\u6280\u672f\u503e\u5411\u4e8e\u4f7f\u7528\u566a\u58f0\u548c\u90e8\u5206\u7684\u6d4b\u91cf\u4fe1\u606f\u6765\u7ea6\u675f\u7269\u7406\u6a21\u578b\uff0c\u4f46\u63a7\u5236\u8fd9\u4e9b\u7cfb\u7edf\u7684\u65b9\u7a0b\u901a\u5e38\u662f\u672a\u77e5\u7684\u3002\u56e0\u6b64\uff0c\u4f7f\u7528\u673a\u5668\u5b66\u4e60\u7b49\u6570\u636e\u9a71\u52a8\u65b9\u6cd5\u6765\u9884\u6d4b\u6b64\u7c7b\u7cfb\u7edf\u975e\u5e38\u91cd\u8981\u3002\u50a8\u5907\u6c60\u8ba1\u7b97\uff08Reservoir Computing\uff09\u662f\u4e00\u79cd\u7b80\u5316\u7684\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\uff0c\u5728\u9884\u6d4b\u590d\u6742\u7cfb\u7edf\u7684\u52a8\u529b\u5b66\u65b9\u9762\u5df2\u7ecf\u663e\u793a\u51fa\u4f18\u8d8a\u7684\u6027\u80fd\u3002\u672c\u6587\u4ee5\u6d1b\u4f26\u5179\u5438\u5f15\u5b50\u4e3a\u4f8b\uff0c\u7b80\u8981\u4ecb\u7ecd\u4e86\u50a8\u5907\u6c60\u8ba1\u7b97\u5728\u8bad\u7ec3\u3001\u9884\u6d4b\u548c\u4f18\u5316\u65b9\u9762\u6240\u9700\u7684\u4ee3\u7801\uff0c\u5e76\u8ba8\u8bba\u4e86\u4f18\u5316\u4ee5\u627e\u5230\u6b63\u786e\u53c2\u6570\u7684\u91cd\u8981\u6027\u3002<\/span><\/strong><\/p>\n


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\u7814\u7a76\u9886\u57df\uff1a\u6df7\u6c8c\u7cfb\u7edf\uff0c\u6d1b\u4f26\u5179\u5438\u5f15\u5b50\uff0c\u50a8\u5907\u6c60\u8ba1\u7b97\uff0c\u5faa\u73af\u795e\u7ecf\u7f51\u7edc<\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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Jason Platt<\/span> <\/span>| \u4f5c\u8005<\/strong><\/span><\/p>\n

\u5218\u5fd7\u822a<\/span> | <\/strong><\/span>\u8bd1\u8005<\/strong><\/span><\/p>\n

\u6881\u91d1<\/span> | \u5ba1\u6821<\/strong><\/span><\/strong><\/span><\/p>\n

\u9093\u4e00\u96ea<\/span> | \u7f16\u8f91<\/strong><\/span><\/strong><\/span><\/p>\n


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\u786e\u5b9a\u6027\u6df7\u6c8c\u8fd9\u4e2a\u51e0\u4e4e\u81ea\u76f8\u77db\u76fe\u7684\u6982\u5ff5\u63cf\u8ff0\u7684\u662f\u8fd9\u6837\u7684\u7cfb\u7edf\uff0c\u5b83\u4eec\u5bf9\u521d\u59cb\u6761\u4ef6\u5341\u5206\u654f\u611f\u4ee5\u81f3\u4e8e\u65e0\u6cd5\u8fdb\u884c\u957f\u671f\u9884\u6d4b\u3002\u56e0\u6b64\uff0c\u5c3d\u7ba1\u52a8\u529b\u5b66\u65b9\u7a0b\u6ca1\u6709\u968f\u673a\u6027\uff0c\u4f46\u5373\u4f7f\u662f\u8ba1\u7b97\u4e2d\u6700\u8f7b\u5fae\u7684\u8bef\u5dee\uff0c\u4f8b\u5982\u8ba1\u7b97\u673a\u7684\u6570\u503c\u7cbe\u5ea6\u8bef\u5dee\uff0c\u4e5f\u4f1a\u5bfc\u81f4\u672a\u6765\u7684\u9884\u6d4b\u8c2c\u4ee5\u5343\u91cc\u3002<\/span><\/section>\n
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\u6df7\u6c8c\u7cfb\u7edf\u7684\u5e94\u7528\u8303\u56f4\u5305\u62ec\u5929\u6c14\u9884\u62a5\u3001\u6d41\u4f53\u4e2d\u7684\u6e4d\u6d41\u3001\u7b49\u79bb\u5b50\u4f53\u52a8\u529b\u5b66\u3001\u5316\u5b66\u53cd\u5e94\u3001\u7fa4\u4f53\u52a8\u529b\u5b66\u3001\u5929\u4f53\u8fd0\u52a8\u548c\u80a1\u7968\u5e02\u573a\u7b49\u7b49\u3002\u867d\u7136\u5f53\u524d\u6280\u672f\u503e\u5411\u4e8e\u4f7f\u7528\u5145\u6ee1\u566a\u58f0\u548c\u90e8\u5206\u7684\u6d4b\u91cf\u4fe1\u606f\u6765\u7ea6\u675f\u7269\u7406\u6a21\u578b<\/span>\uff08\u5982\u5361\u5c14\u66fc\u6ee4\u6ce2\uff09<\/span>\uff0c\u4f46\u901a\u5e38\u63a7\u5236\u8fd9\u4e9b\u7cfb\u7edf\u7684\u65b9\u7a0b\u662f\u672a\u77e5\u7684\u3002\u56e0\u6b64\uff0c\u80fd\u591f\u4f7f\u7528\u673a\u5668\u5b66\u4e60<\/span> (ML) <\/span>\u7b49\u6570\u636e\u9a71\u52a8\u65b9\u6cd5\u6765\u9884\u6d4b\u6b64\u7c7b\u7cfb\u7edf\u975e\u5e38\u91cd\u8981\u3002 <\/span><\/section>\n

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1. \u6d1b\u4f26\u5179\u201c\u8774\u8776\u201d\u5438\u5f15\u5b50<\/strong><\/strong><\/p>\n<\/section>\n

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\u7231\u5fb7\u534e\u00b7\u6d1b\u4f26\u5179<\/span>\uff08Edward Lorenz\uff09<\/span>\u57281963\u5e74\u5f00\u521b\u4e86\u7406\u89e3\u548c\u6d1e\u5bdf\u6df7\u6c8c\u52a8\u529b\u5b66\u7684\u65b0\u65f6\u4ee3\u3002\u6d1b\u4f26\u5179\u5bf9\u5730\u7403\u4f4e\u5c42\u5927\u6c14\u7684\u5bf9\u6d41\u8fdb\u884c\u7b80\u5316\uff0c\u5c06\u786e\u5b9a\u6027\u7684\u3001\u975e\u5468\u671f\u6027\u7684\u884c\u4e3a\u4ee5\u53ca\u201c\u8774\u8776\u6548\u5e94\u201d<\/span>\uff08\u8774\u8776\u6247\u52a8\u7fc5\u8180\u5c31\u80fd\u6539\u53d8\u5929\u6c14\uff09<\/span>\u6982\u5ff5\u5927\u4f17\u5316\u3002\u4ed6\u7684\u7ecf\u5178\u4f8b\u5b50\u88ab\u79f0\u4e3a\u201c\u6d1b\u4f26\u5179\u5438\u5f15\u5b50\u201d<\/span>\uff08Lorenz Attractor\uff09<\/span>\u3002\u8be5\u7cfb\u7edf\u662f\u4e09\u7ef4\u7684<\/span>\uff083D\uff09<\/span>\uff0c\u6d89\u53ca\u4ece\u4e0b\u65b9\u5747\u5300\u52a0\u70ed\u5e76\u5728\u4e0a\u65b9\u51b7\u5374\u7684\u7a00\u8584\u6d41\u4f53\u7684\u7279\u6027\u3002\u4e0b\u56fe\u7ed8\u5236\u7684\u662f\u4e0e\u5bf9\u6d41\u901f\u7387\u548c\u5782\u76f4\u6e29\u5ea6\u53d8\u5316\u76f8\u5173\u7684\u4e24\u4e2a\u53d8\u91cf\u3002\u6d1b\u4f26\u5179\u5438\u5f15\u5b50\u662f\u52a8\u529b\u7cfb\u7edf\u9884\u6d4b\u4e2d\u7684\u4e00\u4e2a\u6807\u51c6\u95ee\u9898\u3002 <\/span><\/section>\n
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\u56fe1. \u8fd9\u91cc\u7684\u51e0\u4f55\u5bf9\u8c61\u88ab\u79f0\u4e3a\u5177\u6709\u5206\u5f62\u7ef4\u6570\u7684\u201c\u5947\u5f02\u5438\u5f15\u5b50\u201d\u3002 \u51c6\u786e\u5730\u8bf4\uff0c\u5438\u5f15\u5b50\u7684\u7ef4\u6570\u662f 2.06\u3002| \u56fe\u7247\u6765\u81ea\u4f5c\u8005<\/span><\/p>\n

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\u6b64\u5904\u7684\u4e09\u7ef4\u6570\u636e u(t) \u6765\u81ea\u4e00\u7ec4\u5e38\u5fae\u5206\u65b9\u7a0b\uff0c\u4e00\u4e2a\u52a8\u529b\u7cfb\u7edf\u3002\u5728\u6b64\uff0c\u6211\u4eec\u6df1\u5165\u8c08\u4e00\u8c08\u52a8\u529b\u7cfb\u7edf\u7406\u8bba\uff0c\u201c\u8774\u8776\u6548\u5e94\u201d\u53ef\u4ee5\u7531\u4e00\u7ec4\u79f0\u4e3a\u674e\u4e9a\u666e\u8bfa\u592b\u6307\u6570<\/span>\uff08Lyapunov Exponents\uff0cLE\uff09<\/span>\u7684\u91cf\u6765\u63cf\u8ff0\u3002\u674e\u4e9a\u666e\u8bfa\u592b\u6307\u6570\u662f\u63cf\u8ff0\u52a8\u529b\u7cfb\u7edf\u4e2d\u6270\u52a8\u5e73\u5747\u589e\u957f\u7387\u7684\u57fa\u672c\u7279\u5f81\u3002<\/span><\/section>\n
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\u674e\u666e\u8bfa\u592b\u6307\u6570\u63cf\u8ff0\u4e86\u6700\u521d\u76f8\u9694\u4e00\u6bb5\u8ddd\u79bb \u03b4x\u2080=|x\u2081-x\u2082| \u7684\u4e24\u70b9 x\u2081 \u548c x\u2082 \u76f8\u5bf9\u4e8e\u5f7c\u6b64\u5982\u4f55\u5728\u65f6\u95f4\u4e0a\u6f14\u5316\u3002\u66f4\u5177\u4f53\u5730\u8bf4\uff1a<\/span><\/section>\n

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\u5176\u4e2d\u03bb\u2081\u662f\u6700\u5927\u7684\u674e\u4e9a\u666e\u8bfa\u592b\u6307\u6570\u3002\u56e0\u6b64\uff0c\u5982\u679c\u03bb\u2081\u4e3a\u6b63\u6570\uff0c\u968f\u7740\u65f6\u95f4\u63a8\u79fb\uff0c\u6700\u521d\u4ec5\u76f8\u5dee\u5f88\u5c0f\u7684\u4e24\u4e2a\u521d\u59cb\u6761\u4ef6\u5c06\u5728\u7ebf\u6027\u5316\u4f53\u7cfb\u4e2d\u4ee5\u6307\u6570\u7ea7\u7684\u901f\u5ea6\u5feb\u901f\u504f\u79bb\u5f7c\u6b64<\/strong>\uff0c\u8fd9\u5c31\u662f\u6df7\u6c8c\u7684\u5b9a\u4e49\u548c\u9884\u6d4b\u56f0\u96be\u7684\u539f\u56e0\u3002<\/span><\/section>\n
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\u56fe2. \u7531\u6b63\u674e\u4e9a\u666e\u8bfa\u592b\u6307\u6570\u5f15\u8d77\u7684\u6270\u52a8\u589e\u957f\u3002| \u56fe\u7247\u6765\u6e90\uff1aYapparina \u5236\u4f5c<\/span><\/p>\n

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\u674e\u4e9a\u666e\u8bfa\u592b\u6307\u6570\u53ef\u4ee5\u76f4\u63a5\u4e0e\u5438\u5f15\u5b50\u7684\u7ef4\u5ea6\u548c\u51e0\u4f55\u5f62\u72b6\u76f8\u5173\uff0c\u5438\u5f15\u5b50\u662f\u53d8\u91cf\u7a7a\u95f4<\/span>\uff08\u79f0\u4e3a\u76f8\u7a7a\u95f4\uff09<\/span>\u4e2d\u6570\u636e\u9a7b\u7559\u7684\u51e0\u4f55\u5bf9\u8c61\u3002\u03bb\u2081 \u4e5f\u662f\u8bef\u5dee\u6307\u6570\u589e\u957f\u7684\u4ee5\u65f6\u95f4\u7684\u9006\u4e3a\u5355\u4f4d\u7684\u5e38\u6570\uff0c\u56e0\u6b64\u674e\u4e9a\u666e\u8bfa\u592b\u65f6\u95f4 \u03bb\u2081*t \u7ed9\u51fa\u4e86\u4e00\u4e2a\u65e0\u91cf\u7eb2\u65f6\u95f4 \uff0c\u53ef\u7528\u4e8e\u5224\u65ad\u9884\u6d4b\u7684\u8d28\u91cf\u3002<\/span><\/section>\n
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2. \u50a8\u5907\u6c60\u8ba1\u7b97<\/strong><\/strong><\/p>\n<\/section>\n

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\u5728\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u6709\u8bb8\u591a\u9884\u6d4b\u590d\u6742\u7cfb\u7edf\u7684\u65b9\u6cd5\uff0c\u7136\u800c\u6700\u8fd1\u6709\u4e00\u79cd\u65b9\u6cd5\u5728\u6027\u80fd\u3001\u7406\u89e3\u548c\u8bad\u7ec3\u7684\u7b80\u5355\u6027\u4ee5\u53ca\u57fa\u7840\u52a8\u529b\u5b66\u7684\u518d\u73b0\u65b9\u9762\u8131\u9896\u800c\u51fa\uff0c\u8fd9\u5c31\u662f\u50a8\u5907\u6c60\u8ba1\u7b97<\/strong><\/span>\uff08Reservoir Computing\uff09<\/span>\u3002\u50a8\u5907\u6c60\u8ba1\u7b97\u662f\u5faa\u73af\u795e\u7ecf\u7f51\u7edc<\/span>\uff08RNN\uff09<\/span>\u7684\u4e00\u79cd\u7b80\u5316\u5f62\u5f0f\u3002\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u662f\u4e00\u79cd\u5177\u6709\u81ea\u6fc0\u52b1\u53cd\u9988\u8fde\u63a5\u7684\u7f51\u7edc\uff0c\u5176\u5185\u90e8\u72b6\u6001\u663e\u5f0f\u5730\u4f9d\u8d56\u4e8e\u5148\u524d\u7684\u5185\u90e8\u6216\u201c\u9690\u85cf\u201d\u72b6\u6001\u548c\u5916\u90e8\u9a71\u52a8\u4fe1\u53f7\u3002\u56e0\u6b64\uff0c\u6570\u636e\u6709\u4e00\u4e2a\u81ea\u7136\u7684\u6392\u5e8f\uff0c\u6bd4\u5982\u65f6\u5e8f\uff0c\u8fd9\u660e\u786e\u5730\u5141\u8bb8\u5c06\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u4f5c\u4e3a\u4e00\u4e2a\u52a8\u529b\u7cfb\u7edf\u6765\u5904\u7406\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u5176\u4ed6\u5f62\u5f0f\u7684\u4eba\u5de5\u795e\u7ecf\u7f51\u7edc\uff0c\u5982\u591a\u5c42\u611f\u77e5\u5668\u4e0d\u5bf9\u6570\u636e\u8fdb\u884c\u6392\u5e8f\uff0c\u56e0\u6b64\u5b83\u4eec\u7684\u81ea\u7136\u6570\u5b66\u63cf\u8ff0\u662f\u4e00\u4e2a\u51fd\u6570\u3002<\/span><\/section>\n
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\u50a8\u5907\u6c60\u8ba1\u7b97\u7684\u4f18\u70b9\u662f\u907f\u514d\u4e86\u76f4\u63a5\u8bad\u7ec3\u50a8\u5907\u6c60\u672c\u8eab\u7684\u6743\u91cd\uff0c\u5b83\u4eec\u662f\u6839\u636e\u63a7\u5236\u8fde\u63a5\u56fe\u5c5e\u6027\u7684\u5c11\u91cf\u53c2\u6570\u8fdb\u884c\u8bbe\u7f6e\uff0c\u8fd9\u4e9b\u6574\u4f53\u53c2\u6570\u662f\u50a8\u5907\u6c60\u8ba1\u7b97\u65b9\u6cd5\u7684\u4f18\u52bf\u6240\u5728\u3002\u8bbe\u7f6e\u8fd9\u4e9b\u53c2\u6570\u53ef\u4ee5\u770b\u4f5c\u662f\u6307\u5b9a\u7c7b\u4f3c\u4e8e\u6c14\u4f53\u7684\u6574\u4f53\u6e29\u5ea6\u6216\u538b\u5f3a\uff0c\u4e0d\u9700\u8981\u8ddf\u8e2a\u5355\u72ec\u6bcf\u4e2a\u7c92\u5b50\u7684\u4f4d\u7f6e\u548c\u901f\u5ea6\uff0c\u5927\u5927\u964d\u4f4e\u4e86\u95ee\u9898\u7684\u590d\u6742\u6027\u3002\u56e0\u6b64\uff0c\u8bad\u7ec3\u4ec5\u9650\u4e8e\u5728\u8f93\u51fa\u7aef\u89e3\u51b3\u7ebf\u6027\u56de\u5f52\u95ee\u9898\uff0c\u4ece\u800c\u5927\u5927\u51cf\u5c11\u4e86\u8bad\u7ec3\u65f6\u95f4\u3002<\/span><\/section>\n

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\u56fe3. \u50a8\u5907\u6c60\u8ba1\u7b97\u4e2d\u7684\u4fe1\u606f\u6d41\u3002\u8f93\u5165u(t) \u901a\u8fc7\u8f93\u5165\u5c42\u5c06\u4fe1\u53f7\u6295\u5c04\u5230\u9ad8\u7ef4\u50a8\u5907\u6c60\u4e2d\uff0c\u4f7f\u6211\u4eec\u80fd\u591f\u7528\u7ebf\u6027\u8bfb\u51fa\u5c42\u6765\u903c\u8fd1\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u3002\u7136\u540e\uff0c\u8bfb\u51fa\u5c42\u9884\u6d4b u(t+1)\u3002\u63a5\u7740\u5c06 u(t+1) \u53cd\u9988\u5230\u8f93\u5165\u5c42\uff0c\u5f97\u5230\u6211\u4eec\u7684\u9884\u6d4b\u3002| \u56fe\u7247\u7531\u4f5c\u8005\u63d0\u4f9b<\/span><\/p>\n

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3. \u50a8\u5907\u6c60\u8ba1\u7b97\u7684\u7ed3\u6784<\/strong><\/strong><\/p>\n<\/section>\n

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\u50a8\u5907\u6c60\u8ba1\u7b97\u7684\u7ed3\u6784\u5305\u62ec\u4e00\u4e2a\u56fa\u5b9a\u7684<\/span>\uff08\u672a\u7ecf\u8bad\u7ec3\u7684\uff09<\/span>\u9ad8\u7ef4\u201c\u50a8\u5907\u6c60\u201d\u548c\u4e00\u4e2a\u5355\u4e00\u7684\u8bad\u7ec3\u8f93\u51fa\u5c42\u3002\u50a8\u5907\u6c60\u7684\u56fa\u5b9a\u6027\u8d28\u51cf\u5c11\u4e86\u8bad\u7ec3\u53c2\u6570\u7a7a\u95f4\u7684\u5927\u5c0f\uff0c\u53ea\u9700\u8bbe\u7f6e\u5c11\u6570\u51e0\u4e2a\u63a7\u5236\u5168\u5c40\u5c5e\u6027\u7684\u53c2\u6570\uff0c\u7136\u540e\u89e3\u51b3\u7ebf\u6027\u56de\u5f52\u95ee\u9898<\/strong>\u3002\u8fd9\u79cd\u5bf9\u53ef\u641c\u7d22\u53c2\u6570\u7a7a\u95f4\u5927\u5c0f\u7684\u51cf\u5c11\uff0c\u4f7f\u4eba\u4eec\u80fd\u591f\u8f7b\u677e\u5730\u7ed5\u8fc7\u7531\u4e8e\u4f7f\u7528\u53cd\u5411\u4f20\u64ad\u65b9\u6cd5\u4ea7\u751f\u7684\u68af\u5ea6\u7206\u70b8\/\u68af\u5ea6\u6d88\u5931\u95ee\u9898\uff0c\u5e76\u5c06\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7684\u8bad\u7ec3\u65f6\u95f4\u52a0\u5feb\u51e0\u4e2a\u6570\u91cf\u7ea7\u3002<\/span><\/section>\n
 <\/span><\/section>\n
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\u88681\uff1a\u63a7\u5236\u50a8\u5907\u6c60\u6574\u4f53\u6027\u8d28\u7684\u53c2\u6570\u4ee5\u53ca\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u65b9\u7a0b\u7684\u66f4\u65b0<\/span><\/p>\n

 <\/span><\/section>\n
\u50a8\u5907\u6c60\u8ba1\u7b97\u7531\u4e09\u5c42\u7ec4\u6210\uff1a\u8f93\u5165\u5c42 Win\uff0c\u5373\u50a8\u5907\u6c60\u672c\u8eab\uff0c\u4ee5\u53ca\u8f93\u51fa\u5c42 Wout\u3002\u50a8\u5907\u6c60\u7531N\u4e2a\u8282\u70b9\u7ec4\u6210\uff0c\u8fd9\u4e9b\u8282\u70b9\u4e00\u822c\u7528\u7b80\u5355\u7684\u975e\u7ebf\u6027\u5143\u7d20<\/span>\uff08\u5982 tanh \u6fc0\u6d3b\u51fd\u6570\uff09<\/span>\u8fdb\u884c\u4f5c\u7528\u3002\u7f51\u7edc\u4e2d\u7684\u8282\u70b9\u901a\u8fc7\u4e00\u4e2a N\u00d7N \u90bb\u63a5\u77e9\u9635 A \u8fde\u63a5\uff0cA \u662f\u968f\u673a\u9009\u62e9\u7684\uff0c\u5176\u8fde\u63a5\u5bc6\u5ea6\u4e3a \u03c1A<\/sub>\uff0c\u975e\u96f6\u5143\u7d20\u5728 [-1, 1] \u4e4b\u95f4\u5747\u5300\u9009\u62e9\uff0c\u91cd\u65b0\u6807\u5ea6\u4f7f\u5f97 A \u7684\u6700\u5927\u7279\u5f81\u503c\u8868\u793a\u8c31\u534a\u5f84\uff08\u03c1SR<\/sub>\uff09\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u901a\u8fc7\u4ee5\u4e0b\u4ee3\u7801\u6211\u4eec\u53ef\u4ee5\u5c1d\u8bd5\u5728 Julia \u4e2d\u521d\u59cb\u5316\u50a8\u5907\u6c60\u8ba1\u7b97\uff0c\u5b83\u88ab\u5b9e\u73b0\u4e3a\u4e00\u4e2a\u5bf9\u8c61\uff0c\u88681\u4e2d\u7684\u8f93\u5165\u53c2\u6570\u4f5c\u4e3a\u521d\u59cb\u5316\u53c2\u6570\u3002\u51fd\u6570\u201cget_connection_matrix\u201d\u4ea7\u751f N\u00d7N \u7684\u50a8\u5907\u6c60\uff0c\u5e76\u6807\u5ea6\u6700\u5927\u7279\u5f81\u503c\u3002<\/span><\/section>\n
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using Random<\/span>
using SparseArrays<\/span>
using Arpack<\/span>

struct rc <\/span>
   “””Structure holding the reservoir computer<\/span>
   <\/span>
   Args:<\/span>
       D::Int : Input system Dimension<\/span>
       N::Int : Reservoir Dimension<\/span>
       SR::Float : Spectral Radius<\/span>
       \u03c1A::Float : Density of A<\/span>
       \u03b1::Float : Leak Rate<\/span>
       \u03c3::Float : Input Strength<\/span>
       random_state::Int : Random seed<\/span>
       \u03c3b::Float : Bias<\/span>
       \u03b2::Float : Tikhonov regularization<\/span>
   “””<\/span>
   D::Int # Input System Dimension<\/span>
   N::Int # Reservoir Dimension<\/span>
   \u03c1SR::Float64 # Spectral Radius of A<\/span>
   \u03c1A::Float64 # Density of A<\/span>
   \u03b1::Float64 # Leak Rate<\/span>
   \u03c3::Float64 # Input Strength<\/span>
   \u03c3b::Float64 # Input Bias<\/span>
   \u03b2::Float64 # Tikhonov Regularization<\/span>
   A::SparseMatrixCSC{Float64, Int64} # Adjacency Matrix<\/span>
   Win::Array{Float64,2} # Input Matrix<\/span>
   Wout::Array{Float64,2} # Output Matrix<\/span>
   rng::MersenneTwister # Random Number Generator<\/span>
   <\/span>
   # Constructor Function<\/span>
   function rc(D, N, \u03c1SR , \u03c1A, \u03b1, \u03c3, \u03c3b, \u03b2; random_state=111111)<\/span>
       rng = MersenneTwister(random_state)<\/span>
       A, Win, Wout = initialize_weights(D, N, \u03c1SR, \u03c1A, \u03c3, rng)<\/span>
       new(D, N, \u03c1SR , \u03c1A, \u03b1, \u03c3, \u03c3b, \u03b2, A, Win, Wout, rng)<\/span>
   end<\/span>
end<\/span>

function initialize_weights(D, N, SR, \u03c1A, \u03c3, rng)<\/span>
   A = get_connection_matrix(N, \u03c1A, \u03c1SR, rng)<\/span>
   Win = rand(rng, Uniform(-\u03c3, \u03c3), N, D)<\/span>
   Wout = Array{Float64}(undef, (D, N)) <\/span>
   return A, Win, Wout<\/span>
end<\/span>

function get_connection_matrix(N, \u03c1A, \u03c1SR, rng)<\/span>
   # Define that numbers selected between [-1, 1]<\/span>
   rf(rng, N) = rand(rng, Uniform(-1.0, 1.0), N)<\/span>
   <\/span>
   # NxN sparse random matrix<\/span>
   A = sprand(rng, N, N, \u03c1A, rf)<\/span>
   <\/span>
   # calculate eigenvalues and rescale<\/span>
   eig, _ = eigs(A, nev=1)<\/span>
   maxeig = abs(eig[1])<\/span>
   A = A.*(\u03c1SR\/maxeig)<\/span>
   return A<\/span>
end<\/span><\/p>\n


<\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n

\u6ce8\uff1a\u50a8\u5907\u6c60\u8ba1\u7b97\u91c7\u7528\u88681\u4e2d\u6240\u793a\u7684\u53c2\u6570\u3002\u6211\u4eec\u770b\u5230\u90bb\u63a5\u77e9\u9635A\u88ab\u5b9e\u73b0\u4e3a\u7a00\u758f\u77e9\u9635\uff0c\u56e0\u4e3a\u5df2\u7ecf\u8bc1\u660e\uff0c\u8fde\u63a5\u5ea6\u975e\u5e38\u4f4e\u7684\u50a8\u5907\u6c60\u5b9e\u9645\u4e0a\u9884\u6d4b\u6548\u679c\u6700\u597d\uff08\u03c1<\/span>A<\/sub>~0.02\uff09\u3002\u6b64\u5916\uff0c\u7a00\u758f\u77e9\u9635\u7684\u77e2\u91cf\u4e58\u6cd5\u4e5f\u975e\u5e38\u9ad8\u6548\u3002\u7136\u540e\uff0cA \u88ab\u5176\u6700\u5927\u7279\u5f81\u503c\u91cd\u65b0\u6807\u5ea6\u3002Win \u5728 [-\u03c3, \u03c3] \u4e4b\u95f4\u968f\u673a\u4ea7\u751f\u3002Wout \u7684\u5185\u5b58\u5728\u521d\u59cb\u5316\u4e2d\u88ab\u5206\u914d\uff0c\u4f46\u50a8\u5907\u6c60\u8ba1\u7b97\u8fd8\u6ca1\u6709\u88ab\u8bad\u7ec3\u3002<\/span>
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\u8f93\u5165\u5c42 Win \u662f\u4e00\u4e2a N\u00d7D \u7ef4\u77e9\u9635\uff0c\u5c06\u8f93\u5165\u4fe1\u53f7 u(t) \u4eceD\u7ef4\u6620\u5c04\u5230N\u7ef4\u50a8\u5907\u6c60\u7a7a\u95f4\u3002Win\u7684\u5143\u7d20\u662f\u5728[-\u03c3, \u03c3]\u4e4b\u95f4\u5747\u5300\u9009\u62e9\u3002<\/span> <\/span><\/section>\n

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4. \u8bad\u7ec3\u50a8\u5907\u6c60\u8ba1\u7b97<\/strong><\/strong><\/p>\n<\/section>\n

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


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\u8c03\u7528\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\/\u50a8\u5907\u6c60\u7684\u9690\u85cf\u72b6\u6001 r(t)\u3002\u66f4\u65b0\u89c4\u5219\u662f<\/span><\/section>\n

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\u5728\u8fd9\u91cc\u6211\u4eec\u770b\u5230\uff0c\u4e0b\u4e00\u4e2a\u50a8\u5907\u6c60\u9690\u85cf\u72b6\u6001 r(t+1) \u53d6\u51b3\u4e8e\u524d\u4e00\u4e2a\u72b6\u6001 r(t) \u548c\u8f93\u5165\u6570\u636e u(t)\u3002\u03b1 \u662f\u201c\u6cc4\u6f0f\u7387\u201d<\/span>\uff08leak rate\uff09<\/span>\uff0c\u51b3\u5b9a\u5728\u4e0b\u4e00\u4e2a\u72b6\u6001\u4e2d\u6709\u591a\u5c11\u6765\u81ea\u524d\u4e00\u4e2a\u72b6\u6001\u7684\u76f4\u63a5\u6df7\u5408\u3002\u8282\u70b9\u4e0a\u7684\u975e\u7ebf\u6027\u5143\u7d20\u7531 tanh \u51fd\u6570\u7ed9\u51fa\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u6548\u679c\u5f88\u597d\uff0c\u56e0\u4e3a\u5b83\u4ee50\u4e3a\u4e2d\u5fc3\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u8bad\u7ec3\u5305\u62ec\u7528\u6570\u636e u(t) \u9a71\u52a8\u50a8\u5907\u6c60\u8ba1\u7b97\uff0c\u5e76\u751f\u6210\u4e00\u7cfb\u5217\u4e0e\u6570\u636e\u76f8\u5339\u914d\u7684\u50a8\u5907\u6c60\u72b6\u6001 r(t)\u3002\u7136\u540e\u6211\u4eec\u901a\u8fc7\u4f18\u5316\u635f\u5931\u6765\u786e\u5b9a u(t) \u548c r(t) \u4e4b\u95f4\u7684\u7ebf\u6027\u6620\u5c04 Wout<\/span><\/section>\n
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<\/section>\n
\u5176\u4e2dr\u662f\u5305\u542br(t)\u7684\u77e9\u9635\uff0cu\u662f\u5305\u542b\u8bad\u7ec3\u6570\u636e\u96c6\u4e2d\u6240\u6709t\u7684u(t)\u7684\u77e9\u9635\uff0c\u03b2\u662f Tikhonov-Miller \u6b63\u5219\u5316\u8d85\u53c2\u6570\u3002\u8fd9\u4e5f\u88ab\u79f0\u4e3a\u201c\u5cad\u56de\u5f52\u201d<\/span>\uff08ridge regression\uff09<\/span>\u3002\u5176\u89e3\u662f<\/span><\/section>\n
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\u5176\u4e2dI\u662fN\u00d7N\u7684\u5355\u4f4d\u77e9\u9635\u3002\u56e0\u6b64\uff0c\u8bad\u7ec3\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u77e9\u9635\u4e58\u6cd5\u95ee\u9898\u3002\u6211\u4eec\u53ef\u4ee5\u770b\u770b\u8fd9\u5728\u4ee3\u7801\u4e2d\u662f\u600e\u6837\u7684\uff1a<\/span><\/section>\n
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function train_RC(rc::rc, u; spinup = 100, readout = false) <\/span>
 “””Train the rc<\/span>
   This function sets the matrix rc.Wout<\/span>
   u \u2208 D\u00d7T<\/span>
   r \u2208 N\u00d7T<\/span>
   Args:<\/span>
       rc : reservoir struct<\/span>
       u : Array of data of shape D\u00d7Time<\/span>
       spinup : Amount of data to use for spinup<\/span>
   Returns:<\/span>
       if readout=true: Wout*r, the readout of the rc training data<\/span>
       else: return the last state of r, can be used to predict<\/span>
             forward from the training data <\/span>
   “””<\/span>
   r = generate(rc, u)<\/span>
   compute_Wout(rc, r[:, 1+spinup:end], u[:, 1+spinup:end])<\/span>
   if readout == true return rc.Wout*r end<\/span>
   return r[:, end]<\/span>
end<\/span>

function driven_rc!(rtp1, rc::rc, rt, ut)<\/span>
 “””Drive the rc with data<\/span>
 <\/span>
 Args:<\/span>
   rtp1 : r(t+1)<\/span>
   rc : reservoir computer<\/span>
   rt : r(t)<\/span>
   ut : u(t)<\/span>
 “””<\/span>
   rtp1[:] .= rc.\u03b1.*tanh.(rc.A*rt .+ rc.Win*ut .+ rc.\u03c3b).+(1 .- rc.\u03b1).*rt<\/span>
end<\/span>

function generate(rc::rc, u)<\/span>
   T = size(u)[2]<\/span>
   r = Array{Float64}(undef, (rc.N, T))<\/span>
   r[:, 1] = zeros(rc.N)<\/span>
   for t in 2:T<\/span>
       driven_esn!(r[:, t], rc, r[:, t-1], u[:, t-1])<\/span>
   end<\/span>
   return r<\/span>
end<\/span>

function compute_Wout(rc::rc, r, u)<\/span>
   rc.Wout[:, :] .= (Symmetric(r*r’+esn.\u03b2*I)(r*u’))’<\/span>
end<\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n

\u6ce8\uff1a\u8bad\u7ec3\u50a8\u5907\u6c60\u8ba1\u7b97\u9700\u8981\u8f93\u5165\u4e00\u4e2a\u6570\u636e\u5e8f\u5217 u(t)\u3002\u6211\u4eec\u751f\u6210\u5bf9\u5e94\u4e8e\u8f93\u5165u(t)\u7684\u9690\u85cf\u72b6\u6001r(t)\uff0c\u4ece\u90a3\u91cc\u53ef\u4ee5\u8ba1\u7b97\u51fa\u8f93\u51fa\u77e9\u9635\u3002<\/span>
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\u50a8\u5907\u6c60\u8ba1\u7b97\u8fd8\u9700\u8981\u4e00\u4e9b\u6b65\u9aa4\u6765\u4e0e\u6570\u636e\u540c\u6b65\u3002\u6211\u4eec\u628a\u8fd9\u79f0\u4e3a\u201c\u65cb\u8f6c\u201d<\/span>\uff08spinning up\uff09<\/span>\u3002<\/span><\/section>\n
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\u73b0\u5728\u8fdb\u884c\u9884\u6d4b\u3002\u6211\u4eec\u7528 Wout*r(t) \u4ee3\u66ff u(t)\uff0c\u628a\u88ab\u52a8\u7684\u50a8\u5907\u6c60\u8ba1\u7b97\u53d8\u6210\u4e00\u4e2a\u81ea\u4e3b\uff08\u53ea\u53d6\u51b3\u4e8er(t)\u7684\uff09\u7cfb\u7edf<\/span><\/section>\n
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\u8fd9\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u201c\u9884\u6d4b\u673a\u5668\u201d\uff0c\u53ea\u8981\u6211\u4eec\u60f3\uff0c\u5c31\u53ef\u4ee5\u7b80\u5355\u5730\u5728\u65f6\u95f4\u4e0a\u5411\u524d\u8fed\u4ee3\u3002\u4ee3\u7801\u5982\u4e0b\uff0c\u6ce8\u610f \u201cauto_rc\u201d\u51fd\u6570\u4e0e\u201cdriven_rc\u201d\u51fd\u6570\u76f8\u540c\uff0c\u4f46 u->Wout*r\u3002<\/span><\/section>\n
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function forecast_RC(rc::rc, nsteps; uspin=nothing, r0 = nothing)<\/span>
 “””Forecast nsteps forward in time from the end of uspin or from rc state r0<\/span>
   Make sure to train the RC before forecasting.  Requires either uspin or r0.<\/span>
   If both are provided will use uspin to set r0.<\/span>
   “””<\/span>
   if !isnothing(uspin)<\/span>
       rspin = generate(rc, uspin)<\/span>
       r0 = rspin[:, end]<\/span>
   end<\/span>
   @assert !isnothing(r0)<\/span>
   <\/span>
   # Initialize array of hidden states and set r(0)=r0<\/span>
   rfc = Array{Float64}(undef, (rc.N, nsteps))<\/span>
   rfc[:, 1] = r0<\/span>
   <\/span>
   # Now iterate the autonomous RC nsteps times for the forecast<\/span>
   for t in 1:nsteps-1<\/span>
      auto_esn!(rfc[:, t+1], rc, rfc[:, t])<\/span>
   end<\/span>
   return esn.Wout*rfc<\/span>
end<\/span>

function auto_rc!(rtp1, rc::rc, rt)<\/span>
   “””Autonomous RC update rule”””<\/span>
   rtp1[:] .= rc.\u03b1.*tanh.(rc.A*rt .+ rc.Win*rc.Wout*rt .+ rc.\u03c3b).+(1 .- rc.\u03b1).*rt<\/span>
end<\/span><\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n

\u6ce8\uff1a\u9884\u6d4b\u53ef\u4ee5\u4ece\u4e00\u5c0f\u7247\u6570\u636e uspin \u6216\u4e00\u4e2a\u50a8\u5907\u6c60\u72b6\u6001 r(0) \u5f00\u59cb\u3002\u7136\u540e\u51fd\u6570\u4f1a\u9884\u6d4b n \u6b65\u4e4b\u540e\u7684\u72b6\u6001\u3002<\/span>
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\u8fd9\u91cc\u6211\u4eec\u4ece\u6d1b\u4f26\u5179\u7cfb\u7edf\u4e2d\u521b\u5efa\u4e00\u4e9b\u8bad\u7ec3\u548c\u6d4b\u8bd5\u6570\u636e\u3002BasicReservoirComputing \u662f\u5177\u6709\u4e0a\u8ff0\u50a8\u5907\u6c60\u8ba1\u7b97\u4ee3\u7801\u7684\u6a21\u5757\u3002\u7cfb\u7edf\u7684\u5927\u5c0f\u4e3aD=3\uff0c\u800c\u50a8\u5907\u6c60\u6709N=200\u4e2a\u8282\u70b9\u3002\u53c2\u6570\u5982\u4e0b \u03c3=0.084\uff1b\u03b1=0.6\uff1bSR=0.8\uff1b\u03c1<\/span>A<\/span>=0.02\uff1b\u03b2=8.5e-8\uff1b\u03c3b=1.6\uff0c\u8fd9\u662f\u901a\u8fc7\u4e0b\u4e00\u8282\u63cf\u8ff0\u7684\u4f18\u5316\u7a0b\u5e8f\u627e\u5230\u7684\u3002\u6211\u6ca1\u6709\u5b9a\u4e49 make_lor63 \u548c plot_prediction\uff0c\u4f46\u5b83\u4eec\u53ef\u4ee5\u5728 github repo \u4e2d\u627e\u5230\u3002\u6574\u4e2a\u8fc7\u7a0b\u53ea\u9700\u8981\u51e0\u79d2\u949f\u3002<\/span><\/span> <\/span><\/section>\n
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using BasicReservoirComputing<\/span>
using Plots<\/span>

function L63_ex()<\/span>
   # make the training and testing data (not defined here)<\/span>
   utrain, utest = make_lor63(train_time = 100, test_time = 20, n_test = 100)<\/span>
   <\/span>
   #define the RC<\/span>
   D=3; N=200; \u03c3=0.084; \u03b1=0.6; SR=0.8; \u03c1A=0.02; \u03b2=8.5e-8; \u03c3b=1.6; \u0394t=0.01<\/span>
   reservoir = rc(D, N, \u03c1A, \u0394t, SR, \u03b1, \u03c3, \u03c3b, \u03b2))<\/span>
   <\/span>
   # Training<\/span>
   nspin = 100<\/span>
   train_RC(reservoir, utrain, spinup=nspin)<\/span>
   <\/span>
   # Forecast and compare to truth<\/span>
   utrue = utest[:, nspin:end]<\/span>
   nsteps = size(utrue)[2]<\/span>
   upred = forecast_RC(rc, nsteps, uspin = utest[:, 1:nspin])<\/span>
   <\/span>
   # now plot results (not defined here)<\/span>
   t = collect(0:nsteps-1)*\u0394t<\/span>
   p = plot_prediction(t, upred, utrue)<\/span>
   display(p)<\/span>
end<\/span>
L63_ex()<\/span><\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n


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\u5728\u4e0b\u9762\u7684\u4f8b\u5b50\u4e2d\uff0c\u6211\u4eec\u770b\u5230\u4e86\u50a8\u5907\u6c60\u8ba1\u7b97\u7684\u5173\u952e\u4f18\u52bf\u4e4b\u4e00\u3002\u4e0d\u4ec5\u9884\u6d4b\u7ed3\u679c\u975e\u5e38\u597d\uff0c\u8fd9\u91cc\u8fbe\u5230\u4e86~9\u4e2a\u674e\u4e9a\u666e\u8bfa\u592b\u65f6\u95f4\uff0c\u800c\u4e14\u5f53\u9884\u6d4b\u6700\u7ec8\u53d1\u6563\u65f6<\/span>\uff08\u7531\u4e8e\u201c\u8774\u8776\u6548\u5e94\u201d\u5fc5\u987b\u5982\u6b64\uff09<\/span>\uff0c\u50a8\u5907\u6c60\u8ba1\u7b97\u4fdd\u6301\u5728\u5438\u5f15\u5b50\u4e0a\uff0c\u5e76\u7ed9\u51fa\u201c\u7269\u7406\u9884\u6d4b\u201d\uff0c\u9884\u6d4b\u5177\u6709\u4e0e\u5b9e\u9645\u6570\u636e\u76f8\u540c\u7684\u7edf\u8ba1\u7279\u5f81\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u51fa\u50a8\u5907\u6c60\u8ba1\u7b97\u7684D\u4e2a\u6700\u5927\u7684\u674e\u4e9a\u666e\u8bfa\u592b\u6307\u6570\uff0c\u5e76\u4e0e\u8f93\u5165\u6570\u636e\u7684\u674e\u4e9a\u666e\u8bfa\u592b\u6307\u6570\u8fdb\u884c\u6bd4\u8f83\u3002\u5f53\u4e24\u8005\u5339\u914d\u65f6\uff0c\u6211\u4eec\u5c31\u4f1a\u5f97\u5230\u8fd9\u6837\u7684\u7279\u5f81\uff0c\u5373\u672a\u6765\u7684\u9884\u6d4b\u4e0e\u6570\u636e\u201c\u6c14\u5019\u5b66\u4e0a\u76f8\u4f3c\u201d\u3002\u8fd9\u610f\u5473\u7740\u9884\u6d4b\u5177\u6709\u4e0e\u8f93\u5165\u6570\u636e\u76f8\u540c\u7684\u7edf\u8ba1\u5c5e\u6027<\/span>\uff08\u5e73\u5747\u503c\u3001\u6807\u51c6\u5dee\u3001\u76f8\u5173\u6027\u3001\u8bef\u5dee\u589e\u957f……\uff09<\/span>\u3002<\/span><\/section>\n
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\u590d\u73b0\u6b63\u786e\u7684\u8bef\u5dee\u589e\u957f\u7edf\u8ba1<\/span>\uff08\u674e\u4e9a\u666e\u8bfa\u592b\u6307\u6570\uff09<\/span>\u7684\u5c5e\u6027\u4e0d\u4ec5\u5bf9\u9884\u6d4b\u7684\u7a33\u5b9a\u6027\u5f88\u91cd\u8981\uff0c\u800c\u4e14\u5bf9\u5c06\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u7eb3\u5165\u5f53\u524d\u6700\u5148\u8fdb\u7684\u9884\u6d4b\u65b9\u6cd5\uff0c\u5982\u96c6\u6210\u5361\u5c14\u66fc\u6ee4\u6ce2\u6216\u5176\u4ed6\u5f62\u5f0f\u7684\u6570\u636e\u540c\u5316\u4e5f\u5f88\u91cd\u8981\u3002\u8fd9\u4e9b\u65b9\u6cd5\u4f7f\u7528\u8d1d\u53f6\u65af\u63a8\u65ad\uff0c\u5c06\u7a00\u758f\u548c\u5608\u6742\u7684\u89c2\u6d4b\u6570\u636e\u7eb3\u5165\u7cfb\u7edf\u7684\u6a21\u578b<\/span>\uff08\u6570\u636e\u9a71\u52a8\u6216\u7269\u7406\uff09<\/span>\u3002\u63a8\u7406\u9700\u8981\u6a21\u578b\u7684\u6270\u52a8\u96c6\u5408\u518d\u73b0\u7269\u7406\u4e0a\u771f\u5b9e\u7684\u6270\u52a8\u9884\u6d4b\uff0c\u56e0\u6b64\u8fd9\u4e00\u7279\u6027\u975e\u5e38\u91cd\u8981\u3002<\/span><\/section>\n
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\u56fe4. \u4ecet=0\u5f00\u59cb\u5bf9X\u3001Y\u548cZ\u8fdb\u884c\u9884\u6d4b\u3002\u6ce8\u610f\u826f\u597d\u7684\u77ed\u671f\u9884\u6d4b\u4ee5\u53ca\u957f\u671f\u7edf\u8ba1\u7684\u518d\u73b0\u3002\u56fe\u7247\u7531\u4f5c\u8005\u63d0\u4f9b\u3002<\/span><\/p>\n

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7. \u6d4b\u8bd5\u50a8\u5907\u6c60\u8ba1\u7b97<\/strong><\/strong><\/p>\n<\/section>\n

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\u4e00\u4e2a\u5e38\u89c1\u7684\u65f6\u95f4\u5e8f\u5217\u9884\u6d4b\u7684\u673a\u5668\u5b66\u4e60\u57fa\u51c6\u662f\u5355\u6b65\u9884\u6d4b\u7684\u5747\u65b9\u8bef\u5dee\u3002\u4e8b\u5b9e\u4e0a\uff0c\u8fd9\u6b63\u662f\u6211\u4eec\u8bad\u7ec3Wout \u4ee5\u83b7\u5f97 u(t)\u2192r(t)\u2192u(t+1) \u7684\u7ebf\u6027\u6620\u5c04\u65b9\u6cd5\u3002\u7136\u800c\uff0c\u4f5c\u4e3a\u4e00\u79cd\u6d4b\u8bd5\u65b9\u6cd5\uff0c\u5355\u6b65\u9884\u6d4b\u5e76\u4e0d\u662f\u4e00\u4e2a\u5408\u9002\u7684\u6307\u6807\uff1b\u5b83\u8fc7\u5206\u5f3a\u8c03\u4e86\u6570\u636e\u4e2d\u7684\u9ad8\u9891\u6a21\u5f0f\uff0c\u800c\u727a\u7272\u4e86\u50a8\u5907\u6c60\u8ba1\u7b97\u9884\u6d4b\u7684\u957f\u671f\u7a33\u5b9a\u6027\u3002<\/span><\/section>\n
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\u6211\u4eec\u9884\u6d4b\u65f6\u95f4\u7684\u6807\u51c6\u6307\u6807\u662f\u6709\u6548\u9884\u6d4b\u65f6\u95f4<\/span> (valid prediction time\uff0cVPT)<\/span>\u3002VPT \u662f\u9884\u6d4b\u51c6\u786e\u5ea6\u8d85\u8fc7\u7ed9\u5b9a\u9608\u503c\u7684\u65f6\u95f4 t\u3002\u4f8b\u5982\uff0c<\/span><\/section>\n
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\u56fe5. \u6d4b\u8bd5\u50a8\u5907\u6c60\u8ba1\u7b97\u7684\u521d\u59cb\u6761\u4ef6\u3002\u8fd9\u4e9b\u9884\u6d4b\u7684\u6709\u6548\u9884\u6d4b\u65f6\u95f4\uff08VPT\uff09\u663e\u793a\u5728\u4e0b\u9762\u7684\u76f4\u65b9\u56fe\u4e2d\u3002\u4ece\u5438\u5f15\u5b50\u7684\u4e2d\u5fc3\u5f00\u59cb\u9884\u6d4b\u672a\u6765\u7684\u884c\u4e3a\u8981\u6bd4\u4ece\u53f6\u7247\u5f00\u59cb\u9884\u6d4b\u96be\u5f97\u591a\u3002\u56fe\u7247\u7531\u4f5c\u8005\u63d0\u4f9b\u3002<\/span><\/p>\n

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\u56fe6. \u76f4\u65b9\u56fe\u663e\u793a\u4e0a\u8ff0100\u4e2a\u521d\u59cb\u6761\u4ef6\u4e0b\u7684\u9884\u6d4b\u65f6\u95f4\u5206\u5e03\u3002\u56fe\u7247\u7531\u4f5c\u8005\u63d0\u4f9b\u3002<\/span><\/p>\n

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\u4e3a\u50a8\u5907\u6c60\u8ba1\u7b97\u627e\u5230\u6b63\u786e\u7684\u53c2\u6570\u662f\u4f7f\u7528\u8be5\u6280\u672f\u7684\u4e3b\u8981\u6311\u6218\u4e4b\u4e00\u3002\u6b63\u5982\u6211\u4eec\u5728\u4e0b\u9762\u770b\u5230\u7684\uff0c\u9884\u6d4b\u80fd\u529b\u5bf9\u6211\u4eec\u9009\u62e9\u7684\u53c2\u6570\u503c\u76f8\u5f53\u654f\u611f\uff0c\u4f7f\u5f97\u8fd9\u662f\u4e00\u4e2a\u6709\u70b9\u56f0\u96be\u7684\u591a\u7ef4\u975e\u7ebf\u6027\u4f18\u5316\u95ee\u9898\u3002\u503c\u5f97\u5e86\u5e78\u7684\u662f\uff0c\u6709\u8bb8\u591a\u7b97\u6cd5\u4e13\u95e8\u5904\u7406\u6b64\u7c7b\u95ee\u9898\uff0c\u4f8b\u5982 CMAES\uff0c\u4f46\u4efb\u4f55\u5168\u5c40\u4f18\u5316\u7b97\u6cd5\u90fd\u53ef\u4ee5\u3002<\/span><\/section>\n
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\u56fe7. \u9884\u6d4b\u5bf9\u50a8\u5907\u6c60\u8ba1\u7b97\u53c2\u6570\u53d8\u5316\u7684\u654f\u611f\u5ea6\u3002\u7ea2\u8272\u8868\u793a\u5e73\u5747VPT\u9ad8\u7684\u533a\u57df\uff0c\u84dd\u8272\u8868\u793aVPT\u4f4e\u7684\u533a\u57df\u3002\u5728\u6574\u4e2a\u53c2\u6570\u7a7a\u95f4\u4e2d\u8fdb\u884c\u641c\u7d22\u662f\u4e00\u4e2a\u4e0d\u7b80\u5355\u7684\u95ee\u9898\u3002\u56fe\u7247\u7531\u4f5c\u8005\u63d0\u4f9b\u3002<\/span><\/p>\n

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\u6211\u5c06\u91c7\u7528\u4e00\u4e2a\u4e09\u6b65\u4f18\u5316\u65b9\u6cd5\u3002\u8be5\u7a0b\u5e8f\u662f<\/span><\/p>\n

1\u3001\u56fa\u5b9a\u53c2\u6570<\/span><\/p>\n

2\u3001\u901a\u8fc7\u7ebf\u6027\u56de\u5f52\u8bad\u7ec3Wout<\/span><\/p>\n

3\u3001\u901a\u8fc7\u4e00\u7cfb\u5217\u957f\u671f\u9884\u6d4b\u8bc4\u4f30\u635f\u5931\u51fd\u6570<\/span><\/section>\n
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\u6211\u7528\u6765\u8bc4\u4f30\u6570\u636e\u62df\u5408\u7684\u635f\u5931\u51fd\u6570\u662f\uff1a<\/span><\/section>\n
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\u5176\u4e2d M \u662f\u7528\u4e8e\u6bd4\u8f83\u7684\u9884\u6d4b\u6570\u76ee\u3002u\u1da0\u662f\u9884\u6d4b\u503c\uff0cu\u662f\u6570\u636e\u3002\u6307\u6570\u9879\u662f\u4e3a\u4e86\u62b5\u6d88\u9884\u671f\u8bef\u5dee\u5728\u65f6\u95f4\u4e0a\u7684\u6307\u6570\u5f0f\u589e\u957f\u3002<\/span><\/section>\n
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\u589e\u52a0 M \u53ef\u4ee5\u5e73\u6ed1\u635f\u5931\u51fd\u6570\uff0c\u5e76\u80fd\u591f\u66f4\u597d\u5730\u6536\u655b\u5230\u5168\u5c40\u6700\u5c0f\u503c\u3002\u7136\u800c\uff0c\u59a5\u534f\u662f\u5fc5\u8981\u7684\uff0c\u56e0\u4e3a\u589e\u52a0 M \u5c06\u5927\u5927\u589e\u52a0\u8ba1\u7b97\u590d\u6742\u6027\u3002\u6211\u4eec\u51ed\u7ecf\u9a8c\u53d1\u73b0\uff0c\u5bf9\u4e8e\u7b80\u5355\u6a21\u578b\u800c\u8a00\uff0c\u4f7f\u7528 15~20 \u4e2a\u5de6\u53f3\u7684\u9884\u6d4b\u901a\u5e38\u5c31\u8db3\u591f\u4e86\uff0c\u53ea\u8981\u8fd9\u4e9b\u9884\u6d4b\u662f\u5728\u7edf\u8ba1\u4e0a\u72ec\u7acb\u7684\u70b9<\/span>\uff08\u5f7c\u6b64\u76f8\u8ddd\u8d85\u8fc7\u51e0\u4e2a\u674e\u4e9a\u666e\u8bfa\u592b\u65f6\u95f4\u5c3a\u5ea6\uff09<\/span>\u8fdb\u884c\u7684\u3002\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u6570\u636e\u4f1a\u5f88\u597d\u5730\u5bf9\u8f93\u5165\u7cfb\u7edf\u5438\u5f15\u5b50\u8fdb\u884c\u91c7\u6837\uff0c\u4f46\u5b9e\u9645\u4e0a\u5e76\u975e\u603b\u662f\u5982\u6b64\u3002<\/span><\/section>\n
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\u4e0b\u9762\u6211\u4eec\u770b\u5230\u8981\u6700\u5c0f\u5316\u7684\u76ee\u6807\u51fd\u6570\u548c\u5355\u6b21\u672a\u6765\u9884\u6d4b\u7684\u635f\u5931\u51fd\u6570\u7684\u5b9a\u4e49\u3002\u8be5\u4ee3\u7801\u4f9d\u8d56\u4e8e\u4e00\u4e2a\u5305\u542b\u975e\u4f18\u5316\u53c2\u6570\u548c\u8bad\u7ec3\u6570\u636e\u7684\u5bf9\u8c61f<\/span>\uff08\u6b64\u5904\u672a\u663e\u793a\uff09<\/span>\u3002\u7136\u540e\uff0c\u6211\u4eec\u4f7f\u7528\u5168\u5c40\u4f18\u5316\u7a0b\u5e8f\u6765\u4f18\u5316\u4e0d\u540c\u7684\u53c2\u6570\u3002<\/span><\/section>\n
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function Loss(data, spinup_steps, rc::esn)<\/span>
       “””Loss function for an individual forecast”””<\/span>
       uspin = data[:, 1:spinup_steps]<\/span>
       utrue = data[:, spinup_steps:end]<\/span>
       Ttrue = size(utrue)[2]<\/span>
       upred = forecast_RC(rc, Ttrue; uspin=uspin)<\/span>
   <\/span>
       err = upred.-utrue<\/span>
       weight = exp.(-(1:Ttrue).\/Ttrue)<\/span>
       return norm(err.*weight’)<\/span>
   end<\/span>

   function obj(\u03b8)<\/span>
       “””The objective function to minimize over the optimization<\/span>
       <\/span>
       Args:<\/span>
           \u03b8 : Array of parameters<\/span>
       <\/span>
       Returns:<\/span>
           objective value<\/span>
       “””<\/span>
       SR, \u03b1, log\u03c3, \u03c3b, log\u03b2 = \u03b8<\/span>
       D, _ = size(f.train_data)<\/span>
       <\/span>
       # build the RC<\/span>
       rc = esn(D, f.N, SR, f.\u03c1A, \u03b1, exp(log\u03c3), \u03c3b, exp(log\u03b2))<\/span>
           <\/span>
       # Train the RC<\/span>
       train_RC(rc, f.train_data)<\/span>
       <\/span>
       # Compute loss over n_valid forecasts<\/span>
       n_valid = length(f.valid_data_list)<\/span>
       loss = zeros(n_valid)<\/span>
       for i in 1:n_valid<\/span>
           loss[i] = Loss(f.valid_data_list[i], f.spinup_steps, rc)<\/span>
       end<\/span>
       <\/span>
       # Objective is sum of loss functions<\/span>
       return sum(loss)<\/span>
   end<\/span><\/span><\/p>\n<\/section>\n<\/section>\n<\/blockquote>\n


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9. \u603b\u7ed3<\/strong><\/strong><\/p>\n<\/section>\n

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\u6211\u4eec\u7b80\u8981\u4ecb\u7ecd\u4e86\u50a8\u5907\u6c60\u8ba1\u7b97\uff0c\u4ee5\u53ca\u8bad\u7ec3\u3001\u9884\u6d4b\u548c\u4f18\u5316\u50a8\u5907\u6c60\u8ba1\u7b97\u4ee5\u9884\u6d4b\u590d\u6742\u7cfb\u7edf\u6240\u9700\u7684\u4ee3\u7801\u3002\u8ba8\u8bba\u4e86\u4f18\u5316\u4ee5\u627e\u5230\u6b63\u786e\u53c2\u6570\u7684\u91cd\u8981\u6027\uff0c\u4ee5\u53ca\u50a8\u5907\u6c60\u8ba1\u7b97\u590d\u73b0\u8f93\u5165\u6570\u636e\u7684\u957f\u671f\u7edf\u8ba1\u6570\u636e\u7684\u80fd\u529b\u3002\u867d\u7136\u6211\u4eec\u4ec5\u4ee5\u6d1b\u4f26\u5179\u7cfb\u7edf\u4e3a\u4f8b\uff0c\u4f46\u50a8\u5907\u6c60\u8ba1\u7b97\u6709\u8bb8\u591a\u66f4\u590d\u6742\u7684\u5e94\u7528\u3002<\/span><\/section>\n
 <\/span><\/section>\n
\u53ef\u4ee5\u5728\u4ee5\u4e0b\u4f4d\u7f6e\u627e\u5230\u4ee3\u7801\u548c\u793a\u4f8b\uff1a<\/span><\/section>\n
https:\/\/github.com\/japlatt\/BasicReservoirComputing<\/span><\/section>\n
 <\/span><\/section>\n
\u5173\u4e8e\u50a8\u5907\u6c60\u8ba1\u7b97\u80cc\u540e\u7406\u8bba\u7684\u66f4\u591a\u4fe1\u606f\uff0c\u4ee5\u53ca\u5bf9\u8bbe\u8ba1\u9009\u62e9\u7684\u66f4\u6df1\u5165\u7684\u8ba8\u8bba\uff0c\u53ef\u4ee5\u5728\u4ee5\u4e0b\u8bba\u6587\u4e2d\u627e\u5230\u3002\u5982\u679c\u4f60\u89c9\u5f97\u8fd9\u4e9b\u4fe1\u606f\u6709\u7528\uff0c\u8bf7\u5f15\u7528\u5b83\u4eec\u3002<\/span><\/section>\n
 <\/span><\/section>\n
1. Jason A. Platt et al. \u201cRobust forecasting using predictive generalized synchronization in reservoir computing\u201d. In: Chaos 31 (2021), p. 123118. URL: https:\/\/doi.org\/10.1063\/5.0066013<\/span><\/section>\n
2. Platt,  J. A., Penny, S. G., Smith, T. A., Chen, T.-C., & Abarbanel, H. D.  I. (2022). A Systematic Exploration of Reservoir Computing for  Forecasting Complex Spatiotemporal Dynamics. arXiv http:\/\/arxiv.org\/abs\/2201.08910<\/span><\/section>\n
 <\/span><\/section>\n
\u5173\u4e8e\u5c06\u5faa\u73af\u795e\u7ecf\u7f51\u7edc\u6574\u5408\u5230\u6570\u636e\u540c\u5316\u7b97\u6cd5\u4e2d\uff0c\u4ee5\u5904\u7406\u7a00\u758f\/\u566a\u58f0\/\u4e0d\u5b8c\u6574\u6570\u636e\u7684\u4fe1\u606f\uff1a<\/span><\/section>\n
 <\/span><\/section>\n
Penny, S. G., Smith, T. A., Chen, T.-C., Platt, J. A.,  Lin, H.-Y., Goodliff, M., & Abarbanel, H. D. I. (2021). Integrating  Recurrent Neural Networks with Data Assimilation for Scalable  Data-Driven State Estimation. arXiv [cs.LG]. Opgehaal van http:\/\/arxiv.org\/abs\/2109.12269<\/span><\/section>\n
 <\/span><\/section>\n
\u8fd9\u7bc7\u6587\u7ae0\u7684\u6700\u521d\u7075\u611f\u6765\u81ea\u4e8e Lu \u7b49\u4eba2018\u5e74\u7684\u5de5\u4f5c\uff1a<\/span><\/section>\n

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Zhixin Lu, Brian R. Hunt, and Edward Ott. \u201cAttractor reconstruction by machine learning\u201d. In: Chaos: An Interdisciplinary Journal of Nonlinear Science 28.6 (2018), p. 061104. DOI: 10.1063\/1.5039508<\/span><\/section>\n


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\u672c\u6587\u7ffb\u8bd1\u81ea\uff1aTowards Data Science<\/span><\/section>\n
\u539f\u6587\u9898\u76ee\uff1a<\/span><\/section>\n
Data Driven Modeling of Complex Systems: A Reservoir Computing Tutorial<\/span><\/section>\n
\u539f\u6587\u94fe\u63a5\uff1a<\/span><\/section>\n
https:\/\/towardsdatascience.com\/data-driven-modeling-of-complex-systems-8a96dc92abf9<\/span><\/section>\n<\/section>\n<\/section>\n<\/blockquote>\n

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\u590d\u6742\u79d1\u5b66\u6700\u65b0\u8bba\u6587<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u96c6\u667a\u6591\u56fe\u9876\u520a\u8bba\u6587\u901f\u9012\u680f\u76ee\u4e0a\u7ebf\u4ee5\u6765\uff0c\u6301\u7eed\u6536\u5f55\u6765\u81eaNature\u3001Science<\/span>\u7b49\u9876\u520a\u7684\u6700\u65b0\u8bba\u6587\uff0c\u8ffd\u8e2a\u590d\u6742\u7cfb\u7edf\u3001\u7f51\u7edc\u79d1\u5b66\u3001\u8ba1\u7b97\u793e\u4f1a\u79d1\u5b66\u7b49\u9886\u57df\u7684\u524d\u6cbf\u8fdb\u5c55\u3002\u73b0\u5728\u6b63\u5f0f\u63a8\u51fa\u8ba2\u9605\u529f\u80fd\uff0c\u6bcf\u5468\u901a\u8fc7\u5fae\u4fe1\u670d\u52a1\u53f7\u300c\u96c6\u667a\u6591\u56fe\u300d\u63a8\u9001\u8bba\u6587\u4fe1\u606f\u3002\u626b\u63cf\u4e0b\u65b9\u4e8c\u7ef4\u7801\u5373\u53ef\u4e00\u952e\u8ba2\u9605\uff1a<\/span>
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