{"id":10021421,"date":"2026-07-30T23:56:34","date_gmt":"2026-07-30T15:56:34","guid":{"rendered":"https:\/\/itmax.cn\/?p=10021421"},"modified":"2026-07-30T23:56:34","modified_gmt":"2026-07-30T15:56:34","slug":"%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9c%e5%85%a5%e9%97%a8%ef%bc%9a%e4%bb%8e-xor-%e9%97%ae%e9%a2%98%e7%90%86%e8%a7%a3%e5%89%8d%e9%a6%88%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9c","status":"publish","type":"post","link":"https:\/\/itmax.cn\/?p=10021421","title":{"rendered":"\u795e\u7ecf\u7f51\u7edc\u5165\u95e8\uff1a\u4ece XOR \u95ee\u9898\u7406\u89e3\u524d\u9988\u795e\u7ecf\u7f51\u7edc"},"content":{"rendered":"<blockquote><p>\u672c\u6587\u6863\u57fa\u4e8eXOR\u95ee\u9898\uff0c\u5faa\u5e8f\u6e10\u8fdb\u5730\u8bb2\u89e3\u795e\u7ecf\u7f51\u7edc\u7684\u57fa\u672c\u539f\u7406\u3001\u6570\u5b66\u63a8\u5bfc\u548c\u4ee3\u7801\u5b9e\u73b0\u3002<br \/>\u9002\u5408\u6709\u4e00\u5b9a\u7f16\u7a0b\u57fa\u7840\uff0c\u4f46\u521d\u6b21\u63a5\u89e6\u795e\u7ecf\u7f51\u7edc\u7684\u7a0b\u5e8f\u5458\u9605\u8bfb\u3002<\/p><\/blockquote>\n<hr>\n<h2>\u76ee\u5f55<\/h2>\n<ol>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#1-\u795e\u7ecf\u7f51\u7edc\u662f\u4ec0\u4e48\" target=\"_blank\" rel=\"noopener noreferrer\">\u795e\u7ecf\u7f51\u7edc\u662f\u4ec0\u4e48\uff1f<\/a><\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#2-xor-\u95ee\u9898\u4e3a\u4ec0\u4e48\u9700\u8981\u795e\u7ecf\u7f51\u7edc\" target=\"_blank\" rel=\"noopener noreferrer\">XOR \u95ee\u9898\uff1a\u4e3a\u4ec0\u4e48\u9700\u8981\u795e\u7ecf\u7f51\u7edc\uff1f<\/a><\/li>\n<li>\n<p><a href=\"https:\/\/segmentfault.com\/a\/#3-\u7f51\u7edc\u7ed3\u6784\u4e09\u5c42\u7684\u79d8\u5bc6\" target=\"_blank\" rel=\"noopener noreferrer\">\u7f51\u7edc\u7ed3\u6784\uff1a\u4e09\u5c42\u7684\u79d8\u5bc6<\/a><\/p>\n<ul>\n<li>3.1 <a href=\"https:\/\/segmentfault.com\/a\/#31-\u5c42-layer-\u7684\u6982\u5ff5\" target=\"_blank\" rel=\"noopener noreferrer\">\u5c42 (Layer) \u7684\u6982\u5ff5<\/a><\/li>\n<li>3.2 <a href=\"https:\/\/segmentfault.com\/a\/#32-\u4e3a\u4ec0\u4e48\u9690\u85cf\u5c42\u9700\u8981-4-\u4e2a\u795e\u7ecf\u5143\" target=\"_blank\" rel=\"noopener noreferrer\">\u4e3a\u4ec0\u4e48\u9690\u85cf\u5c42\u9700\u8981 4 \u4e2a\u795e\u7ecf\u5143\uff1f<\/a><\/li>\n<li>3.3 <a href=\"https:\/\/segmentfault.com\/a\/#33-\u7f51\u7edc\u53c2\u6570\u6570\u91cf\" target=\"_blank\" rel=\"noopener noreferrer\">\u7f51\u7edc\u53c2\u6570\u6570\u91cf<\/a><\/li>\n<li>3.4 <a href=\"https:\/\/segmentfault.com\/a\/#34-neuralnet-\u7ed3\u6784\u4f53\u7684\u8bbe\u8ba1\u610f\u56fe\u628a\u5c42\u6620\u5c04\u6210\u6570\u636e\u7ed3\u6784\" target=\"_blank\" rel=\"noopener noreferrer\">NeuralNet \u7ed3\u6784\u4f53\u7684\u8bbe\u8ba1\u610f\u56fe<\/a><\/li>\n<li>3.5 <a href=\"https:\/\/segmentfault.com\/a\/#35-\u7ef4\u5ea6\u4e3a\u4ec0\u4e48\u8fd9\u4e48\u5b9a\u4ece\u5f62\u72b6\u770b\u4ee3\u7801\" target=\"_blank\" rel=\"noopener noreferrer\">\u7ef4\u5ea6\u4e3a\u4ec0\u4e48\u8fd9\u4e48\u5b9a\uff1f<\/a><\/li>\n<li>3.6 <a href=\"https:\/\/segmentfault.com\/a\/#36-\u4e3a\u4ec0\u4e48\u4e0d\u76f4\u63a5\u7528\u52a8\u6001\u5185\u5b58\u9759\u6001\u6570\u7ec4\u7684\u53d6\u820d\" target=\"_blank\" rel=\"noopener noreferrer\">\u9759\u6001\u6570\u7ec4\u7684\u53d6\u820d<\/a><\/li>\n<li>3.7 <a href=\"https:\/\/segmentfault.com\/a\/#37-\u4e00\u4e2a\u66f4\u901a\u7528\u7684\u5c42\u62bd\u8c61\u957f\u4ec0\u4e48\u6837\" target=\"_blank\" rel=\"noopener noreferrer\">\u901a\u7528\u5c42\u62bd\u8c61<\/a><\/li>\n<\/ul>\n<\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#4-\u795e\u7ecf\u5143\u5185\u90e8\u52a0\u6743\u6c42\u548c\u4e0e\u6fc0\u6d3b\u51fd\u6570\" target=\"_blank\" rel=\"noopener noreferrer\">\u795e\u7ecf\u5143\u5185\u90e8\uff1a\u52a0\u6743\u6c42\u548c\u4e0e\u6fc0\u6d3b\u51fd\u6570<\/a><\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#5-\u6b63\u5411\u4f20\u64ad\u4fe1\u53f7\u5982\u4f55\u6d41\u52a8\" target=\"_blank\" rel=\"noopener noreferrer\">\u6b63\u5411\u4f20\u64ad\uff1a\u4fe1\u53f7\u5982\u4f55\u6d41\u52a8<\/a><\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#6-\u635f\u5931\u51fd\u6570\u5982\u4f55\u8861\u91cf\u5bf9\u9519\" target=\"_blank\" rel=\"noopener noreferrer\">\u635f\u5931\u51fd\u6570\uff1a\u5982\u4f55\u8861\u91cf\u5bf9\u9519<\/a><\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#7-\u53cd\u5411\u4f20\u64ad\u8bef\u5dee\u5982\u4f55\u9006\u6d41\" target=\"_blank\" rel=\"noopener noreferrer\">\u53cd\u5411\u4f20\u64ad\uff1a\u8bef\u5dee\u5982\u4f55\u9006\u6d41<\/a><\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#8-\u68af\u5ea6\u4e0b\u964d\u5982\u4f55\u8c03\u6574\u53c2\u6570\" target=\"_blank\" rel=\"noopener noreferrer\">\u68af\u5ea6\u4e0b\u964d\uff1a\u5982\u4f55\u8c03\u6574\u53c2\u6570<\/a><\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#9-\u8bad\u7ec3\u5faa\u73af\u91cd\u590d\u7684\u529b\u91cf\" target=\"_blank\" rel=\"noopener noreferrer\">\u8bad\u7ec3\u5faa\u73af\uff1a\u91cd\u590d\u7684\u529b\u91cf<\/a><\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#10-\u4ee3\u7801\u9010\u6bb5\u7cbe\u8bfb\" target=\"_blank\" rel=\"noopener noreferrer\">\u4ee3\u7801\u9010\u6bb5\u7cbe\u8bfb<\/a><\/li>\n<li><a href=\"https:\/\/segmentfault.com\/a\/#11-\u8fd0\u884c\u4e0e\u89e3\u8bfb\u7ed3\u679c\" target=\"_blank\" rel=\"noopener noreferrer\">\u8fd0\u884c\u4e0e\u89e3\u8bfb\u7ed3\u679c<\/a><\/li>\n<li>\n<p><a href=\"https:\/\/segmentfault.com\/a\/#12-\u5ef6\u4f38\u601d\u8003\u4e0e\u6539\u8fdb\u65b9\u5411\" target=\"_blank\" rel=\"noopener noreferrer\">\u5ef6\u4f38\u601d\u8003\u4e0e\u6539\u8fdb\u65b9\u5411<\/a><\/p>\n<ul>\n<li>12.1 <a href=\"https:\/\/segmentfault.com\/a\/#121-\u4ee3\u7801\u53ef\u4ee5\u6539\u8fdb\u7684\u5730\u65b9\" target=\"_blank\" rel=\"noopener noreferrer\">\u4ee3\u7801\u6539\u8fdb<\/a><\/li>\n<li>12.2 <a href=\"https:\/\/segmentfault.com\/a\/#122-\u6570\u5b66\u5c42\u9762\u7684\u6539\u8fdb\" target=\"_blank\" rel=\"noopener noreferrer\">\u6570\u5b66\u5c42\u9762\u6539\u8fdb<\/a><\/li>\n<li>12.3 <a href=\"https:\/\/segmentfault.com\/a\/#123-\u66f4\u4e00\u822c\u7684\u795e\u7ecf\u7f51\u7edc\u6a21\u5f0f\" target=\"_blank\" rel=\"noopener noreferrer\">\u66f4\u4e00\u822c\u7684\u795e\u7ecf\u7f51\u7edc\u6a21\u5f0f<\/a><\/li>\n<li>12.4 <a href=\"https:\/\/segmentfault.com\/a\/#124-\u4ece-c-\u5230\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6\" target=\"_blank\" rel=\"noopener noreferrer\">\u4ece C \u5230\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6<\/a><\/li>\n<li>12.5 <a href=\"https:\/\/segmentfault.com\/a\/#125-\u4ece-neuralnet-\u5230\u901a\u7528\u6846\u67b6\u7ed3\u6784\u4f53\u6f14\u8fdb\u8def\u7ebf\" target=\"_blank\" rel=\"noopener noreferrer\">\u4ece NeuralNet \u5230\u901a\u7528\u6846\u67b6<\/a><\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<hr>\n<h2>1. \u795e\u7ecf\u7f51\u7edc\u662f\u4ec0\u4e48\uff1f<\/h2>\n<h3>1.1 \u76f4\u89c2\u7406\u89e3<\/h3>\n<p><strong>\u795e\u7ecf\u7f51\u7edc (Neural Network)<\/strong> \u662f\u4e00\u79cd\u53d7\u751f\u7269\u795e\u7ecf\u7cfb\u7edf\u542f\u53d1\u7684\u8ba1\u7b97\u6a21\u578b\u3002\u5b83\u7531\u5927\u91cf\u7b80\u5355\u7684\u5904\u7406\u5355\u5143\uff08\u79f0\u4e3a<strong>\u795e\u7ecf\u5143<\/strong>\uff0cneuron\uff09\u76f8\u4e92\u8fde\u63a5\u800c\u6210\uff0c\u901a\u8fc7\u8c03\u6574\u8fde\u63a5\u5f3a\u5ea6\u6765\u4ece\u6570\u636e\u4e2d&#8221;\u5b66\u4e60&#8221;\u6a21\u5f0f\u3002<\/p>\n<p>\u4f60\u53ef\u4ee5\u628a\u5b83\u60f3\u8c61\u6210\u4e00\u4e2a\u6781\u5176\u590d\u6742\u7684<strong>\u51fd\u6570\u903c\u8fd1\u5668<\/strong>\uff1a<\/p>\n<blockquote><p>\u795e\u7ecf\u7f51\u7edc\u53ef\u4ee5\u8fd1\u4f3c\u4efb\u4f55\u6570\u5b66\u51fd\u6570 \u2014\u2014 \u53ea\u8981\u7ed9\u8db3\u6570\u636e\u548c\u5408\u9002\u7684\u7f51\u7edc\u7ed3\u6784\u3002<\/p><\/blockquote>\n<h3>1.2 \u751f\u7269\u7c7b\u6bd4<\/h3>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u751f\u7269\u795e\u7ecf\u5143<\/th>\n<th align=\"left\">\u4eba\u5de5\u795e\u7ecf\u5143<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">\u6811\u7a81\uff08\u63a5\u6536\u4fe1\u53f7\uff09<\/td>\n<td align=\"left\">\u8f93\u5165\u503c $x_1, x_2, &#8230;, x_n$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u7a81\u89e6\uff08\u8fde\u63a5\u5f3a\u5ea6\uff09<\/td>\n<td align=\"left\">\u6743\u91cd $w_1, w_2, &#8230;, w_n$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u7ec6\u80de\u4f53\uff08\u6574\u5408\u4fe1\u53f7\uff09<\/td>\n<td align=\"left\">\u52a0\u6743\u6c42\u548c $\/sum w_i x_i + b$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u8f74\u7a81\uff08\u8f93\u51fa\u4fe1\u53f7\uff09<\/td>\n<td align=\"left\">\u6fc0\u6d3b\u51fd\u6570\u8f93\u51fa $a = f(\/sum w_i x_i + b)$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>1.3 \u795e\u7ecf\u7f51\u7edc\u7684\u6838\u5fc3\u601d\u60f3<\/h3>\n<pre><code>\u8f93\u5165\u6570\u636e \u2192 \u52a0\u6743\u6c42\u548c \u2192 \u6fc0\u6d3b\u51fd\u6570 \u2192 \u8f93\u51fa\u7ed3\u679c \u2192 \u8ba1\u7b97\u8bef\u5dee \u2192 \u53cd\u5411\u8c03\u6574\u6743\u91cd \u2192 \u91cd\u590d<\/code><\/pre>\n<p>\u795e\u7ecf\u7f51\u7edc&#8221;\u5b66\u4e60&#8221;\u7684\u672c\u8d28\u662f\uff1a<strong>\u4e0d\u65ad\u8c03\u6574\u6743\u91cd (weights) \u548c\u504f\u7f6e (biases)\uff0c\u4f7f\u5f97\u8f93\u51fa\u7ed3\u679c\u5c3d\u53ef\u80fd\u63a5\u8fd1\u6b63\u786e\u7b54\u6848<\/strong>\u3002<\/p>\n<hr>\n<h2>2. XOR \u95ee\u9898\uff1a\u4e3a\u4ec0\u4e48\u9700\u8981\u795e\u7ecf\u7f51\u7edc\uff1f<\/h2>\n<h3>2.1 \u4ec0\u4e48\u662f XOR\uff1f<\/h3>\n<p><strong>XOR\uff08\u5f02\u6216\uff0ceXclusive OR\uff09<\/strong> \u662f\u4e00\u4e2a\u4e8c\u5143\u903b\u8f91\u8fd0\u7b97\uff1a<\/p>\n<table>\n<thead>\n<tr>\n<th align=\"center\">\u8f93\u5165 A<\/th>\n<th align=\"center\">\u8f93\u5165 B<\/th>\n<th align=\"center\">A XOR B<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"center\">0<\/td>\n<td align=\"center\">0<\/td>\n<td align=\"center\">0<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">0<\/td>\n<td align=\"center\">1<\/td>\n<td align=\"center\">1<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">1<\/td>\n<td align=\"center\">0<\/td>\n<td align=\"center\">1<\/td>\n<\/tr>\n<tr>\n<td align=\"center\">1<\/td>\n<td align=\"center\">1<\/td>\n<td align=\"center\">0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u89c4\u5219\u5f88\u7b80\u5355\uff1a<strong>\u4e24\u4e2a\u8f93\u5165\u4e0d\u540c\u65f6\u8f93\u51fa 1\uff0c\u76f8\u540c\u65f6\u8f93\u51fa 0<\/strong>\u3002<\/p>\n<h3>2.2 \u4e3a\u4ec0\u4e48 XOR \u9700\u8981\u795e\u7ecf\u7f51\u7edc\uff1f<\/h3>\n<p>\u5982\u679c\u6211\u4eec\u5c06 XOR \u7684\u56db\u79cd\u60c5\u51b5\u753b\u5728\u4e8c\u7ef4\u5e73\u9762\u4e0a\uff1a<\/p>\n<pre><code>\u8f93\u51fa 0: (0,0) \u548c (1,1)\n\u8f93\u51fa 1: (0,1) \u548c (1,0)<\/code><\/pre>\n<p>\u4f60\u4f1a\u53d1\u73b0\uff1a<strong>\u65e0\u6cd5\u7528\u4e00\u6761\u76f4\u7ebf\u5c06\u4e24\u7c7b\u70b9\u5206\u5f00<\/strong>\u3002<\/p>\n<ul>\n<li>AND\uff08\u4e0e\uff09\u3001OR\uff08\u6216\uff09\u662f<strong>\u7ebf\u6027\u53ef\u5206<\/strong>\u7684 \u2192 \u53ef\u4ee5\u7528\u4e00\u6761\u76f4\u7ebf\u5206\u5f00<\/li>\n<li>XOR \u662f<strong>\u7ebf\u6027\u4e0d\u53ef\u5206<\/strong>\u7684 \u2192 \u5355\u5c42\u611f\u77e5\u673a\uff08\u6ca1\u6709\u9690\u85cf\u5c42\uff09\u6c38\u8fdc\u65e0\u6cd5\u89e3\u51b3<\/li>\n<\/ul>\n<p>\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48 XOR \u6210\u4e3a\u795e\u7ecf\u7f51\u7edc\u9886\u57df\u7684\u7ecf\u5178\u5165\u95e8\u95ee\u9898\uff1a\u5b83\u8bc1\u660e\u4e86<strong>\u5355\u5c42\u7f51\u7edc\u7684\u5c40\u9650\u6027<\/strong>\uff0c\u4e5f\u5c55\u793a\u4e86<strong>\u591a\u5c42\u7f51\u7edc\uff08\u542b\u9690\u85cf\u5c42\uff09\u7684\u5f3a\u5927<\/strong>\u3002<\/p>\n<blockquote><p>1969 \u5e74\uff0cMinsky \u548c Papert \u5728\u300a\u611f\u77e5\u673a\u300b\u4e00\u4e66\u4e2d\u6307\u51fa\u5355\u5c42\u611f\u77e5\u673a\u65e0\u6cd5\u89e3\u51b3 XOR \u95ee\u9898\uff0c\u8fd9\u76f4\u63a5\u5bfc\u81f4\u4e86\u4eba\u5de5\u667a\u80fd\u7684\u7b2c\u4e00\u4e2a&#8221;\u5bd2\u51ac&#8221;\u3002\u76f4\u5230 1980 \u5e74\u4ee3\u53cd\u5411\u4f20\u64ad\u7b97\u6cd5\u7684\u666e\u53ca\uff0c\u591a\u5c42\u7f51\u7edc\u624d\u89e3\u51b3\u4e86\u8fd9\u4e00\u95ee\u9898\u3002<\/p><\/blockquote>\n<h3>2.3 \u89e3\u51b3\u65b9\u6cd5<\/h3>\n<p>\u6211\u4eec\u7684\u7b56\u7565\u662f<strong>\u589e\u52a0\u4e00\u5c42\u9690\u85cf\u5c42<\/strong>\uff0c\u8ba9\u7f51\u7edc\u5148\u5b66\u4e60\u4e00\u4e2a\u975e\u7ebf\u6027\u7684\u7279\u5f81\u8868\u793a\uff0c\u518d\u505a\u5206\u7c7b\u3002<\/p>\n<pre><code>\u8f93\u5165\u5c42 (2\u4e2a)  \u2192  \u9690\u85cf\u5c42 (4\u4e2a)  \u2192  \u8f93\u51fa\u5c42 (1\u4e2a)\n    x1 \u2500\u2500\u2500\u2510\n          \u251c\u2500\u2500\u2500 h1 \u2500\u2500\u2500\u2510\n    x2 \u2500\u2500\u2500\u2524          \u2502\n          \u251c\u2500\u2500\u2500 h2 \u2500\u2500\u2500\u2524\n          \u2502          \u251c\u2500\u2500\u2500 y\n          \u251c\u2500\u2500\u2500 h3 \u2500\u2500\u2500\u2524\n          \u2502          \u2502\n          \u2514\u2500\u2500\u2500 h4 \u2500\u2500\u2500\u2518<\/code><\/pre>\n<p>\u9690\u85cf\u5c42\u7684\u4f5c\u7528\u76f8\u5f53\u4e8e<strong>\u7279\u5f81\u63d0\u53d6\u5668<\/strong> \u2014\u2014 \u5c06\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u4e00\u4e2a\u65b0\u7684\u7279\u5f81\u7a7a\u95f4\uff0c\u5728\u8fd9\u4e2a\u65b0\u7a7a\u95f4\u4e2d\u6570\u636e\u53d8\u5f97\u7ebf\u6027\u53ef\u5206\u3002<\/p>\n<hr>\n<h2>3. \u7f51\u7edc\u7ed3\u6784\uff1a\u4e09\u5c42\u7684\u79d8\u5bc6<\/h2>\n<h3>3.1 \u5c42 (Layer) \u7684\u6982\u5ff5<\/h3>\n<p>\u672c\u7a0b\u5e8f\u4f7f\u7528\u4e00\u4e2a<strong>\u4e09\u5c42\u524d\u9988\u795e\u7ecf\u7f51\u7edc<\/strong>\uff1a<\/p>\n<pre><code>\u8f93\u5165\u5c42 (Input Layer)     \u9690\u85cf\u5c42 (Hidden Layer)     \u8f93\u51fa\u5c42 (Output Layer)\n    size=2                    size=4                    size=1<\/code><\/pre>\n<ul>\n<li><strong>\u8f93\u5165\u5c42<\/strong>\uff1a\u63a5\u6536\u539f\u59cb\u6570\u636e\uff0c\u4e0d\u8fdb\u884c\u8ba1\u7b97\u3002\u6bcf\u4e2a\u8f93\u5165\u8282\u70b9\u4ee3\u8868\u4e00\u4e2a\u7279\u5f81\u3002<\/li>\n<li><strong>\u9690\u85cf\u5c42<\/strong>\uff1a\u4e2d\u95f4\u8ba1\u7b97\u5c42\uff0c\u63d0\u53d6\u548c\u7ec4\u5408\u7279\u5f81\u3002\u4e4b\u6240\u4ee5\u53eb&#8221;\u9690\u85cf&#8221;\uff0c\u662f\u56e0\u4e3a\u5b83\u7684\u503c\u4e0d\u662f\u76f4\u63a5\u53ef\u89c1\u7684\u8f93\u5165\u6216\u8f93\u51fa\u3002<\/li>\n<li><strong>\u8f93\u51fa\u5c42<\/strong>\uff1a\u4ea7\u751f\u6700\u7ec8\u9884\u6d4b\u7ed3\u679c\u3002<\/li>\n<\/ul>\n<blockquote><p><strong>\u4e3a\u4ec0\u4e48\u53eb&#8221;\u524d\u9988&#8221; (Feedforward)\uff1f<\/strong><br \/>\u56e0\u4e3a\u4fe1\u53f7\u53ea\u6cbf\u7740\u4e00\u4e2a\u65b9\u5411\u6d41\u52a8\uff1a\u4ece\u8f93\u5165\u5c42\u5230\u9690\u85cf\u5c42\u518d\u5230\u8f93\u51fa\u5c42\uff0c\u6ca1\u6709\u5faa\u73af\u6216\u53cd\u9988\u8fde\u63a5\u3002\u4e0e\u4e4b\u76f8\u5bf9\u7684\u662f&#8221;\u5faa\u73af\u795e\u7ecf\u7f51\u7edc&#8221; (RNN)\uff0c\u5b83\u6709\u53cd\u9988\u8fde\u63a5\u3002<\/p><\/blockquote>\n<h3>3.2 \u4e3a\u4ec0\u4e48\u9690\u85cf\u5c42\u9700\u8981 4 \u4e2a\u795e\u7ecf\u5143\uff1f<\/h3>\n<p>\u8fd9\u4e0d\u662f\u56fa\u5b9a\u7684\u6570\u5b57\u3002\u6211\u4eec\u9009\u62e9 4 \u4e2a\u662f\u56e0\u4e3a\uff1a<\/p>\n<ul>\n<li>XOR \u6709 4 \u79cd\u8f93\u5165\u6a21\u5f0f $(0,0), (0,1), (1,0), (1,1)$<\/li>\n<li>\u7406\u8bba\u4e0a 4 \u4e2a\u9690\u85cf\u795e\u7ecf\u5143\u8db3\u591f\u4e3a\u6bcf\u79cd\u6a21\u5f0f\u5206\u914d\u4e00\u4e2a\u7279\u5f81\u8868\u793a<\/li>\n<li>\u5b9e\u8df5\u4e2d 2~4 \u4e2a\u5373\u53ef\u89e3\u51b3 XOR \u95ee\u9898<\/li>\n<\/ul>\n<blockquote><p>\u9690\u85cf\u5c42\u795e\u7ecf\u5143\u6570\u91cf\u662f\u4e00\u4e2a<strong>\u8d85\u53c2\u6570 (hyperparameter)<\/strong>\uff0c\u9700\u8981\u6839\u636e\u95ee\u9898\u590d\u6742\u5ea6\u6765\u8c03\u6574\u3002\u592a\u5c11\u7684\u795e\u7ecf\u5143\u5b66\u4e0d\u4f1a\u6a21\u5f0f\uff0c\u592a\u591a\u7684\u795e\u7ecf\u5143\u4f1a\u5bfc\u81f4\u8fc7\u62df\u5408\u6216\u6d6a\u8d39\u8ba1\u7b97\u8d44\u6e90\u3002<\/p><\/blockquote>\n<h3>3.3 \u7f51\u7edc\u53c2\u6570\u6570\u91cf<\/h3>\n<p>\u5c42\u4e0e\u5c42\u4e4b\u95f4\u7684<strong>\u5168\u8fde\u63a5 (fully connected)<\/strong> \u610f\u5473\u7740\u6bcf\u4e2a\u795e\u7ecf\u5143\u90fd\u4e0e\u524d\u4e00\u5c42\u7684\u6240\u6709\u795e\u7ecf\u5143\u76f8\u8fde\u3002<\/p>\n<pre><code>\u8f93\u5165\u5c42 \u2192 \u9690\u85cf\u5c42: 4 \u00d7 2 = 8 \u4e2a\u6743\u91cd + 4 \u4e2a\u504f\u7f6e = 12 \u4e2a\u53c2\u6570\n\u9690\u85cf\u5c42 \u2192 \u8f93\u51fa\u5c42: 1 \u00d7 4 = 4 \u4e2a\u6743\u91cd + 1 \u4e2a\u504f\u7f6e = 5 \u4e2a\u53c2\u6570\n\u603b\u8ba1: 17 \u4e2a\u53c2\u6570<\/code><\/pre>\n<p>\u8fd9 17 \u4e2a\u53c2\u6570\u5c31\u662f\u795e\u7ecf\u7f51\u7edc\u9700\u8981&#8221;\u5b66\u4e60&#8221;\u7684\u5168\u90e8\u5185\u5bb9\u3002\u4f60\u53ef\u4ee5\u60f3\u8c61\uff1a\u795e\u7ecf\u7f51\u7edc\u7684\u8bad\u7ec3\u672c\u8d28\u4e0a\u5c31\u662f\u5728 17 \u7ef4\u7a7a\u95f4\u4e2d\u5bfb\u627e\u6700\u4f18\u53c2\u6570\u7ec4\u5408\u3002<\/p>\n<hr>\n<h2>3.4 <code>NeuralNet<\/code> \u7ed3\u6784\u4f53\u7684\u8bbe\u8ba1\u610f\u56fe\uff1a\u628a&#8221;\u5c42&#8221;\u6620\u5c04\u6210\u6570\u636e\u7ed3\u6784<\/h2>\n<p>\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6211\u4eec\u8ba8\u8bba\u7684\u90fd\u662f\u7f51\u7edc\u7ed3\u6784\u548c\u6570\u5b66\u516c\u5f0f\u3002\u771f\u6b63\u628a\u8fd9\u4e9b\u4e1c\u897f\u653e\u8fdb\u4ee3\u7801\u91cc\u7684\u5173\u952e\uff0c\u5c31\u662f <code>xor_nn.c<\/code> \u4e2d\u7684\u8fd9\u4e2a\u7ed3\u6784\u4f53\uff1a<\/p>\n<pre><code class=\"c\">typedef struct {\n    \/* --- \u7b2c\u4e00\u5c42 (\u8f93\u5165\u5c42 -&gt; \u9690\u85cf\u5c42) \u7684\u53c2\u6570 --- *\/\n    double hidden_weights[HIDDEN_SIZE][INPUT_SIZE];  \/* \u6743\u91cd\u77e9\u9635\uff1a4\u00d72 *\/\n    double hidden_biases[HIDDEN_SIZE];              \/* \u504f\u7f6e\u5411\u91cf\uff1a4 *\/\n\n    \/* --- \u7b2c\u4e8c\u5c42 (\u9690\u85cf\u5c42 -&gt; \u8f93\u51fa\u5c42) \u7684\u53c2\u6570 --- *\/\n    double output_weights[OUTPUT_SIZE][HIDDEN_SIZE]; \/* \u6743\u91cd\u77e9\u9635\uff1a1\u00d74 *\/\n    double output_biases[OUTPUT_SIZE];              \/* \u504f\u7f6e\u5411\u91cf\uff1a1 *\/\n\n    \/* --- \u6b63\u5411\u4f20\u64ad\u7684\u4e2d\u95f4\u7ed3\u679c --- *\/\n    double hidden_z[HIDDEN_SIZE], hidden_a[HIDDEN_SIZE];  \/* \u9690\u85cf\u5c42\uff1a\u51c0\u8f93\u5165 z \u4e0e\u6fc0\u6d3b\u503c a *\/\n    double output_z[OUTPUT_SIZE], output_a[OUTPUT_SIZE];  \/* \u8f93\u51fa\u5c42\uff1a\u51c0\u8f93\u5165 z \u4e0e\u6fc0\u6d3b\u503c a *\/\n\n    \/* --- \u53cd\u5411\u4f20\u64ad\u7684\u8bef\u5dee\u9879 --- *\/\n    double output_delta[OUTPUT_SIZE];  \/* \u8f93\u51fa\u5c42\u8bef\u5dee *\/\n    double hidden_delta[HIDDEN_SIZE];  \/* \u9690\u85cf\u5c42\u8bef\u5dee *\/\n} NeuralNet;<\/code><\/pre>\n<p>\u4e3a\u4ec0\u4e48\u8fd9\u6837\u8bbe\u8ba1\uff1f\u4e00\u53e5\u8bdd\u6982\u62ec\uff1a<strong>\u5b83\u7684\u5b57\u6bb5\u5e03\u5c40\u5b8c\u5168\u5bf9\u5e94\u4e86\u6570\u5b66\u63a8\u5bfc\u4e2d&#8221;\u5c42\u2192\u5c42&#8221;\u7684\u8ba1\u7b97\u56fe<\/strong>\u3002\u4e0b\u9762\u6211\u4eec\u4ece\u4e09\u4e2a\u89d2\u5ea6\u62c6\u89e3\uff1a\u5c42\u4e0e\u77e9\u9635\u7684\u5bf9\u5e94\u3001\u4e2d\u95f4\u53d8\u91cf\u7684\u7f13\u5b58\u8bbe\u8ba1\u3001\u53ef\u5b66\u4e60\u53c2\u6570\u4e0e\u4e34\u65f6\u53d8\u91cf\u7684\u5206\u79bb\u3002<\/p>\n<h3>3.4.1 \u5c42\u4e0e\u77e9\u9635\u7684\u5bf9\u5e94\uff1a\u4e8c\u7ef4\u6570\u7ec4\u5982\u4f55\u8868\u793a\u5168\u8fde\u63a5<\/h3>\n<p>\u5728\u6570\u5b66\u4e0a\uff0c\u7b2c 1 \u5c42\uff08\u8f93\u5165\u2192\u9690\u85cf\uff09\u7684\u53d8\u6362\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n<p>$$\/vec{z_1} = W_1 \/cdot \/vec{x} + \/vec{b_1}$$<\/p>\n<p>\u5176\u4e2d\uff1a<\/p>\n<ul>\n<li>$\/vec{x}$ \u662f\u5f62\u72b6 $(2, 1)$ \u7684\u5217\u5411\u91cf<\/li>\n<li>$W_1$ \u662f\u5f62\u72b6 $(4, 2)$ \u7684\u77e9\u9635<\/li>\n<li>$\/vec{b_1}$ \u662f\u5f62\u72b6 $(4, 1)$ \u7684\u504f\u7f6e\u5411\u91cf<\/li>\n<li>$\/vec{z_1}$ \u662f\u5f62\u72b6 $(4, 1)$ \u7684\u9690\u85cf\u5c42\u51c0\u8f93\u5165<\/li>\n<\/ul>\n<p>\u5728 C \u8bed\u8a00\u91cc\uff0c\u7528<strong>\u4e8c\u7ef4\u6570\u7ec4<\/strong> <code>hidden_weights[HIDDEN_SIZE][INPUT_SIZE]<\/code>\uff08\u5373 <code>hidden_weights[4][2]<\/code>\uff09\u6765\u5b58 $W_1$ \u662f\u6700\u81ea\u7136\u7684\uff1a<\/p>\n<pre><code>hidden_weights[i][j]  \u8868\u793a  \u9690\u85cf\u5c42\u7b2c i \u4e2a\u795e\u7ecf\u5143 \u63a5\u6536 \u8f93\u5165\u5c42\u7b2c j \u4e2a\u795e\u7ecf\u5143 \u7684\u6743\u91cd<\/code><\/pre>\n<p>\u540c\u7406\uff0c<code>output_weights[OUTPUT_SIZE][HIDDEN_SIZE]<\/code> \u5373 <code>output_weights[1][4]<\/code>\uff0c\u8868\u793a\u8f93\u51fa\u5c42\uff081 \u4e2a\u795e\u7ecf\u5143\uff09\u63a5\u6536\u9690\u85cf\u5c42 4 \u4e2a\u795e\u7ecf\u5143\u7684\u6743\u91cd\u3002\u8fd9\u79cd\u5b89\u6392\u548c\u77e9\u9635\u4e58\u6cd5\u7684\u987a\u5e8f\u4e00\u81f4\uff1a<\/p>\n<pre><code class=\"c\">for (int neuron_idx = 0; neuron_idx &lt; HIDDEN_SIZE; neuron_idx++) {\n    double weighted_sum = 0.0;\n    for (int input_idx = 0; input_idx &lt; INPUT_SIZE; input_idx++)\n        weighted_sum += net-&gt;hidden_weights[neuron_idx][input_idx] * input[input_idx];  \/\/ \u77e9\u9635 hidden_weights \u7684\u7b2c neuron_idx \u884c \u4e0e \u5411\u91cf input \u505a\u70b9\u79ef\n    weighted_sum += net-&gt;hidden_biases[neuron_idx];\n    net-&gt;hidden_z[neuron_idx] = weighted_sum;\n    net-&gt;hidden_a[neuron_idx] = sigmoid(weighted_sum);\n}<\/code><\/pre>\n<p>\u5982\u679c\u6362\u6210\u884c\u4f18\u5148 \/ \u5217\u4f18\u5148\u7684\u5b58\u50a8\u65b9\u5f0f\uff0c\u5185\u5c42\u5faa\u73af\u7684\u4e0b\u6807\u987a\u5e8f\u5c31\u8981\u8ddf\u7740\u53d8\u3002\u8fd9\u91cc\u9009\u62e9 <code>W[dst][src]<\/code>\uff0c\u610f\u601d\u662f<strong>\u76ee\u6807\u795e\u7ecf\u5143\u4f5c\u4e3a\u7b2c\u4e00\u7ef4\uff0c\u6e90\u795e\u7ecf\u5143\u4f5c\u4e3a\u7b2c\u4e8c\u7ef4<\/strong>\uff0c\u597d\u5904\u662f\uff1a<\/p>\n<ul>\n<li>\u5916\u5c42\u904d\u5386\u76ee\u6807\u795e\u7ecf\u5143\uff0c\u5185\u5c42\u7d2f\u52a0\u5176\u6240\u6709\u8f93\u5165\uff0c\u7b26\u5408\u4eba\u7c7b\u7684&#8221;\u5148\u5b9a\u4f4d\u795e\u7ecf\u5143\uff0c\u518d\u770b\u5b83\u6709\u54ea\u4e9b\u8fde\u63a5&#8221;\u7684\u76f4\u89c9\u3002<\/li>\n<li>\u5728 CPU \u7f13\u5b58\u4e0a\uff0c\u6309\u884c\u4f18\u5148\u8bbf\u95ee <code>hidden_weights[i][*]<\/code> \u4e5f\u901a\u5e38\u662f\u8fde\u7eed\u7684\uff0c\u6548\u7387\u8f83\u597d\u3002<\/li>\n<\/ul>\n<blockquote><p><strong>\u5c0f\u63d0\u793a<\/strong>\uff1a\u5f88\u591a\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6\uff08\u5982 NumPy\u3001PyTorch\uff09\u9ed8\u8ba4\u4e60\u60ef\u7528 <code>W[out, in]<\/code>\u3002C \u8bed\u8a00\u6570\u7ec4 <code>W[i][j]<\/code> \u7684\u8bed\u4e49\u4e5f\u6b63\u597d\u5bf9\u5e94 <code>out=i, in=j<\/code>\u3002<\/p><\/blockquote>\n<h3>3.4.2 \u4e3a\u4ec0\u4e48\u504f\u7f6e\u662f\u5355\u72ec\u7684\u4e00\u7ef4\u6570\u7ec4\uff1f<\/h3>\n<p>\u4f60\u53ef\u80fd\u4f1a\u60f3\uff1a\u628a\u504f\u7f6e\u653e\u8fdb\u6743\u91cd\u77e9\u9635\u7684\u6700\u540e\u4e00\u5217\u4e0d\u5c31\u884c\u4e86\u5417\uff1f\u4e5f\u5c31\u662f\u7ed9\u6bcf\u4e2a\u6837\u672c\u8ffd\u52a0\u4e00\u4e2a\u6052\u4e3a 1 \u7684\u8f93\u5165\uff0c\u8fd9\u6837\u53ef\u4ee5\u7edf\u4e00\u5199\u6210\u4e00\u6b21\u77e9\u9635\u4e58\u6cd5\u3002\u8fd9\u79cd\u6280\u5de7\u786e\u5b9e\u5728\u6570\u5b66\u63a8\u5bfc\u4e2d\u5f88\u5e38\u89c1\uff0c\u4f46\u5728\u5165\u95e8\u4ee3\u7801\u91cc\uff1a<\/p>\n<ol>\n<li><strong>\u5206\u5f00\u5b58\u50a8\u66f4\u76f4\u89c2<\/strong>\uff1a\u6743\u91cd $W$ \u8d1f\u8d23&#8221;\u8f93\u5165\u5bf9\u8f93\u51fa\u7684\u5f71\u54cd&#8221;\uff0c\u504f\u7f6e $b$ \u8d1f\u8d23&#8221;\u57fa\u7ebf\u6fc0\u6d3b\u6c34\u5e73&#8221;\uff0c\u4e24\u8005\u804c\u8d23\u6e05\u6670\u3002<\/li>\n<li><strong>\u4ee3\u7801\u66f4\u5bf9\u79f0<\/strong>\uff1a\u6bcf\u4e2a\u795e\u7ecf\u5143\u90fd\u6709 <code>W[i][*]<\/code> \u548c <code>b[i]<\/code>\uff0c\u516c\u5f0f $\/sum W x + b$ \u53ef\u4ee5\u76f4\u63a5\u5bf9\u5e94\u4e24\u884c\u4ee3\u7801\u3002<\/li>\n<li><strong>\u907f\u514d\u8f93\u5165\u9020\u5047\u6837\u672c<\/strong>\uff1a\u4e0d\u9700\u8981\u628a\u8bad\u7ec3\u6570\u636e\u6269\u5c55\u6210 <code>{x1, x2, 1}<\/code>\uff0cXOR \u7684\u8f93\u5165\u4ecd\u662f\u81ea\u7136\u7684\u4e8c\u7ef4\u6570\u7ec4\u3002<\/li>\n<\/ol>\n<p>\u56e0\u6b64\uff0c\u504f\u7f6e\u5355\u72ec\u7528 <code>hidden_biases[HIDDEN_SIZE]<\/code>\u3001<code>output_biases[OUTPUT_SIZE]<\/code> \u5b58\u50a8\uff0c\u662f<strong>\u53ef\u8bfb\u6027\u4f18\u5148<\/strong>\u7684\u8bbe\u8ba1\u3002<\/p>\n<h3>3.4.3 \u4e2d\u95f4\u53d8\u91cf <code>z<\/code>\u3001<code>a<\/code>\u3001<code>delta<\/code> \u4e3a\u4ec0\u4e48\u4e5f\u653e\u5728\u7ed3\u6784\u4f53\u91cc\uff1f<\/h3>\n<p>\u795e\u7ecf\u7f51\u7edc\u7684\u8ba1\u7b97\u4e0d\u662f\u4e00\u6b21\u6027\u7684\uff1a<\/p>\n<ul>\n<li>\u6b63\u5411\u4f20\u64ad\u65f6\u8ba1\u7b97 <code>hidden_z<\/code>, <code>hidden_a<\/code>, <code>output_z<\/code>, <code>output_a<\/code><\/li>\n<li>\u53cd\u5411\u4f20\u64ad\u65f6\u53c8\u8981\u7528 <code>hidden_a<\/code>\uff08\u8f93\u51fa\u5c42\u6743\u91cd\u68af\u5ea6\uff09\u548c <code>output_a<\/code>\uff08\u635f\u5931\u5bf9\u8f93\u51fa\u6c42\u5bfc\uff09<\/li>\n<li>\u66f4\u65b0\u6743\u91cd\u65f6\u8fd8\u8981\u7528 <code>hidden_delta<\/code>, <code>output_delta<\/code> \u548c <code>hidden_a<\/code> \/ <code>x<\/code><\/li>\n<\/ul>\n<p>\u5982\u679c\u628a\u8fd9\u4e9b\u4e34\u65f6\u53d8\u91cf\u90fd\u4f5c\u4e3a\u51fd\u6570\u53c2\u6570\u4f20\u6765\u4f20\u53bb\uff0c\u4ee3\u7801\u4f1a\u53d8\u6210\u8fd9\u6837\uff1a<\/p>\n<pre><code class=\"c\">void backward(NeuralNet *network, double *input, double target,\n              double *hidden_z, double *hidden_a, double *output_z, double *output_a,\n              double *hidden_delta, double *output_delta);<\/code><\/pre>\n<p>\u975e\u5e38\u5197\u957f\uff0c\u800c\u4e14\u5bb9\u6613\u4f20\u9519\u3002\u628a\u5b83\u4eec\u585e\u8fdb <code>NeuralNet<\/code> \u7ed3\u6784\u4f53\uff0c\u76f8\u5f53\u4e8e\u7ed9\u7f51\u7edc\u52a0\u4e86\u4e00\u5c42&#8221;\u5de5\u4f5c\u5185\u5b58&#8221;\uff1a<\/p>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u5b57\u6bb5<\/th>\n<th align=\"left\">\u6b63\u5411\u4f20\u64ad\u5199\u5165<\/th>\n<th align=\"left\">\u53cd\u5411\u4f20\u64ad\u8bfb\u53d6<\/th>\n<th align=\"left\">\u66f4\u65b0\u6743\u91cd\u8bfb\u53d6<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\"><code>hidden_z<\/code><\/td>\n<td align=\"left\">\u662f<\/td>\n<td align=\"left\">\u7528\u4e8e\u8ba1\u7b97 <code>sigmoid_derivative(hidden_z)<\/code>\uff08\u672c\u4ee3\u7801\u7528 <code>hidden_a<\/code> \u95f4\u63a5\u5b58\uff09<\/td>\n<td align=\"left\">\u5426<\/td>\n<\/tr>\n<tr>\n<td align=\"left\"><code>hidden_a<\/code><\/td>\n<td align=\"left\">\u662f<\/td>\n<td align=\"left\">\u662f\uff08\u8f93\u51fa\u5c42\u68af\u5ea6 = \u03b42 \u00b7 hidden_a\uff09<\/td>\n<td align=\"left\">\u662f\uff08\u66f4\u65b0 output_weights\uff09<\/td>\n<\/tr>\n<tr>\n<td align=\"left\"><code>output_z<\/code><\/td>\n<td align=\"left\">\u662f<\/td>\n<td align=\"left\">\u662f\uff08\u8ba1\u7b97 sigmoid \u5bfc\u6570\uff09<\/td>\n<td align=\"left\">\u5426<\/td>\n<\/tr>\n<tr>\n<td align=\"left\"><code>output_a<\/code><\/td>\n<td align=\"left\">\u662f<\/td>\n<td align=\"left\">\u662f\uff08\u8ba1\u7b97\u635f\u5931\u548c \u03b42\uff09<\/td>\n<td align=\"left\">\u5426<\/td>\n<\/tr>\n<tr>\n<td align=\"left\"><code>hidden_delta<\/code><\/td>\n<td align=\"left\">\u5426<\/td>\n<td align=\"left\">\u5199\u5165<\/td>\n<td align=\"left\">\u662f\uff08\u66f4\u65b0 hidden_weights\uff09<\/td>\n<\/tr>\n<tr>\n<td align=\"left\"><code>output_delta<\/code><\/td>\n<td align=\"left\">\u5426<\/td>\n<td align=\"left\">\u5199\u5165<\/td>\n<td align=\"left\">\u662f\uff08\u66f4\u65b0 output_weights\uff09<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<blockquote><p><strong>\u8bbe\u8ba1\u6a21\u5f0f<\/strong>\uff1a\u8fd9\u662f\u4e00\u79cd&#8221;\u7528\u7a7a\u95f4\u6362\u4fbf\u5229 + \u7528\u5c01\u88c5\u6362\u6b63\u786e\u6027&#8221;\u7684\u7b56\u7565\u3002\u7c7b\u4f3c PyTorch \u91cc <code>Tensor<\/code> \u4f1a\u6301\u6709 <code>data<\/code> \u548c <code>grad<\/code>\uff0c\u8bad\u7ec3\u65f6\u524d\u5411\u4fdd\u5b58\u6fc0\u6d3b\uff0c\u53cd\u5411\u76f4\u63a5\u590d\u7528\u3002<\/p><\/blockquote>\n<h3>3.4.4 \u53ef\u5b66\u4e60\u53c2\u6570 vs \u4e34\u65f6\u53d8\u91cf\u7684\u5206\u79bb<\/h3>\n<p>\u518d\u770b\u4e00\u773c\u7ed3\u6784\u4f53\uff0c\u4f60\u4f1a\u53d1\u73b0\u5b83\u5929\u7136\u5206\u6210\u4e86\u4e24\u5927\u7ec4\uff1a<\/p>\n<pre><code class=\"c\">typedef struct {\n    \/* ===== \u53ef\u5b66\u4e60\u53c2\u6570\uff08\u8bad\u7ec3\u8981\u6539\u7684\uff09 ===== *\/\n    double hidden_weights[HIDDEN_SIZE][INPUT_SIZE];  double hidden_biases[HIDDEN_SIZE];\n    double output_weights[OUTPUT_SIZE][HIDDEN_SIZE]; double output_biases[OUTPUT_SIZE];\n\n    \/* ===== \u524d\u5411\/\u53cd\u5411\u8ba1\u7b97\u7684\u4e34\u65f6\u72b6\u6001\uff08\u8bad\u7ec3\u8981\u7528\uff0c\u4f46\u672c\u8eab\u4e0d\u4fdd\u5b58\uff09 ===== *\/\n    double hidden_z[HIDDEN_SIZE], hidden_a[HIDDEN_SIZE];\n    double output_z[OUTPUT_SIZE], output_a[OUTPUT_SIZE];\n    double output_delta[OUTPUT_SIZE], hidden_delta[HIDDEN_SIZE];\n} NeuralNet;<\/code><\/pre>\n<p>\u8fd9\u4e24\u7ec4\u6570\u636e\u7684&#8221;\u751f\u547d\u5468\u671f&#8221;\u4e0d\u540c\uff1a<\/p>\n<ul>\n<li><strong>\u53ef\u5b66\u4e60\u53c2\u6570<\/strong>\uff1a\u521d\u59cb\u5316\u540e\u4e00\u76f4\u5b58\u5728\uff0c\u76f4\u5230\u8bad\u7ec3\u7ed3\u675f\u3002\u5b83\u4eec\u5c31\u662f\u6a21\u578b\u7684&#8221;\u77e5\u8bc6&#8221;\u3002<\/li>\n<li><strong>\u4e34\u65f6\u72b6\u6001<\/strong>\uff1a\u6bcf\u4e2a\u6837\u672c\u524d\u5411\u3001\u53cd\u5411\u3001\u66f4\u65b0\u540e\uff0c\u4e0a\u4e00\u8f6e\u6837\u672c\u7684\u503c\u5c31\u88ab\u8986\u76d6\u4e86\u3002\u5b83\u4eec\u53ea\u670d\u52a1\u4e8e\u5f53\u524d\u8fd9\u4e00\u6b65\u8ba1\u7b97\u3002<\/li>\n<\/ul>\n<p>\u8fd9\u79cd\u5206\u79bb\u7684\u597d\u5904\u662f\uff1a<\/p>\n<ul>\n<li>\u4fdd\u5b58\u6a21\u578b\u65f6\uff0c\u53ea\u9700\u628a <code>hidden_weights, hidden_biases, output_weights, output_biases<\/code> \u5199\u8fdb\u6587\u4ef6\uff1b<\/li>\n<li>\u63a8\u7406\uff08inference\uff09\u65f6\uff0c\u53ea\u9700\u8981 <code>hidden_weights, hidden_biases, output_weights, output_biases<\/code> \u548c\u4e34\u65f6\u7684 <code>hidden_z, hidden_a, output_z, output_a<\/code>\uff0c\u4e0d\u9700\u8981 <code>delta<\/code>\uff1b<\/li>\n<li>\u591a\u4eba\u534f\u4f5c\u6216\u8c03\u8bd5\u65f6\uff0c\u4e00\u773c\u5c31\u80fd\u770b\u51fa&#8221;\u54ea\u4e9b\u662f\u771f\u6b63\u7684\u53c2\u6570\uff0c\u54ea\u4e9b\u53ea\u662f\u8ba1\u7b97\u8fc7\u7a0b\u91cf&#8221;\u3002<\/li>\n<\/ul>\n<h2>3.5 \u7ef4\u5ea6\u4e3a\u4ec0\u4e48\u8fd9\u4e48\u5b9a\uff1f\u4ece\u5f62\u72b6\u770b\u4ee3\u7801<\/h2>\n<p><code>NeuralNet<\/code> \u7684\u7ef4\u5ea6\u4e0d\u662f\u968f\u610f\u5199\u7684\uff0c\u5b83\u4eec\u7531\u4e09\u4e2a\u5b8f\u5b9a\u4e49\u51b3\u5b9a\uff1a<\/p>\n<pre><code class=\"c\">#define INPUT_SIZE   2\n#define HIDDEN_SIZE  4\n#define OUTPUT_SIZE  1<\/code><\/pre>\n<p>\u628a\u5b83\u4eec\u5199\u6210\u8868\u683c\uff0c\u53ef\u4ee5\u770b\u5230\u6574\u4e2a\u7f51\u7edc\u7684&#8221;\u5f62\u72b6&#8221;\uff08shape\uff09\uff1a<\/p>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u6570\u636e<\/th>\n<th align=\"left\">\u6570\u5b66\u5f62\u72b6<\/th>\n<th align=\"left\">C \u6570\u7ec4\u7ef4\u5ea6<\/th>\n<th align=\"right\">\u5143\u7d20\u603b\u6570<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">\u8f93\u5165 $x$<\/td>\n<td align=\"left\">$(2, 1)$<\/td>\n<td align=\"left\"><code>x[INPUT_SIZE]<\/code><\/td>\n<td align=\"right\">2<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u9690\u85cf\u5c42\u6743\u91cd $W_1$<\/td>\n<td align=\"left\">$(4, 2)$<\/td>\n<td align=\"left\"><code>hidden_weights[4][2]<\/code><\/td>\n<td align=\"right\">8<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u9690\u85cf\u5c42\u504f\u7f6e $b_1$<\/td>\n<td align=\"left\">$(4, 1)$<\/td>\n<td align=\"left\"><code>hidden_biases[4]<\/code><\/td>\n<td align=\"right\">4<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u9690\u85cf\u5c42\u51c0\u8f93\u5165 $z_1$<\/td>\n<td align=\"left\">$(4, 1)$<\/td>\n<td align=\"left\"><code>hidden_z[4]<\/code><\/td>\n<td align=\"right\">4<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u9690\u85cf\u5c42\u6fc0\u6d3b $a_1$<\/td>\n<td align=\"left\">$(4, 1)$<\/td>\n<td align=\"left\"><code>hidden_a[4]<\/code><\/td>\n<td align=\"right\">4<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u8f93\u51fa\u5c42\u6743\u91cd $W_2$<\/td>\n<td align=\"left\">$(1, 4)$<\/td>\n<td align=\"left\"><code>output_weights[1][4]<\/code><\/td>\n<td align=\"right\">4<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u8f93\u51fa\u5c42\u504f\u7f6e $b_2$<\/td>\n<td align=\"left\">$(1, 1)$<\/td>\n<td align=\"left\"><code>output_biases[1]<\/code><\/td>\n<td align=\"right\">1<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u8f93\u51fa\u5c42\u51c0\u8f93\u5165 $z_2$<\/td>\n<td align=\"left\">$(1, 1)$<\/td>\n<td align=\"left\"><code>output_z[1]<\/code><\/td>\n<td align=\"right\">1<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u8f93\u51fa\u5c42\u6fc0\u6d3b $a_2$ \/ \u9884\u6d4b\u503c<\/td>\n<td align=\"left\">$(1, 1)$<\/td>\n<td align=\"left\"><code>output_a[1]<\/code><\/td>\n<td align=\"right\">1<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u9690\u85cf\u5c42\u8bef\u5dee $\/delta_1$<\/td>\n<td align=\"left\">$(4, 1)$<\/td>\n<td align=\"left\"><code>hidden_delta[4]<\/code><\/td>\n<td align=\"right\">4<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u8f93\u51fa\u5c42\u8bef\u5dee $\/delta_2$<\/td>\n<td align=\"left\">$(1, 1)$<\/td>\n<td align=\"left\"><code>output_delta[1]<\/code><\/td>\n<td align=\"right\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u8fd9\u4e2a\u5f62\u72b6\u77e9\u9635\u662f\u7406\u89e3\u4ee3\u7801\u7684\u5173\u952e\uff1a\u6bcf\u5f53\u4f60\u5199\u5faa\u73af\u65f6\uff0c\u5148\u60f3\u6e05\u695a&#8221;\u8fd9\u4e2a\u6570\u7ec4\u4ee3\u8868\u54ea\u4e2a\u5411\u91cf\/\u77e9\u9635\uff0c\u5f62\u72b6\u662f\u4ec0\u4e48&#8221;\u3002\u4f8b\u5982\uff1a<\/p>\n<ul>\n<li>\u8f93\u51fa\u5c42\u53ea\u6709 1 \u4e2a\u795e\u7ecf\u5143\uff0c\u6240\u4ee5 <code>output_a[0]<\/code> \u5c31\u662f\u9884\u6d4b\u503c\uff1b<\/li>\n<li>\u9690\u85cf\u5c42\u6709 4 \u4e2a\u795e\u7ecf\u5143\uff0c\u6240\u4ee5 <code>hidden_a[0]<\/code> \u5230 <code>hidden_a[3]<\/code> \u5206\u522b\u5bf9\u5e94 4 \u4e2a&#8221;\u7279\u5f81\u63d0\u53d6\u5668&#8221;\u7684\u8f93\u51fa\uff1b<\/li>\n<li>\u66f4\u65b0 hidden_weights \u65f6\uff0c\u5916\u5c42\u5faa\u73af\u662f <code>i &lt; HIDDEN_SIZE<\/code>\uff0c\u5185\u5c42\u662f <code>j &lt; INPUT_SIZE<\/code>\uff0c\u6b63\u597d\u8986\u76d6 4\u00d72 \u7684\u77e9\u9635\u3002<\/li>\n<\/ul>\n<blockquote><p><strong>\u8bb0\u5fc6\u53e3\u8bc0<\/strong>\uff1a<code>hidden_weights[i][j]<\/code> \u4e2d <code>i<\/code> \u662f\u76ee\u6807\uff08\u9690\u85cf\u5c42\u795e\u7ecf\u5143\uff09\uff0c<code>j<\/code> \u662f\u6765\u6e90\uff08\u8f93\u5165\u7279\u5f81\uff09\u3002\u7ef4\u5ea6\u603b\u662f <code>[\u76ee\u6807\u5c42\u5927\u5c0f][\u6765\u6e90\u5c42\u5927\u5c0f]<\/code>\u3002<\/p><\/blockquote>\n<h2>3.6 \u4e3a\u4ec0\u4e48\u4e0d\u76f4\u63a5\u7528\u52a8\u6001\u5185\u5b58\uff1f\u9759\u6001\u6570\u7ec4\u7684\u53d6\u820d<\/h2>\n<p>\u672c\u4ee3\u7801\u7528\u7684\u662f\u9759\u6001\u6570\u7ec4\uff08\u7f16\u8bd1\u671f\u5df2\u77e5\u5927\u5c0f\uff09\uff0c\u800c\u4e0d\u662f <code>malloc<\/code> \u52a8\u6001\u5206\u914d\u3002\u539f\u56e0\uff1a<\/p>\n<ol>\n<li><strong>\u7b80\u5355\u3001\u53ef\u9884\u6d4b<\/strong>\uff1a\u5bf9\u4e8e\u56fa\u5b9a\u5c42\u5927\u5c0f\u7684 XOR \u7f51\u7edc\uff0c\u7f16\u8bd1\u65f6\u5c31\u77e5\u9053\u6240\u6709\u5185\u5b58\u9700\u6c42\uff0c\u65e0\u9700\u5904\u7406\u5185\u5b58\u6cc4\u6f0f\u3002<\/li>\n<li><strong>\u7f13\u5b58\u53cb\u597d<\/strong>\uff1a\u7ed3\u6784\u4f53\u5728\u6808\u6216\u5168\u5c40\u533a\u8fde\u7eed\u5206\u914d\uff0c\u8bbf\u95ee\u901f\u5ea6\u901a\u5e38\u4f18\u4e8e\u5206\u6563\u7684\u5806\u6307\u9488\u3002<\/li>\n<li><strong>\u6559\u5b66\u6e05\u6670<\/strong>\uff1a\u8bfb\u8005\u4e0d\u9700\u8981\u5206\u5fc3\u5b66\u4e60\u5185\u5b58\u7ba1\u7406\uff0c\u80fd\u4e13\u6ce8\u4e8e\u795e\u7ecf\u7f51\u7edc\u7b97\u6cd5\u672c\u8eab\u3002<\/li>\n<\/ol>\n<p>\u4f46\u9759\u6001\u6570\u7ec4\u7684\u4ee3\u4ef7\u662f<strong>\u7f3a\u4e4f\u7075\u6d3b\u6027<\/strong>\uff1a<\/p>\n<ul>\n<li>\u5982\u679c\u4f60\u60f3\u628a\u9690\u85cf\u5c42\u6539\u6210 8 \u4e2a\u795e\u7ecf\u5143\uff0c\u5fc5\u987b\u91cd\u65b0\u7f16\u8bd1\u3002<\/li>\n<li>\u5982\u679c\u4f60\u60f3\u52a0\u7b2c\u4e09\u4e2a\u9690\u85cf\u5c42\uff0c\u7ed3\u6784\u4f53\u65e0\u6cd5\u76f4\u63a5\u590d\u7528\uff0c\u9700\u8981\u91cd\u65b0\u8bbe\u8ba1\u3002<\/li>\n<\/ul>\n<p>\u56e0\u6b64\uff0c\u672c\u4ee3\u7801\u662f<strong>\u9488\u5bf9\u56fa\u5b9a\u7f51\u7edc\u7ed3\u6784\u7684\u6559\u80b2\u5b9e\u73b0<\/strong>\uff0c\u4e0d\u662f\u901a\u7528\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6\u3002\u7406\u89e3\u4e86\u9759\u6001\u6570\u7ec4\u7248\u672c\u540e\uff0c\u518d\u6539\u6210\u52a8\u6001\u5c42\u7ed3\u6784\u4f1a\u66f4\u81ea\u7136\u3002<\/p>\n<h2>3.7 \u4e00\u4e2a\u66f4\u901a\u7528\u7684\u5c42\u62bd\u8c61\u957f\u4ec0\u4e48\u6837\uff1f<\/h2>\n<p>\u5982\u679c\u4f60\u628a\u9690\u85cf\u5c42\u6570\u91cf\u3001\u6bcf\u5c42\u5927\u5c0f\u90fd\u53d8\u6210\u53ef\u914d\u7f6e\u7684\uff0c\u5c31\u53ef\u4ee5\u628a <code>NeuralNet<\/code> \u62bd\u8c61\u6210\u4e00\u7ec4<strong>\u5c42 (Layer)<\/strong>\u3002\u6bcf\u4e2a\u5c42\u53ea\u5173\u5fc3\uff1a<\/p>\n<ul>\n<li>\u8f93\u5165\u7ef4\u5ea6 <code>input_size<\/code><\/li>\n<li>\u8f93\u51fa\u7ef4\u5ea6 <code>output_size<\/code><\/li>\n<li>\u6743\u91cd\u77e9\u9635 <code>weights<\/code>\uff08\u5927\u5c0f <code>output_size \u00d7 input_size<\/code>\uff09<\/li>\n<li>\u504f\u7f6e\u5411\u91cf <code>bias<\/code>\uff08\u5927\u5c0f <code>output_size<\/code>\uff09<\/li>\n<li>\u672c\u5c42\u7684\u51c0\u8f93\u5165 <code>z<\/code> \u548c\u6fc0\u6d3b <code>a<\/code><\/li>\n<li>\u672c\u5c42\u7684\u8bef\u5dee\u9879 <code>delta<\/code><\/li>\n<li>\u4e00\u4e2a\u6fc0\u6d3b\u51fd\u6570\u6307\u9488<\/li>\n<\/ul>\n<pre><code class=\"c\">typedef struct {\n    int input_size;\n    int output_size;\n\n    double *weights;   \/\/ \u4e00\u7ef4\u5c55\u5e73\uff0c\u957f\u5ea6 = output_size * input_size\n    double *bias;      \/\/ \u957f\u5ea6 = output_size\n\n    double *z;         \/\/ \u51c0\u8f93\u5165\uff0c\u957f\u5ea6 = output_size\n    double *a;         \/\/ \u6fc0\u6d3b\u503c\uff0c\u957f\u5ea6 = output_size\n    double *delta;     \/\/ \u8bef\u5dee\u9879\uff0c\u957f\u5ea6 = output_size\n\n    double (*activation)(double value);\n    double (*activation_derivative)(double activation_value);\n} Layer;<\/code><\/pre>\n<p>\u6574\u4e2a\u7f51\u7edc\u5c31\u662f\u4e00\u4e2a\u5c42\u6570\u7ec4\uff1a<\/p>\n<pre><code class=\"c\">typedef struct {\n    Layer *layers;\n    int num_layers;     \/\/ \u4f8b\u5982 2\uff1a\u8f93\u5165\u2192\u9690\u85cf\u3001\u9690\u85cf\u2192\u8f93\u51fa\n} DeepNet;<\/code><\/pre>\n<p>\u6b64\u65f6\uff1a<\/p>\n<ul>\n<li>\u524d\u5411\u4f20\u64ad\u5c31\u662f\u6309\u987a\u5e8f\u904d\u5386\u6bcf\u4e00\u5c42 <code>forward(layers[i])<\/code>\uff1b<\/li>\n<li>\u53cd\u5411\u4f20\u64ad\u5148\u8ba1\u7b97\u6700\u540e\u4e00\u5c42 <code>delta<\/code>\uff0c\u518d\u5411\u524d\u9010\u5c42\u4f20\u64ad\uff1b<\/li>\n<li>\u52a0\u5c42\u3001\u6539\u5c42\u5927\u5c0f\u53ea\u9700\u8981\u6539\u914d\u7f6e\u6570\u7ec4\uff0c\u4e0d\u9700\u8981\u6539\u7ed3\u6784\u4f53\u3002<\/li>\n<\/ul>\n<p>\u672c\u6559\u7a0b\u7684 <code>NeuralNet<\/code> \u53ef\u4ee5\u770b\u4f5c\u8fd9\u4e2a\u901a\u7528\u5c42\u7ed3\u6784\u7684<strong>\u7279\u4f8b<\/strong>\u2014\u2014\u4e24\u5c42\u3001\u9690\u85cf\u5c42 4 \u4e2a\u795e\u7ecf\u5143\u3001\u56fa\u5b9a sigmoid\u3002\u5148\u5403\u900f\u7279\u4f8b\uff0c\u518d\u63a8\u5e7f\u5230\u901a\u7528\u6846\u67b6\uff0c\u662f\u5b66\u4e60\u795e\u7ecf\u7f51\u7edc\u6700\u7a33\u59a5\u7684\u8def\u5f84\u3002<\/p>\n<hr>\n<h2>4. \u795e\u7ecf\u5143\u5185\u90e8\uff1a\u52a0\u6743\u6c42\u548c\u4e0e\u6fc0\u6d3b\u51fd\u6570<\/h2>\n<h3>4.1 \u5355\u4e2a\u795e\u7ecf\u5143\u7684\u8ba1\u7b97\u8fc7\u7a0b<\/h3>\n<p>\u4e00\u4e2a\u795e\u7ecf\u5143\u505a\u4e24\u4ef6\u4e8b\uff1a<\/p>\n<ol>\n<li><strong>\u52a0\u6743\u6c42\u548c<\/strong>\uff1a\u5c06\u6240\u6709\u8f93\u5165\u4e58\u4ee5\u5bf9\u5e94\u6743\u91cd\u540e\u6c42\u548c\uff0c\u518d\u52a0\u4e0a\u504f\u7f6e<\/li>\n<li><strong>\u6fc0\u6d3b<\/strong>\uff1a\u5c06\u6c42\u548c\u7ed3\u679c\u901a\u8fc7\u6fc0\u6d3b\u51fd\u6570\uff0c\u4ea7\u751f\u6700\u7ec8\u8f93\u51fa<\/li>\n<\/ol>\n<pre><code>        \u8f93\u51651 \u2500\u2500 w1 \u2500\u2500\u2510\n        \u8f93\u51652 \u2500\u2500 w2 \u2500\u2500\u2524\n                      \u251c\u2500\u2500\u2192 \u03a3(wi\u00b7xi) + b \u2500\u2500\u2192 f(\u03a3) \u2500\u2500\u2192 \u8f93\u51fa\n        \u8f93\u51653 \u2500\u2500 w3 \u2500\u2500\u2524\n         ...          \u2502\n        \u8f93\u5165n \u2500\u2500 wn \u2500\u2500\u2518<\/code><\/pre>\n<p>\u6570\u5b66\u8868\u8fbe\u5f0f\uff1a<\/p>\n<p>$$z = \/sum_{i=1}^{n} w_i x_i + b$$<\/p>\n<p>$$a = f(z)$$<\/p>\n<p>\u5176\u4e2d $f$ \u662f\u6fc0\u6d3b\u51fd\u6570\u3002<\/p>\n<h3>4.2 \u6743\u91cd\u548c\u504f\u7f6e\u7684\u51e0\u4f55\u610f\u4e49<\/h3>\n<p><strong>\u6743\u91cd (Weight)<\/strong> $w_i$ \u8868\u793a\u5bf9\u5e94\u8f93\u5165 $x_i$ \u5bf9\u795e\u7ecf\u5143\u8f93\u51fa\u7684<strong>\u5f71\u54cd\u7a0b\u5ea6<\/strong>\uff1a<\/p>\n<ul>\n<li>\u6743\u91cd\u4e3a\u6b63\uff1a\u8f93\u5165\u589e\u5927\u65f6\u8f93\u51fa\u589e\u5927\uff08\u5174\u594b\u6027\u8fde\u63a5\uff09<\/li>\n<li>\u6743\u91cd\u4e3a\u8d1f\uff1a\u8f93\u5165\u589e\u5927\u65f6\u8f93\u51fa\u51cf\u5c0f\uff08\u6291\u5236\u6027\u8fde\u63a5\uff09<\/li>\n<li>\u6743\u91cd\u7684\u7edd\u5bf9\u503c\u8d8a\u5927\uff1a\u8be5\u8f93\u5165\u7684\u5f71\u54cd\u8d8a\u663e\u8457<\/li>\n<\/ul>\n<p><strong>\u504f\u7f6e (Bias)<\/strong> $b$ \u76f8\u5f53\u4e8e\u6570\u5b66\u76f4\u7ebf\u65b9\u7a0b $y = wx + b$ \u4e2d\u7684 $b$\uff08\u622a\u8ddd\u9879\uff09\uff0c\u5b83\u51b3\u5b9a\u4e86\u795e\u7ecf\u5143<strong>\u5728\u6ca1\u6709\u4efb\u4f55\u6709\u6548\u8f93\u5165\u65f6\u7684\u8f93\u51fa\u503e\u5411<\/strong>\u3002<\/p>\n<p>\u6362\u4e2a\u89d2\u5ea6\u7406\u89e3\uff1a<\/p>\n<p>$$z = \/sum w_i x_i + b = \/vec{w} \/cdot \/vec{x} + b$$<\/p>\n<p>\u8fd9\u662f\u4e00\u4e2a\u8d85\u5e73\u9762\uff08\u5728\u9ad8\u7ef4\u7a7a\u95f4\u4e2d\u7684&#8221;\u5206\u754c\u7ebf&#8221;\uff09\uff0c\u6743\u91cd $w$ \u51b3\u5b9a\u4e86\u5e73\u9762\u7684\u65b9\u5411\uff0c\u504f\u7f6e $b$ \u51b3\u5b9a\u4e86\u5e73\u9762\u7684\u4f4d\u7f6e\u3002<\/p>\n<h3>4.3 \u6fc0\u6d3b\u51fd\u6570\uff1a\u4e3a\u4ec0\u4e48\u9700\u8981&#8221;\u975e\u7ebf\u6027&#8221;\uff1f<\/h3>\n<p>\u5982\u679c\u6ca1\u6709\u6fc0\u6d3b\u51fd\u6570\uff0c\u65e0\u8bba\u5806\u53e0\u591a\u5c11\u5c42\u795e\u7ecf\u7f51\u7edc\uff0c\u6700\u7ec8\u90fd\u7b49\u4ef7\u4e8e\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff1a<\/p>\n<p>$$W_2(W_1x + b_1) + b_2 = (W_2W_1)x + (W_2b_1 + b_2) = W&#8217;x + b&#8217;$$<\/p>\n<p>\u8fd9\u4ecd\u7136\u662f\u7ebf\u6027\u51fd\u6570\uff01<strong>\u591a\u5c42\u7ebf\u6027 = \u5355\u5c42\u7ebf\u6027<\/strong>\u3002<\/p>\n<p>\u6fc0\u6d3b\u51fd\u6570\u5f15\u5165<strong>\u975e\u7ebf\u6027<\/strong>\uff0c\u8ba9\u7f51\u7edc\u80fd\u591f\u5b66\u4e60\u590d\u6742\u7684\u6a21\u5f0f\u3002\u5e38\u89c1\u7684\u6fc0\u6d3b\u51fd\u6570\u6709\uff1a<\/p>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u6fc0\u6d3b\u51fd\u6570<\/th>\n<th align=\"left\">\u516c\u5f0f<\/th>\n<th align=\"center\">\u8f93\u51fa\u8303\u56f4<\/th>\n<th align=\"left\">\u7279\u70b9<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">Sigmoid<\/td>\n<td align=\"left\">$1\/(1+e^{-x})$<\/td>\n<td align=\"center\">(0, 1)<\/td>\n<td align=\"left\">\u5e73\u6ed1\uff0c\u9002\u5408\u4e8c\u5206\u7c7b\u8f93\u51fa\u5c42<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">Tanh<\/td>\n<td align=\"left\">$(e^x-e^{-x})\/(e^x+e^{-x})$<\/td>\n<td align=\"center\">(-1, 1)<\/td>\n<td align=\"left\">\u96f6\u4e2d\u5fc3\uff0c\u6bd4 Sigmoid \u68af\u5ea6\u66f4\u597d<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">ReLU<\/td>\n<td align=\"left\">$\/max(0, x)$<\/td>\n<td align=\"center\">[0, \u221e)<\/td>\n<td align=\"left\">\u8ba1\u7b97\u5feb\uff0c\u7f13\u89e3\u68af\u5ea6\u6d88\u5931\uff0c\u6700\u5e38\u7528<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">Leaky ReLU<\/td>\n<td align=\"left\">$\/max(0.01x, x)$<\/td>\n<td align=\"center\">(-\u221e, \u221e)<\/td>\n<td align=\"left\">ReLU \u7684\u6539\u8fdb\uff0c\u8d1f\u503c\u4e0d\u6d88\u5931<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>4.4 Sigmoid \u51fd\u6570\u8be6\u89e3<\/h3>\n<p>\u672c\u7a0b\u5e8f\u4f7f\u7528\u4e86 Sigmoid \u6fc0\u6d3b\u51fd\u6570\uff1a<\/p>\n<p>$$sigmoid(x) = \/frac{1}{1 + e^{-x}}$$<\/p>\n<p>\u5176\u56fe\u50cf\u5448 S \u5f62\uff1a<\/p>\n<pre><code>\u8f93\u51fa\n 1.0 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n     |\n     |          ****\n 0.5 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500**\n     |   **\n 0.0 \u2500\u2500**\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2192 \u8f93\u5165<\/code><\/pre>\n<p><strong>Sigmoid \u7684\u7279\u6027\uff1a<\/strong><\/p>\n<ul>\n<li>\u5c06\u4efb\u610f\u5b9e\u6570\u6620\u5c04\u5230 $(0, 1)$ \u533a\u95f4<\/li>\n<li>$x = 0$ \u65f6\u8f93\u51fa $0.5$<\/li>\n<li>$x$ \u5f88\u5927\u65f6\u8f93\u51fa\u63a5\u8fd1 $1$\uff08\u795e\u7ecf\u5143&#8221;\u6fc0\u6d3b&#8221;\uff09<\/li>\n<li>$x$ \u5f88\u5c0f\u65f6\u8f93\u51fa\u63a5\u8fd1 $0$\uff08\u795e\u7ecf\u5143&#8221;\u6291\u5236&#8221;\uff09<\/li>\n<li>\u5bfc\u6570\u8ba1\u7b97\u7b80\u5355\uff1a$sigmoid'(x) = sigmoid(x) \/cdot (1 &#8211; sigmoid(x))$<\/li>\n<\/ul>\n<p><strong>\u4e3a\u4ec0\u4e48\u7528\u4e8e\u4e8c\u5206\u7c7b\uff1f<\/strong><br \/>\u56e0\u4e3a\u8f93\u51fa\u5728 0 \u5230 1 \u4e4b\u95f4\uff0c\u53ef\u4ee5\u89e3\u91ca\u4e3a&#8221;\u662f\u6b63\u7c7b\u7684\u6982\u7387&#8221;\u3002<\/p>\n<p><strong>Sigmoid \u7684\u7f3a\u70b9\uff1a<\/strong><\/p>\n<ul>\n<li>\u5f53 $x$ \u5f88\u5927\u6216\u5f88\u5c0f\u65f6\uff0c\u68af\u5ea6\u63a5\u8fd1 0\uff08\u68af\u5ea6\u6d88\u5931\u95ee\u9898\uff09<\/li>\n<li>\u8f93\u51fa\u4e0d\u662f\u96f6\u4e2d\u5fc3\u7684\uff0c\u53ef\u80fd\u5f71\u54cd\u4f18\u5316\u6548\u7387<\/li>\n<\/ul>\n<blockquote><p>\u73b0\u4ee3\u795e\u7ecf\u7f51\u7edc\u901a\u5e38\u9690\u85cf\u5c42\u7528 ReLU\uff0c\u8f93\u51fa\u5c42\u6839\u636e\u4efb\u52a1\u9009\u62e9\uff08\u4e8c\u5206\u7c7b\u7528 Sigmoid\uff0c\u591a\u5206\u7c7b\u7528 Softmax\uff0c\u56de\u5f52\u7528\u7ebf\u6027\uff09\u3002<\/p><\/blockquote>\n<h3>4.5 \u4ee3\u7801\u5bf9\u7167<\/h3>\n<pre><code class=\"c\">\/* \u5355\u4e2a\u795e\u7ecf\u5143\u7684\u8ba1\u7b97\u6d41\u7a0b *\/\ndouble weighted_sum = 0.0;                              \/\/ 1. \u521d\u59cb\u5316\u7d2f\u52a0\u5668\nfor (int input_idx = 0; input_idx &lt; INPUT_SIZE; input_idx++)\n    weighted_sum += net-&gt;hidden_weights[neuron_idx][input_idx] * input[input_idx];               \/\/ 2. \u52a0\u6743\u6c42\u548c\nweighted_sum += net-&gt;hidden_biases[neuron_idx];                             \/\/ 3. \u52a0\u4e0a\u504f\u7f6e\nnet-&gt;hidden_z[neuron_idx] = weighted_sum;                              \/\/ 4. \u4fdd\u5b58\u51c0\u8f93\u5165 z\nnet-&gt;hidden_a[neuron_idx] = sigmoid(weighted_sum);                     \/\/ 5. \u6fc0\u6d3b\u5f97\u5230\u8f93\u51fa a<\/code><\/pre>\n<hr>\n<h2>5. \u6b63\u5411\u4f20\u64ad\uff1a\u4fe1\u53f7\u5982\u4f55\u6d41\u52a8<\/h2>\n<h3>5.1 \u5b9a\u4e49<\/h3>\n<p><strong>\u6b63\u5411\u4f20\u64ad (Forward Propagation)<\/strong> \u662f\u6307\u8f93\u5165\u6570\u636e\u4ece\u8f93\u5165\u5c42\u51fa\u53d1\uff0c\u9010\u5c42\u5411\u524d\u4f20\u9012\uff0c\u6700\u7ec8\u4ea7\u751f\u8f93\u51fa\u7684\u8fc7\u7a0b\u3002<\/p>\n<h3>5.2 \u6570\u5b66\u63a8\u5bfc<\/h3>\n<p><strong>\u7b2c 1 \u6b65\uff1a\u8f93\u5165\u5c42 \u2192 \u9690\u85cf\u5c42<\/strong><\/p>\n<p>\u5bf9\u4e8e\u9690\u85cf\u5c42\u7b2c $i$ \u4e2a\u795e\u7ecf\u5143\uff1a<\/p>\n<p>$$z1_i = \/sum_{j=1}^{2} W1_{i,j} \/cdot x_j + b1_i$$<\/p>\n<p>$$a1_i = \/sigma(z1_i)$$<\/p>\n<p>\u5176\u4e2d $\/sigma$ \u8868\u793a sigmoid \u51fd\u6570\u3002<\/p>\n<p>\u7528\u77e9\u9635\u5f62\u5f0f\u8868\u793a\uff1a<\/p>\n<p>$$\/vec{z1} = W_1 \/cdot \/vec{x} + \/vec{b_1}$$<\/p>\n<p>$$\/vec{a1} = \/sigma(\/vec{z1})$$<\/p>\n<p><strong>\u7b2c 2 \u6b65\uff1a\u9690\u85cf\u5c42 \u2192 \u8f93\u51fa\u5c42<\/strong><\/p>\n<p>\u5bf9\u4e8e\u8f93\u51fa\u5c42\uff08\u53ea\u6709\u4e00\u4e2a\u795e\u7ecf\u5143\uff09\uff1a<\/p>\n<p>$$z2 = \/sum_{j=1}^{4} W2_{1,j} \/cdot a1_j + b2$$<\/p>\n<p>$$a2 = \/sigma(z2)$$<\/p>\n<p>\u77e9\u9635\u5f62\u5f0f\uff1a<\/p>\n<p>$$z2 = W_2 \/cdot \/vec{a1} + b_2$$<\/p>\n<p>$$a2 = \/sigma(z2)$$<\/p>\n<h3>5.3 \u76f4\u89c2\u7406\u89e3<\/h3>\n<p>\u6b63\u5411\u4f20\u64ad\u5c31\u662f<strong>\u8f93\u5165\u4fe1\u53f7\u5728\u7f51\u7edc\u4e2d\u9010\u5c42\u4f20\u9012<\/strong>\u7684\u8fc7\u7a0b\u3002\u6bcf\u4e00\u5c42\u90fd\uff1a<\/p>\n<ol>\n<li>\u63a5\u6536\u4e0a\u4e00\u5c42\u7684\u8f93\u51fa<\/li>\n<li>\u7528\u5f53\u524d\u5c42\u7684\u6743\u91cd\u8fdb\u884c\u52a0\u6743\u6c42\u548c<\/li>\n<li>\u52a0\u4e0a\u504f\u7f6e<\/li>\n<li>\u901a\u8fc7\u6fc0\u6d3b\u51fd\u6570\u538b\u7f29<\/li>\n<li>\u4f20\u9012\u7ed9\u4e0b\u4e00\u5c42<\/li>\n<\/ol>\n<pre style=\"display:none;\"><code class=\"mermaid\">flowchart LR\n    x1[&quot;x\u2081&quot;] --&gt; h1[&quot;h\u2081: z\u2081\u2081=\u03a3w\u00b7x+b \u2192 \u03c3(z\u2081\u2081)&quot;]\n    x1 --&gt; h2[&quot;h\u2082&quot;]\n    x2[&quot;x\u2082&quot;] --&gt; h1\n    x2 --&gt; h2\n    x1 --&gt; h3[&quot;h\u2083&quot;]\n    x1 --&gt; h4[&quot;h\u2084&quot;]\n    x2 --&gt; h3\n    x2 --&gt; h4\n    h1 --&gt; y[&quot;y: z\u2082=\u03a3w\u00b7a\u2081+b \u2192 \u03c3(z\u2082)&quot;]\n    h2 --&gt; y\n    h3 --&gt; y\n    h4 --&gt; y<\/code><\/pre>\n<h3>5.4 \u4e3a\u4ec0\u4e48\u53eb&#8221;\u524d\u9988&#8221;\uff1f<\/h3>\n<p>\u524d\u9988 (Feedforward) \u7684\u542b\u4e49\u662f\u4fe1\u606f<strong>\u5355\u5411\u6d41\u52a8<\/strong>\uff0c\u6ca1\u6709\u56de\u8def\uff1a<\/p>\n<pre><code>x \u2192 z1 \u2192 \u03c3(z1) = a1 \u2192 z2 \u2192 \u03c3(z2) = a2 \u2192 \u0177<\/code><\/pre>\n<p>\u4e0e\u4e4b\u5bf9\u5e94\u7684\u662f\u5faa\u73af\u795e\u7ecf\u7f51\u7edc (RNN)\uff0c\u5b83\u7684\u9690\u85cf\u5c42\u4f1a\u63a5\u6536\u81ea\u5df1\u7684\u5386\u53f2\u8f93\u51fa\u4f5c\u4e3a\u5f53\u524d\u8f93\u5165\u7684\u4e00\u90e8\u5206\uff0c\u5f62\u6210\u5faa\u73af\u3002<\/p>\n<h3>5.5 \u4ee3\u7801\u4e2d\u7684\u5b9e\u73b0<\/h3>\n<pre><code class=\"c\">void forward(NeuralNet *network, double *input) {\n    \/\/ \u7b2c\u4e00\u6b65\uff1a\u8f93\u5165\u5c42 \u2192 \u9690\u85cf\u5c42\n    for (int neuron_idx = 0; neuron_idx &lt; HIDDEN_SIZE; neuron_idx++) {\n        double weighted_sum = 0.0;\n        for (int input_idx = 0; input_idx &lt; INPUT_SIZE; input_idx++)\n            weighted_sum += network-&gt;hidden_weights[neuron_idx][input_idx] * input[input_idx];    \/\/ \u52a0\u6743\u6c42\u548c\n        weighted_sum += network-&gt;hidden_biases[neuron_idx];                                       \/\/ \u52a0\u504f\u7f6e\n        network-&gt;hidden_z[neuron_idx] = weighted_sum;                                        \/\/ \u4fdd\u5b58\u51c0\u8f93\u5165\n        network-&gt;hidden_a[neuron_idx] = sigmoid(weighted_sum);                               \/\/ \u6fc0\u6d3b\n    }\n\n    \/\/ \u7b2c\u4e8c\u6b65\uff1a\u9690\u85cf\u5c42 \u2192 \u8f93\u51fa\u5c42\n    for (int output_idx = 0; output_idx &lt; OUTPUT_SIZE; output_idx++) {\n        double weighted_sum = 0.0;\n        for (int hidden_idx = 0; hidden_idx &lt; HIDDEN_SIZE; hidden_idx++)\n            weighted_sum += network-&gt;output_weights[output_idx][hidden_idx] * network-&gt;hidden_a[hidden_idx];  \/\/ \u4ee5\u4e0a\u4e00\u5c42\u7684\u6fc0\u6d3b\u503c\u4f5c\u4e3a\u8f93\u5165\n        weighted_sum += network-&gt;output_biases[output_idx];\n        network-&gt;output_z[output_idx] = weighted_sum;\n        network-&gt;output_a[output_idx] = sigmoid(weighted_sum);                               \/\/ \u6700\u7ec8\u8f93\u51fa\n    }\n}<\/code><\/pre>\n<hr>\n<h2>6. \u635f\u5931\u51fd\u6570\uff1a\u5982\u4f55\u8861\u91cf\u5bf9\u9519<\/h2>\n<h3>6.1 \u4ec0\u4e48\u662f\u635f\u5931\u51fd\u6570\uff1f<\/h3>\n<p><strong>\u635f\u5931\u51fd\u6570 (Loss Function)<\/strong> \u91cf\u5316\u4e86\u6a21\u578b\u7684\u9884\u6d4b\u503c\u4e0e\u771f\u5b9e\u503c\u4e4b\u95f4\u7684\u5dee\u8ddd\u3002\u635f\u5931\u8d8a\u5c0f\uff0c\u8bf4\u660e\u9884\u6d4b\u8d8a\u51c6\u786e\u3002<\/p>\n<p>\u8bad\u7ec3\u795e\u7ecf\u7f51\u7edc\u7684\u76ee\u6807\u5c31\u662f<strong>\u6700\u5c0f\u5316\u635f\u5931\u51fd\u6570<\/strong>\u3002<\/p>\n<h3>6.2 \u5747\u65b9\u8bef\u5dee (Mean Squared Error, MSE)<\/h3>\n<p>\u672c\u7a0b\u5e8f\u4f7f\u7528<strong>\u5747\u65b9\u8bef\u5dee<\/strong>\u7684\u4e00\u534a\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\uff1a<\/p>\n<p>$$L = \/frac{1}{2}(\/hat{y} &#8211; y)^2$$<\/p>\n<p>\u5176\u4e2d $\/hat{y}$ \u662f\u9884\u6d4b\u503c\uff0c$y$ \u662f\u771f\u5b9e\u76ee\u6807\u503c\u3002<\/p>\n<p><strong>\u4e3a\u4ec0\u4e48\u7528\u5e73\u65b9\uff1f<\/strong><\/p>\n<ul>\n<li>\u8bef\u5dee\u4e3a\u6b63\u65f6\u60e9\u7f5a\u4e3a\u6b63\uff0c\u8bef\u5dee\u4e3a\u8d1f\u65f6\u60e9\u7f5a\u4e5f\u4e3a\u6b63\uff08\u5e73\u65b9\u6d88\u9664\u4e86\u7b26\u53f7\uff09<\/li>\n<li>\u653e\u5927\u8f83\u5927\u8bef\u5dee\uff08\u5e73\u65b9\u653e\u5927\u4e86\u504f\u79bb\u7a0b\u5ea6\uff09<\/li>\n<li>\u6570\u5b66\u4e0a\u53ef\u5bfc\uff0c\u65b9\u4fbf\u68af\u5ea6\u8ba1\u7b97<\/li>\n<\/ul>\n<p><strong>\u4e3a\u4ec0\u4e48\u4e58\u4ee5 $\/frac{1}{2}$\uff1f<\/strong><br \/>\u8fd9\u662f\u4e3a\u4e86\u540e\u7eed\u6c42\u5bfc\u65f6\u62b5\u6d88\u5e73\u65b9\u9879\u5e26\u6765\u7684\u7cfb\u6570 2\uff0c\u4f7f\u68af\u5ea6\u8868\u8fbe\u5f0f\u66f4\u7b80\u6d01\uff1a<\/p>\n<p>$$\/frac{\/partial L}{\/partial \/hat{y}} = \/hat{y} &#8211; y$$<\/p>\n<p>\u800c\u4e0d\u662f $2(\/hat{y} &#8211; y)$\u3002<\/p>\n<h3>6.3 \u5e73\u5747\u635f\u5931<\/h3>\n<p>\u5f53\u6709\u591a\u4e2a\u8bad\u7ec3\u6837\u672c\u65f6\uff0c\u6211\u4eec\u5bf9\u6240\u6709\u6837\u672c\u7684\u635f\u5931\u53d6\u5e73\u5747\uff1a<\/p>\n<p>$$\/text{Average Loss} = \/frac{1}{n} \/sum_{s=1}^{n} \/frac{1}{2}(\/hat{y}_s &#8211; y_s)^2$$<\/p>\n<p>\u8fd9\u79f0\u4e3a<strong>\u5747\u65b9\u8bef\u5dee (MSE)<\/strong>\u3002<\/p>\n<h3>6.4 \u4ee3\u7801\u4e2d\u7684\u635f\u5931\u8ba1\u7b97<\/h3>\n<pre><code class=\"c\">double difference = network-&gt;output_a[0] - targets[sample_idx];   \/\/ \u9884\u6d4b\u503c - \u76ee\u6807\u503c\nloss += 0.5 * difference * difference;                    \/\/ \u7d2f\u52a0\u635f\u5931\n\/\/ \u6bcf 1000 \u8f6e\u6253\u5370\u5e73\u5747\u635f\u5931\nprintf(&quot;Epoch %d, Loss = %.6f\/n&quot;, epoch, loss \/ num_samples);<\/code><\/pre>\n<p>\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u4f60\u4f1a\u770b\u5230\u635f\u5931\u9010\u6e10\u4e0b\u964d\uff1a<\/p>\n<pre><code>Epoch 0,     Loss = 0.134335\nEpoch 1000,  Loss = 0.003974\nEpoch 10000, Loss = 0.000091\nEpoch 999000, Loss = 0.000001<\/code><\/pre>\n<p>\u635f\u5931\u63a5\u8fd1 0 \u610f\u5473\u7740\u7f51\u7edc\u5df2\u7ecf\u5b66\u4f1a\u4e86 XOR \u51fd\u6570\u3002<\/p>\n<h3>6.5 \u5176\u4ed6\u5e38\u89c1\u635f\u5931\u51fd\u6570<\/h3>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u635f\u5931\u51fd\u6570<\/th>\n<th align=\"left\">\u516c\u5f0f<\/th>\n<th align=\"left\">\u9002\u7528\u573a\u666f<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">\u5747\u65b9\u8bef\u5dee (MSE)<\/td>\n<td align=\"left\">$\/frac{1}{n}\/sum(\/hat{y}-y)^2$<\/td>\n<td align=\"left\">\u56de\u5f52\u95ee\u9898<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u5e73\u5747\u7edd\u5bf9\u8bef\u5dee (MAE)<\/td>\n<td align=\"left\">$\/frac{1}{n}\/sum\/<\/td>\n<td align=\"left\">\/hat{y}-y \/<\/td>\n<td>$<\/td>\n<td>\u56de\u5f52\u95ee\u9898\uff08\u5bf9\u5f02\u5e38\u503c\u66f4\u9c81\u68d2\uff09<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u4ea4\u53c9\u71b5 (Cross Entropy)<\/td>\n<td align=\"left\">$-[y\/log\/hat{y} + (1-y)\/log(1-\/hat{y})]$<\/td>\n<td align=\"left\">\u4e8c\u5206\u7c7b\u95ee\u9898<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">Hinge Loss<\/td>\n<td align=\"left\">$\/max(0, 1 &#8211; y \/cdot \/hat{y})$<\/td>\n<td align=\"left\">\u652f\u6301\u5411\u91cf\u673a<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr>\n<h2>7. \u53cd\u5411\u4f20\u64ad\uff1a\u8bef\u5dee\u5982\u4f55\u9006\u6d41<\/h2>\n<h3>7.1 \u6838\u5fc3\u601d\u60f3<\/h3>\n<p><strong>\u53cd\u5411\u4f20\u64ad (Backpropagation)<\/strong> \u662f\u8bad\u7ec3\u795e\u7ecf\u7f51\u7edc\u7684\u6838\u5fc3\u7b97\u6cd5\u3002\u5b83\u7684\u4f5c\u7528\u662f\uff1a<\/p>\n<blockquote><p>\u8ba1\u7b97\u6bcf\u4e2a\u53c2\u6570\uff08\u6743\u91cd\u548c\u504f\u7f6e\uff09\u5bf9\u6700\u7ec8\u635f\u5931\u7684&#8221;\u8d21\u732e\u5ea6&#8221;\uff08\u5373\u68af\u5ea6\uff09\uff0c\u4ece\u800c\u77e5\u9053\u6bcf\u4e2a\u53c2\u6570\u5e94\u8be5\u671d\u54ea\u4e2a\u65b9\u5411\u8c03\u6574\u3002<\/p><\/blockquote>\n<p>\u53cd\u5411\u4f20\u64ad\u7684\u672c\u8d28\u662f<strong>\u94fe\u5f0f\u6cd5\u5219 (Chain Rule)<\/strong> \u7684\u5e94\u7528 \u2014\u2014 \u8fd9\u662f\u5fae\u79ef\u5206\u4e2d\u8ba1\u7b97\u590d\u5408\u51fd\u6570\u5bfc\u6570\u7684\u65b9\u6cd5\u3002<\/p>\n<h3>7.2 \u76f4\u89c2\u7406\u89e3<\/h3>\n<pre><code>\u6b63\u5411\u4f20\u64ad\uff1a    x \u2192 z1 \u2192 a1 \u2192 z2 \u2192 a2 \u2192 \u0177 \u2192 \u635f\u5931 L\n\u53cd\u5411\u4f20\u64ad\uff1a    x \u2190 z1 \u2190 a1 \u2190 z2 \u2190 a2 \u2190 \u0177 \u2190 \u635f\u5931 L\n                     \u2191                      \u2191\n                \u8bef\u5dee\u4ece\u9690\u85cf\u5c42         \u8bef\u5dee\u4ece\u8f93\u51fa\u5c42\n                \u53cd\u5411\u4f20\u9012\u800c\u6765         \u76f4\u63a5\u8ba1\u7b97\u5f97\u5230<\/code><\/pre>\n<p><strong>\u5173\u952e insight\uff1a<\/strong> \u8f93\u51fa\u5c42\u7684\u8bef\u5dee\u53ef\u4ee5\u76f4\u63a5\u6bd4\u8f83\u9884\u6d4b\u503c\u548c\u76ee\u6807\u503c\u5f97\u5230\u3002\u4f46\u9690\u85cf\u5c42\u7684\u8bef\u5dee\u4e0d\u80fd\u76f4\u63a5\u5f97\u5230 \u2014\u2014 \u5b83\u9700\u8981\u901a\u8fc7\u8f93\u51fa\u5c42\u7684\u8bef\u5dee<strong>\u53cd\u5411\u4f20\u64ad<\/strong>\u56de\u6765\u3002<\/p>\n<h3>7.3 \u6570\u5b66\u63a8\u5bfc\uff08\u94fe\u5f0f\u6cd5\u5219\uff09<\/h3>\n<p>\u6211\u4eec\u5148\u68b3\u7406\u6574\u4e2a\u8ba1\u7b97\u56fe\uff1a<\/p>\n<p>$$\/begin{aligned}<br \/>z1_i &#038;= \/sum_j W1_{i,j} x_j + b1_i \/<br \/>a1_i &#038;= \/sigma(z1_i) \/<br \/>z2   &#038;= \/sum_j W2_{1,j} a1_j + b2 \/<br \/>a2   &#038;= \/sigma(z2) \/<br \/>L    &#038;= \/frac{1}{2}(a2 &#8211; y)^2<br \/>\/end{aligned}$$<\/p>\n<h4>7.3.1 \u8f93\u51fa\u5c42\u7684\u68af\u5ea6<\/h4>\n<p>\u9996\u5148\u8ba1\u7b97\u8f93\u51fa\u5c42\u7684\u8bef\u5dee\u9879 $\/delta_2$\uff08\u8f93\u51fa\u5c42\u68af\u5ea6\uff09\uff1a<\/p>\n<p>$$\/delta_2 = \/frac{\/partial L}{\/partial z2} = \/frac{\/partial L}{\/partial a2} \/cdot \/frac{\/partial a2}{\/partial z2}$$<\/p>\n<p>\u5176\u4e2d\uff1a<\/p>\n<p>$$\/frac{\/partial L}{\/partial a2} = a2 &#8211; y$$<\/p>\n<p>$$\/frac{\/partial a2}{\/partial z2} = \/sigma'(z2) = \/sigma(z2) \/cdot (1 &#8211; \/sigma(z2)) = a2 \/cdot (1 &#8211; a2)$$<\/p>\n<p>\u6240\u4ee5\uff1a<\/p>\n<p>$$\/delta_2 = (a2 &#8211; y) \/cdot a2 \/cdot (1 &#8211; a2)$$<\/p>\n<p>\u6709\u4e86 $\/delta_2$\uff0c\u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u8f93\u51fa\u5c42\u53c2\u6570\u7684\u68af\u5ea6\uff1a<\/p>\n<p>$$\/frac{\/partial L}{\/partial W2_{1,j}} = \/frac{\/partial L}{\/partial z2} \/cdot \/frac{\/partial z2}{\/partial W2_{1,j}} = \/delta_2 \/cdot a1_j$$<\/p>\n<p>$$\/frac{\/partial L}{\/partial b2} = \/frac{\/partial L}{\/partial z2} \/cdot \/frac{\/partial z2}{\/partial b2} = \/delta_2$$<\/p>\n<h4>7.3.2 \u9690\u85cf\u5c42\u7684\u68af\u5ea6<\/h4>\n<p>\u73b0\u5728\u5c06\u8bef\u5dee\u4f20\u64ad\u5230\u9690\u85cf\u5c42\u3002\u9690\u85cf\u5c42\u7b2c $i$ \u4e2a\u795e\u7ecf\u5143\u7684\u8bef\u5dee\u9879 $\/delta_{1,i}$\uff1a<\/p>\n<p>$$\/delta_{1,i} = \/frac{\/partial L}{\/partial z1_i} = \/frac{\/partial L}{\/partial a1_i} \/cdot \/frac{\/partial a1_i}{\/partial z1_i}$$<\/p>\n<p>\u5176\u4e2d $\/frac{\/partial a1_i}{\/partial z1_i} = \/sigma'(z1_i) = a1_i \/cdot (1 &#8211; a1_i)$\u3002<\/p>\n<p>\u800c $\/frac{\/partial L}{\/partial a1_i}$ \u9700\u8981\u4ece\u8f93\u51fa\u5c42\u4f20\u56de\u6765\uff1a<\/p>\n<p>$$\/frac{\/partial L}{\/partial a1_i} = \/frac{\/partial L}{\/partial z2} \/cdot \/frac{\/partial z2}{\/partial a1_i} = \/delta_2 \/cdot W2_{1,i}$$<\/p>\n<p>\u6240\u4ee5\uff1a<\/p>\n<p>$$\/delta_{1,i} = \/delta_2 \/cdot W2_{1,i} \/cdot a1_i \/cdot (1 &#8211; a1_i)$$<\/p>\n<p>\u6709\u4e86 $\/delta_{1,i}$\uff0c\u9690\u85cf\u5c42\u53c2\u6570\u7684\u68af\u5ea6\u4e3a\uff1a<\/p>\n<p>$$\/frac{\/partial L}{\/partial W1_{i,j}} = \/delta_{1,i} \/cdot x_j$$<\/p>\n<p>$$\/frac{\/partial L}{\/partial b1_i} = \/delta_{1,i}$$<\/p>\n<h3>7.4 \u5c42\u4e0e\u5c42\u4e4b\u95f4\u7684\u901a\u7528\u89c4\u5f8b<\/h3>\n<p>\u4ed4\u7ec6\u89c2\u5bdf\u4e0a\u9762\u7684\u63a8\u5bfc\uff0c\u4f60\u4f1a\u53d1\u73b0\u4e00\u4e2a\u4f18\u7f8e\u7684\u6a21\u5f0f\uff1a<\/p>\n<pre><code>\u8bef\u5dee\u4ece\u4e0a\u5c42\u4f20\u4e0b\u6765\u7684\u503c = (\u4e0a\u5c42\u7684\u8bef\u5dee\u9879) \u00d7 (\u8fde\u63a5\u6743\u91cd)\n\u672c\u5c42\u8bef\u5dee\u9879 = \u4ece\u4e0a\u5c42\u4f20\u4e0b\u6765\u7684\u503c \u00d7 \u672c\u5c42\u6fc0\u6d3b\u51fd\u6570\u7684\u5bfc\u6570\n\u53c2\u6570\u68af\u5ea6 = \u672c\u5c42\u8bef\u5dee\u9879 \u00d7 \u672c\u5c42\u7684\u8f93\u5165\u503c<\/code><\/pre>\n<p>\u8fd9\u6b63\u662f\u53cd\u5411\u4f20\u64ad\u7684\u901a\u7528\u516c\u5f0f\uff0c\u53ef\u4ee5\u63a8\u5e7f\u5230\u4efb\u610f\u6df1\u5ea6\u7684\u7f51\u7edc\u3002<\/p>\n<h3>7.5 \u4ee3\u7801\u5b9e\u73b0<\/h3>\n<pre><code class=\"c\">void backward(NeuralNet *network, double *input, double target) {\n    \/\/ 1. \u8ba1\u7b97\u8f93\u51fa\u5c42\u8bef\u5dee\u9879\n    double error = network-&gt;output_a[0] - target;\n    network-&gt;output_delta[0] = error * sigmoid_derivative(network-&gt;output_a[0]);\n    \/\/ 2. \u5c06\u8bef\u5dee\u4f20\u64ad\u5230\u9690\u85cf\u5c42\n    for (int neuron_idx = 0; neuron_idx &lt; HIDDEN_SIZE; neuron_idx++) {\n        double propagated_error = 0.0;\n        \/\/ \u8bef\u5dee = \u8f93\u51fa\u5c42\u8bef\u5dee\u9879 \u00d7 \u8fde\u63a5\u6743\u91cd\n        propagated_error += network-&gt;output_weights[0][neuron_idx] * network-&gt;output_delta[0];\n        \/\/ \u9690\u85cf\u5c42\u8bef\u5dee\u9879 = \u4f20\u56de\u7684\u8bef\u5dee \u00d7 \u6fc0\u6d3b\u51fd\u6570\u5bfc\u6570\n        network-&gt;hidden_delta[neuron_idx] = propagated_error * sigmoid_derivative(network-&gt;hidden_a[neuron_idx]);\n    }\n}<\/code><\/pre>\n<hr>\n<h2>8. \u68af\u5ea6\u4e0b\u964d\uff1a\u5982\u4f55\u8c03\u6574\u53c2\u6570<\/h2>\n<h3>8.1 \u6838\u5fc3\u601d\u60f3<\/h3>\n<p><strong>\u68af\u5ea6\u4e0b\u964d (Gradient Descent)<\/strong> \u662f\u4e00\u79cd\u8fed\u4ee3\u4f18\u5316\u7b97\u6cd5\uff0c\u7528\u4e8e\u6700\u5c0f\u5316\u635f\u5931\u51fd\u6570\u3002<\/p>\n<p>\u60f3\u8c61\u4f60\u5728\u4e00\u4e2a\u5c71\u8c37\u4e2d\uff08\u635f\u5931\u51fd\u6570\u66f2\u9762\uff09\uff0c\u4f60\u7684\u76ee\u6807\u662f\u8d70\u5230\u8c37\u5e95\uff08\u6700\u5c0f\u503c\u70b9\uff09\u3002\u68af\u5ea6\u544a\u8bc9\u4f60\u5728\u5f53\u524d\u4f4d\u7f6e\u54ea\u4e2a\u65b9\u5411\u6700\u9661\u5ced\uff08\u4e0a\u5347\u65b9\u5411\uff09\uff0c\u4f60\u5f80\u53cd\u65b9\u5411\u8d70\u4e00\u6b65\uff0c\u5c31\u80fd\u4e0b\u964d\u4e00\u70b9\u3002<\/p>\n<p>$$\u65b0\u53c2\u6570 = \u65e7\u53c2\u6570 &#8211; \u5b66\u4e60\u7387 \/times \u68af\u5ea6$$<\/p>\n<pre style=\"display:none;\"><code class=\"mermaid\">flowchart TD\n    A[\u968f\u673a\u521d\u59cb\u5316\u53c2\u6570] --&gt; B[\u6b63\u5411\u4f20\u64ad: \u8ba1\u7b97\u9884\u6d4b\u503c\u548c\u635f\u5931]\n    B --&gt; C[\u53cd\u5411\u4f20\u64ad: \u8ba1\u7b97\u68af\u5ea6]\n    C --&gt; D[\u66f4\u65b0\u53c2\u6570: W -= lr \u00d7 \u68af\u5ea6]\n    D --&gt; E{\u635f\u5931\u8db3\u591f\u5c0f?}\n    E -- \u5426 --&gt; B\n    E -- \u662f --&gt; F[\u8bad\u7ec3\u5b8c\u6210]<\/code><\/pre>\n<h3>8.2 \u5b66\u4e60\u7387<\/h3>\n<p><strong>\u5b66\u4e60\u7387 (Learning Rate, lr)<\/strong> \u63a7\u5236\u6bcf\u6b21\u53c2\u6570\u66f4\u65b0\u7684\u6b65\u957f\uff1a<\/p>\n<ul>\n<li><strong>\u5b66\u4e60\u7387\u592a\u5927<\/strong>\uff1a\u53c2\u6570\u66f4\u65b0\u5e45\u5ea6\u8fc7\u5927\uff0c\u53ef\u80fd\u8d8a\u8fc7\u6700\u4f18\u70b9\uff0c\u5728\u8c37\u5e95\u9644\u8fd1\u9707\u8361\u751a\u81f3\u53d1\u6563<\/li>\n<\/ul>\n<pre><code>       \u2571\u2572       \u2571\u2572\n      \u2571  \u2572     \u2571  \u2572\n     \u2571    \u2572   \u2571    \u2572\n    \u2571      \u2572_\u2571      \u2572<\/code><\/pre>\n<ul>\n<li><strong>\u5b66\u4e60\u7387\u592a\u5c0f<\/strong>\uff1a\u66f4\u65b0\u901f\u5ea6\u6162\uff0c\u9700\u8981\u5f88\u591a\u6b21\u8fed\u4ee3\u624d\u80fd\u6536\u655b<\/li>\n<\/ul>\n<pre><code>     \u2192 \u2192 \u2192 \u2192 \u2192 \u2192 \u2192 \u2192\n     \u2192 \u2192 \u2192 \u2192 \u2192 \u2192 \u2192 \u2192<\/code><\/pre>\n<ul>\n<li><strong>\u5b66\u4e60\u7387\u9002\u4e2d<\/strong>\uff1a\u7a33\u5b9a\u4e0b\u964d\uff0c\u9ad8\u6548\u6536\u655b<\/li>\n<\/ul>\n<pre><code>     \u2192 \u2192 \u2192\n       \u2192 \u2192 \u2192\n         \u2192 \u2192 \u2192<\/code><\/pre>\n<p>\u672c\u7a0b\u5e8f\u4f7f\u7528 $lr = 0.5$\uff0c\u5bf9\u4e8e XOR \u95ee\u9898\u6765\u8bf4\u662f\u4e00\u4e2a\u5408\u9002\u7684\u503c\u3002<\/p>\n<h3>8.3 \u68af\u5ea6\u4e0b\u964d\u7684\u53d8\u4f53<\/h3>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u65b9\u6cd5<\/th>\n<th align=\"center\">\u6bcf\u6b21\u66f4\u65b0\u4f7f\u7528\u7684\u6837\u672c\u6570<\/th>\n<th align=\"left\">\u7279\u70b9<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\"><strong>\u6279\u91cf\u68af\u5ea6\u4e0b\u964d (BGD)<\/strong><\/td>\n<td align=\"center\">\u5168\u90e8\u6837\u672c<\/td>\n<td align=\"left\">\u7a33\u5b9a\u4f46\u6162<\/td>\n<\/tr>\n<tr>\n<td align=\"left\"><strong>\u968f\u673a\u68af\u5ea6\u4e0b\u964d (SGD)<\/strong><\/td>\n<td align=\"center\">1 \u4e2a\u6837\u672c<\/td>\n<td align=\"left\">\u5feb\u4f46\u9707\u8361\u5927<\/td>\n<\/tr>\n<tr>\n<td align=\"left\"><strong>\u5c0f\u6279\u91cf\u68af\u5ea6\u4e0b\u964d (Mini-batch)<\/strong><\/td>\n<td align=\"center\">16~256 \u4e2a\u6837\u672c<\/td>\n<td align=\"left\">\u6298\u4e2d\uff0c\u6700\u5e38\u7528<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u672c\u7a0b\u5e8f\u4f7f\u7528\u7684\u662f <strong>SGD<\/strong>\uff08\u6bcf\u770b\u5230 1 \u4e2a\u6837\u672c\u5c31\u66f4\u65b0\u4e00\u6b21\u53c2\u6570\uff09\u3002<\/p>\n<h3>8.4 \u4ee3\u7801\u5b9e\u73b0<\/h3>\n<pre><code class=\"c\">void update_weights(NeuralNet *network, double *input, double learning_rate) {\n    \/\/ \u66f4\u65b0\u8f93\u51fa\u5c42: output_weights -= lr \u00d7 \u03b42 \u00d7 hidden_a\n    for (int hidden_idx = 0; hidden_idx &lt; HIDDEN_SIZE; hidden_idx++)\n        network-&gt;output_weights[0][hidden_idx] -= learning_rate * network-&gt;output_delta[0] * network-&gt;hidden_a[hidden_idx];\n    network-&gt;output_biases[0] -= learning_rate * network-&gt;output_delta[0];\n\n    \/\/ \u66f4\u65b0\u9690\u85cf\u5c42: hidden_weights -= lr \u00d7 \u03b41 \u00d7 input\n    for (int neuron_idx = 0; neuron_idx &lt; HIDDEN_SIZE; neuron_idx++) {\n        for (int input_idx = 0; input_idx &lt; INPUT_SIZE; input_idx++)\n            network-&gt;hidden_weights[neuron_idx][input_idx] -= learning_rate * network-&gt;hidden_delta[neuron_idx] * input[input_idx];\n        network-&gt;hidden_biases[neuron_idx] -= learning_rate * network-&gt;hidden_delta[neuron_idx];\n    }\n}<\/code><\/pre>\n<h3>8.5 \u5b8c\u6574\u7684\u53c2\u6570\u66f4\u65b0\u516c\u5f0f\u6c47\u603b<\/h3>\n<p>\u6574\u7406\u6240\u6709\u53c2\u6570\u7684\u66f4\u65b0\u89c4\u5219\uff1a<\/p>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u53c2\u6570<\/th>\n<th align=\"left\">\u66f4\u65b0\u516c\u5f0f<\/th>\n<th align=\"left\">\u542b\u4e49<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">$W2_{1,j}$<\/td>\n<td align=\"left\">$W2_{1,j} \/leftarrow W2_{1,j} &#8211; lr \/cdot \/delta_2 \/cdot a1_j$<\/td>\n<td align=\"left\">\u8f93\u51fa\u5c42\u6743\u91cd<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">$b2$<\/td>\n<td align=\"left\">$b2 \/leftarrow b2 &#8211; lr \/cdot \/delta_2$<\/td>\n<td align=\"left\">\u8f93\u51fa\u5c42\u504f\u7f6e<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">$W1_{i,j}$<\/td>\n<td align=\"left\">$W1_{i,j} \/leftarrow W1_{i,j} &#8211; lr \/cdot \/delta_{1,i} \/cdot x_j$<\/td>\n<td align=\"left\">\u9690\u85cf\u5c42\u6743\u91cd<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">$b1_i$<\/td>\n<td align=\"left\">$b1_i \/leftarrow b1_i &#8211; lr \/cdot \/delta_{1,i}$<\/td>\n<td align=\"left\">\u9690\u85cf\u5c42\u504f\u7f6e<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u5176\u4e2d\uff1a$\/delta_2 = (a2 &#8211; y) \/cdot a2 \/cdot (1 &#8211; a2)$\uff0c$\/delta_{1,i} = \/delta_2 \/cdot W2_{1,i} \/cdot a1_i \/cdot (1 &#8211; a1_i)$\u3002<\/p>\n<hr>\n<h2>9. \u8bad\u7ec3\u5faa\u73af\uff1a\u91cd\u590d\u7684\u529b\u91cf<\/h2>\n<h3>9.1 \u8bad\u7ec3\u6d41\u7a0b<\/h3>\n<pre><code class=\"c\">void train(NeuralNet *network, double inputs[][INPUT_SIZE], double targets[],\n           int num_samples, int epochs, double learning_rate) {\n    for (int epoch = 0; epoch &lt; epochs; epoch++) {       \/\/ \u5916\u5c42\uff1a\u8fed\u4ee3\u591a\u4e2a\u8f6e\u6b21\n        double loss = 0.0;\n        for (int sample_idx = 0; sample_idx &lt; num_samples; sample_idx++) {        \/\/ \u5185\u5c42\uff1a\u904d\u5386\u6240\u6709\u6837\u672c\n            forward(network, inputs[sample_idx]);          \/\/ \u6b65\u9aa4 1: \u6b63\u5411\u4f20\u64ad\n            backward(network, inputs[sample_idx], targets[sample_idx]); \/\/ \u6b65\u9aa4 2: \u53cd\u5411\u4f20\u64ad\n            update_weights(network, inputs[sample_idx], learning_rate);  \/\/ \u6b65\u9aa4 3: \u66f4\u65b0\u6743\u91cd\n\n            double difference = network-&gt;output_a[0] - targets[sample_idx];\n            loss += 0.5 * difference * difference;          \/\/ \u7d2f\u8ba1\u635f\u5931\n        }\n        if (epoch % 1000 == 0)\n            printf(&quot;Epoch %d, Loss = %.6f\/n&quot;, epoch, loss \/ num_samples);\n    }\n}<\/code><\/pre>\n<h3>9.2 Epoch \u662f\u4ec0\u4e48\uff1f<\/h3>\n<p>\u4e00\u4e2a <strong>epoch<\/strong>\uff08\u8f6e\u6b21\uff09\u7b49\u4e8e<strong>\u904d\u5386\u6240\u6709\u8bad\u7ec3\u6837\u672c\u4e00\u6b21<\/strong>\u3002<\/p>\n<ul>\n<li>\u8bad\u7ec3 1,000,000 \u8f6e = \u6bcf\u8f6e 4 \u4e2a\u6837\u672c = \u603b\u5171 4,000,000 \u6b21\u53c2\u6570\u66f4\u65b0<\/li>\n<li>\u6bcf 1000 \u8f6e\u6253\u5370\u4e00\u6b21\u635f\u5931\uff0c\u53ef\u4ee5\u770b\u5230\u635f\u5931\u9010\u6e10\u4e0b\u964d\u7684\u8d8b\u52bf<\/li>\n<\/ul>\n<h3>9.3 \u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u7684\u635f\u5931\u53d8\u5316<\/h3>\n<p>\u5178\u578b\u7684\u8bad\u7ec3\u66f2\u7ebf\uff1a<\/p>\n<pre><code>\u635f\u5931\n\u2502\n\u2502   *\n\u2502    *\n\u2502     ****\n\u2502         ***\n\u2502            ****\n\u2502                 ***\n\u2502                     ****\n\u2502                          ****\n\u2502                               ********\n\u2502                                        ************\n\u2514\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2192 \u8bad\u7ec3\u8f6e\u6b21<\/code><\/pre>\n<ul>\n<li><strong>\u521d\u671f<\/strong>\uff1a\u635f\u5931\u5feb\u901f\u4e0b\u964d\uff08\u53c2\u6570\u4ece\u968f\u673a\u503c\u5feb\u901f\u8c03\u6574\u5230\u5408\u7406\u8303\u56f4\uff09<\/li>\n<li><strong>\u4e2d\u671f<\/strong>\uff1a\u635f\u5931\u7f13\u6162\u4e0b\u964d\uff08\u7cbe\u7ec6\u8c03\u6574\uff09<\/li>\n<li><strong>\u540e\u671f<\/strong>\uff1a\u635f\u5931\u8d8b\u4e8e\u5e73\u7a33\uff08\u63a5\u8fd1\u6700\u4f18\u89e3\uff09<\/li>\n<\/ul>\n<h3>9.4 \u4ec0\u4e48\u65f6\u5019\u505c\u6b62\u8bad\u7ec3\uff1f<\/h3>\n<p>\u672c\u7a0b\u5e8f\u786c\u7f16\u7801\u4e86 1,000,000 \u8f6e\u8bad\u7ec3\u3002\u5b9e\u9645\u4e2d\u901a\u5e38\u4f7f\u7528\u4ee5\u4e0b\u7b56\u7565\uff1a<\/p>\n<ol>\n<li><strong>Early Stopping<\/strong>\uff1a\u5f53\u635f\u5931\u4e0d\u518d\u660e\u663e\u4e0b\u964d\u65f6\u505c\u6b62<\/li>\n<li><strong>\u8fbe\u5230\u76ee\u6807\u7cbe\u5ea6<\/strong>\uff1a\u5f53\u9a8c\u8bc1\u96c6\u7cbe\u5ea6\u8fbe\u5230\u9608\u503c\u65f6\u505c\u6b62<\/li>\n<li><strong>\u6700\u5927\u8f6e\u6b21\u9650\u5236<\/strong>\uff1a\u8bbe\u7f6e\u4e00\u4e2a\u4e0a\u9650\uff0c\u9632\u6b62\u65e0\u9650\u8bad\u7ec3<\/li>\n<\/ol>\n<hr>\n<h2>10. \u4ee3\u7801\u9010\u6bb5\u7cbe\u8bfb<\/h2>\n<h3>10.1 \u5934\u6587\u4ef6\u548c\u5b8f\u5b9a\u4e49<\/h3>\n<pre><code class=\"c\">#include &lt;stdio.h&gt;    \/\/ \u6807\u51c6 IO\uff1aprintf\n#include &lt;stdlib.h&gt;   \/\/ \u6807\u51c6\u5e93\uff1arand(), srand()\n#include &lt;math.h&gt;     \/\/ \u6570\u5b66\u5e93\uff1aexp()\n#include &lt;time.h&gt;     \/\/ \u65f6\u95f4\u5e93\uff08\u672c\u7a0b\u5e8f\u672a\u76f4\u63a5\u4f7f\u7528\uff09\n\n#define INPUT_SIZE   2   \/\/ \u8f93\u5165\u5c42\uff1a2 \u4e2a\u795e\u7ecf\u5143\uff08x\u2081, x\u2082\uff09\n#define HIDDEN_SIZE  4   \/\/ \u9690\u85cf\u5c42\uff1a4 \u4e2a\u795e\u7ecf\u5143\n#define OUTPUT_SIZE  1   \/\/ \u8f93\u51fa\u5c42\uff1a1 \u4e2a\u795e\u7ecf\u5143\uff08\u9884\u6d4b\u503c\uff09<\/code><\/pre>\n<p>\u4f7f\u7528 <code>#define<\/code> \u5b9a\u4e49\u5e38\u91cf\u662f\u4e00\u79cd\u826f\u597d\u7684\u7f16\u7a0b\u4e60\u60ef\u2014\u2014\u5982\u679c\u9700\u8981\u6539\u53d8\u7f51\u7edc\u7ed3\u6784\uff0c\u53ea\u9700\u8981\u4fee\u6539\u8fd9\u4e9b\u6570\u5b57\u5373\u53ef\u3002<\/p>\n<h3>10.2 \u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\u4f53<\/h3>\n<pre><code class=\"c\">typedef struct {\n    \/\/ \u6743\u91cd\u548c\u504f\u7f6e\uff08\u53ef\u5b66\u4e60\u53c2\u6570\uff09\n    double hidden_weights[HIDDEN_SIZE][INPUT_SIZE];   \/\/ [4][2]: \u9690\u85cf\u5c42\u6743\u91cd\n    double hidden_biases[HIDDEN_SIZE];               \/\/ [4]:    \u9690\u85cf\u5c42\u504f\u7f6e\n    double output_weights[OUTPUT_SIZE][HIDDEN_SIZE];  \/\/ [1][4]: \u8f93\u51fa\u5c42\u6743\u91cd\n    double output_biases[OUTPUT_SIZE];               \/\/ [1]:    \u8f93\u51fa\u5c42\u504f\u7f6e\n\n    \/\/ \u4e2d\u95f4\u7ed3\u679c\uff08\u5728\u6b63\u5411\u548c\u53cd\u5411\u4f20\u64ad\u4e2d\u7528\u5230\uff09\n    double hidden_z[HIDDEN_SIZE], hidden_a[HIDDEN_SIZE];       \/\/ \u9690\u85cf\u5c42\u7684\u51c0\u8f93\u5165\u548c\u6fc0\u6d3b\u503c\n    double output_z[OUTPUT_SIZE], output_a[OUTPUT_SIZE];       \/\/ \u8f93\u51fa\u5c42\u7684\u51c0\u8f93\u5165\u548c\u6fc0\u6d3b\u503c\n    double output_delta[OUTPUT_SIZE], hidden_delta[HIDDEN_SIZE]; \/\/ \u8bef\u5dee\u9879\n} NeuralNet;<\/code><\/pre>\n<p><strong>\u4e3a\u4ec0\u4e48\u5c06\u4e2d\u95f4\u7ed3\u679c\u4e5f\u5b58\u50a8\u5728\u7ed3\u6784\u4f53\u4e2d\uff1f<\/strong><\/p>\n<p>\u8fd9\u662f\u4e00\u79cd&#8221;\u7528\u7a7a\u95f4\u6362\u4fbf\u5229&#8221;\u7684\u505a\u6cd5\uff1a<\/p>\n<ul>\n<li><code>hidden_z<\/code>, <code>hidden_a<\/code>, <code>output_z<\/code>, <code>output_a<\/code> \u5728\u6b63\u5411\u4f20\u64ad\u4e2d\u8ba1\u7b97\uff0c\u5728\u53cd\u5411\u4f20\u64ad\u4e2d\u8981\u7528\u5230<\/li>\n<li><code>hidden_delta<\/code>, <code>output_delta<\/code> \u5728\u53cd\u5411\u4f20\u64ad\u4e2d\u8ba1\u7b97<\/li>\n<li>\u5c06\u5b83\u4eec\u5b58\u5728\u7ed3\u6784\u4f53\u4e2d\uff0c\u907f\u514d\u5728\u51fd\u6570\u95f4\u4f20\u9012\u5927\u91cf\u4e34\u65f6\u6570\u7ec4<\/li>\n<\/ul>\n<p><strong>\u66f4\u73b0\u4ee3\u7684\u505a\u6cd5<\/strong>\u662f\u5728\u6bcf\u6b21\u524d\u5411\u4f20\u64ad\u65f6\u91cd\u65b0\u8ba1\u7b97\u5e76\u7acb\u5373\u7528\u4e8e\u53cd\u5411\u4f20\u64ad\uff0c\u8fd9\u6837\u53ef\u4ee5\u8282\u7701\u5185\u5b58\u4f46\u589e\u52a0\u4ee3\u7801\u590d\u6742\u5ea6\u3002<\/p>\n<h3>10.3 \u521d\u59cb\u5316\u51fd\u6570<\/h3>\n<pre><code class=\"c\">void init_net(NeuralNet *network) {\n    srand(42);  \/\/ \u56fa\u5b9a\u968f\u673a\u79cd\u5b50\uff0c\u786e\u4fdd\u7ed3\u679c\u53ef\u590d\u73b0\n\n    \/\/ \u521d\u59cb\u5316\u9690\u85cf\u5c42\u53c2\u6570\n    for (int neuron_idx = 0; neuron_idx &lt; HIDDEN_SIZE; neuron_idx++) {\n        for (int input_idx = 0; input_idx &lt; INPUT_SIZE; input_idx++)\n            network-&gt;hidden_weights[neuron_idx][input_idx] = ((double)rand() \/ RAND_MAX) * 2.0 - 1.0;\n        network-&gt;hidden_biases[neuron_idx] = ((double)rand() \/ RAND_MAX) * 2.0 - 1.0;\n    }\n\n    \/\/ \u521d\u59cb\u5316\u8f93\u51fa\u5c42\u53c2\u6570\uff08\u540c\u7406\uff09\n    \/\/ ...\n}<\/code><\/pre>\n<p><strong>\u521d\u59cb\u5316\u7684\u91cd\u8981\u6027<\/strong>\uff1a<\/p>\n<ul>\n<li>\u5982\u679c\u6240\u6709\u6743\u91cd\u521d\u59cb\u5316\u4e3a 0\uff0c\u6240\u6709\u795e\u7ecf\u5143\u7684\u8f93\u51fa\u76f8\u540c\uff0c\u68af\u5ea6\u4e5f\u76f8\u540c\uff0c\u7f51\u7edc\u65e0\u6cd5\u5b66\u4e60<\/li>\n<li>\u5982\u679c\u6743\u91cd\u592a\u5927\uff0c\u7ecf\u8fc7 sigmoid \u540e\u4f1a\u8fdb\u5165\u9971\u548c\u533a\uff0c\u68af\u5ea6\u63a5\u8fd1 0\uff0c\u5b66\u4e60\u505c\u6ede<\/li>\n<li>\u5408\u7406\u7684\u968f\u673a\u521d\u59cb\u5316\u53ef\u4ee5\u6253\u7834\u5bf9\u79f0\u6027\uff0c\u8ba9\u4e0d\u540c\u795e\u7ecf\u5143\u5b66\u5230\u4e0d\u540c\u7279\u5f81<\/li>\n<\/ul>\n<p>\u672c\u7a0b\u5e8f\u4f7f\u7528 $[-1, 1]$ \u5747\u5300\u5206\u5e03\u521d\u59cb\u5316\uff0c\u8fd9\u662f\u4e00\u4e2a\u7b80\u5355\u6709\u6548\u7684\u65b9\u6cd5\u3002\u66f4\u5e38\u7528\u7684\u521d\u59cb\u5316\u65b9\u6cd5\u5305\u62ec\uff1a<\/p>\n<ul>\n<li><strong>Xavier\/Glorot \u521d\u59cb\u5316<\/strong>\uff1a\u9002\u5408 sigmoid\/tanh<\/li>\n<li><strong>He \u521d\u59cb\u5316<\/strong>\uff1a\u9002\u5408 ReLU<\/li>\n<\/ul>\n<h3>10.4 \u6307\u9488\u8bed\u6cd5\u89e3\u91ca<\/h3>\n<p>\u5bf9\u4e8e C \u8bed\u8a00\u65b0\u624b\uff0c\u8fd9\u91cc\u89e3\u91ca\u4e00\u4e0b\u4ee3\u7801\u4e2d\u7684\u6307\u9488\u7528\u6cd5\uff1a<\/p>\n<pre><code class=\"c\">void init_net(NeuralNet *network) {\n    \/\/ network \u662f\u4e00\u4e2a\u6307\u9488\uff0c\u6307\u5411 NeuralNet \u7ed3\u6784\u4f53\n    \/\/ network-&gt;hidden_weights \u7b49\u4ef7\u4e8e (*network).hidden_weights\uff0c\u8bbf\u95ee\u6307\u9488\u6307\u5411\u7684\u7ed3\u6784\u4f53\u7684\u6210\u5458\n    network-&gt;hidden_weights[neuron_idx][input_idx] = ...;\n}\n\nint main() {\n    NeuralNet network;      \/\/ \u521b\u5efa\u7ed3\u6784\u4f53\u53d8\u91cf\n    init_net(&amp;network);     \/\/ &amp; \u53d6\u5730\u5740\u8fd0\u7b97\u7b26\uff0c\u4f20\u9012\u53d8\u91cf\u7684\u5730\u5740\u7ed9\u51fd\u6570\n}<\/code><\/pre>\n<p><strong>\u4e3a\u4ec0\u4e48\u7528\u6307\u9488\uff1f<\/strong><\/p>\n<ul>\n<li>C \u8bed\u8a00\u51fd\u6570\u53c2\u6570\u662f<strong>\u4f20\u503c<\/strong>\u7684\uff0c\u5982\u679c\u4e0d\u4f20\u6307\u9488\uff0c\u51fd\u6570\u4f1a\u590d\u5236\u6574\u4e2a\u7ed3\u6784\u4f53<\/li>\n<li><code>NeuralNet<\/code> \u5305\u542b\u591a\u4e2a\u6570\u7ec4\uff0c\u590d\u5236\u5f00\u9500\u5f88\u5927<\/li>\n<li>\u7528\u6307\u9488\u53ef\u4ee5\u907f\u514d\u590d\u5236\uff0c\u76f4\u63a5\u4fee\u6539\u539f\u7ed3\u6784\u4f53<\/li>\n<\/ul>\n<hr>\n<h2>11. \u8fd0\u884c\u4e0e\u89e3\u8bfb\u7ed3\u679c<\/h2>\n<h3>11.1 \u7f16\u8bd1\u548c\u8fd0\u884c<\/h3>\n<pre><code class=\"bash\">gcc -Wall -o xor_nn.exe xor_nn.c -lm\n.\/xor_nn.exe<\/code><\/pre>\n<p>\u53c2\u6570\u8bf4\u660e\uff1a<\/p>\n<ul>\n<li><code>-Wall<\/code>\uff1a\u5f00\u542f\u6240\u6709\u8b66\u544a<\/li>\n<li><code>-o xor_nn.exe<\/code>\uff1a\u6307\u5b9a\u8f93\u51fa\u6587\u4ef6\u540d<\/li>\n<li><code>-lm<\/code>\uff1a\u94fe\u63a5\u6570\u5b66\u5e93\uff08\u56e0\u4e3a\u4f7f\u7528\u4e86 <code>exp()<\/code>\uff09<\/li>\n<\/ul>\n<h3>11.2 \u8f93\u51fa\u89e3\u8bfb<\/h3>\n<pre><code>Epoch 0,     Loss = 0.134335     \u2190 \u521d\u59cb\u635f\u5931\uff08\u968f\u673a\u53c2\u6570\uff09\nEpoch 1000,  Loss = 0.003974     \u2190 \u635f\u5931\u5feb\u901f\u4e0b\u964d\nEpoch 2000,  Loss = 0.000836     \u2190 \u7ee7\u7eed\u4e0b\u964d\nEpoch 3000,  Loss = 0.000436\n...\nEpoch 10000, Loss = 0.000091     \u2190 \u635f\u5931\u964d\u5230\u5f88\u4f4e\n...\nEpoch 999000, Loss = 0.000001    \u2190 \u51e0\u4e4e\u4e3a 0\n\nTesting XOR:\n0 XOR 0 = 0.001062  (expected 0)    \u2190 \u63a5\u8fd1 0 \u2713\n0 XOR 1 = 0.998870  (expected 1)    \u2190 \u63a5\u8fd1 1 \u2713\n1 XOR 0 = 0.998873  (expected 1)    \u2190 \u63a5\u8fd1 1 \u2713\n1 XOR 1 = 0.001154  (expected 0)    \u2190 \u63a5\u8fd1 0 \u2713<\/code><\/pre>\n<p><strong>\u7ed3\u679c\u5206\u6790<\/strong>\uff1a<\/p>\n<ul>\n<li>\u9884\u6d4b\u503c\u4e0d\u662f\u7cbe\u786e\u7684 0 \u6216 1\uff0c\u800c\u662f\u63a5\u8fd1 0 \u6216 1 \u7684\u5c0f\u6570<\/li>\n<li>\u5982\u679c\u8bbe\u7f6e\u9608\u503c\u4e3a 0.5\uff08\u5927\u4e8e 0.5 \u5224\u4e3a 1\uff0c\u5426\u5219\u5224\u4e3a 0\uff09\uff0c\u5219\u5168\u90e8\u6b63\u786e<\/li>\n<li>\u8bad\u7ec3\u66f4\u591a\u8f6e\u6b21\u6216\u8c03\u6574\u5b66\u4e60\u7387\u53ef\u4ee5\u8fdb\u4e00\u6b65\u63d0\u9ad8\u7cbe\u5ea6<\/li>\n<\/ul>\n<h3>11.3 \u5982\u4f55\u5224\u65ad\u7f51\u7edc\u662f\u5426\u5b66\u5230\u4e86\uff1f<\/h3>\n<p>\u5b66\u4e60\u6210\u529f\u7684\u6807\u5fd7\uff1a<\/p>\n<ul>\n<li>\u635f\u5931\u6301\u7eed\u4e0b\u964d\u5230\u5f88\u4f4e\uff08\u5982 $10^{-6}$ \u91cf\u7ea7\uff09<\/li>\n<li>\u6d4b\u8bd5\u7ed3\u679c\u7b26\u5408\u9884\u671f<\/li>\n<\/ul>\n<p>\u5b66\u4e60\u5931\u8d25\u7684\u53ef\u80fd\u539f\u56e0\uff1a<\/p>\n<ul>\n<li>\u635f\u5931\u4e0d\u4e0b\u964d\u6216\u4e0a\u5347 \u2192 \u5b66\u4e60\u7387\u592a\u5927\uff0c\u6216\u4ee3\u7801\u6709 bug<\/li>\n<li>\u635f\u5931\u4e0b\u964d\u4f46\u6d4b\u8bd5\u7ed3\u679c\u5dee \u2192 \u8fc7\u62df\u5408\uff08\u5728 XOR \u5c0f\u6570\u636e\u96c6\u4e0a\u4e0d\u5e38\u89c1\uff09<\/li>\n<li>\u635f\u5931\u5361\u5728\u8f83\u9ad8\u503c \u2192 \u9677\u5165\u5c40\u90e8\u6700\u4f18\uff0c\u6216\u7f51\u7edc\u5bb9\u91cf\u4e0d\u8db3<\/li>\n<\/ul>\n<hr>\n<h2>12. \u5ef6\u4f38\u601d\u8003\u4e0e\u6539\u8fdb\u65b9\u5411<\/h2>\n<h3>12.1 \u4ee3\u7801\u53ef\u4ee5\u6539\u8fdb\u7684\u5730\u65b9<\/h3>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u6539\u8fdb\u65b9\u5411<\/th>\n<th align=\"left\">\u8bf4\u660e<\/th>\n<th align=\"left\">\u4ee3\u7801\u4fee\u6539<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">\u52a8\u6001\u5b66\u4e60\u7387<\/td>\n<td align=\"left\">\u8bad\u7ec3\u521d\u671f\u7528\u5927\u5b66\u4e60\u7387\u5feb\u901f\u4e0b\u964d\uff0c\u540e\u671f\u7528\u5c0f\u5b66\u4e60\u7387\u7cbe\u7ec6\u8c03\u6574<\/td>\n<td align=\"left\">\u6dfb\u52a0\u5b66\u4e60\u7387\u8870\u51cf\u903b\u8f91<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u52a8\u91cf (Momentum)<\/td>\n<td align=\"left\">\u52a0\u901f\u6536\u655b\uff0c\u51cf\u5c11\u9707\u8361<\/td>\n<td align=\"left\">\u5f15\u5165\u52a8\u91cf\u9879 $v = \/gamma v + lr \/cdot g$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">Mini-batch<\/td>\n<td align=\"left\">\u6bcf\u6b21\u7528\u591a\u4e2a\u6837\u672c\u8ba1\u7b97\u68af\u5ea6\uff0c\u66f4\u7a33\u5b9a<\/td>\n<td align=\"left\">\u6bcf\u6b21\u53d6\u4e00\u6279\u6837\u672c\uff0c\u68af\u5ea6\u53d6\u5e73\u5747<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u9a8c\u8bc1\u96c6<\/td>\n<td align=\"left\">\u7528\u4e00\u90e8\u5206\u6570\u636e\u9a8c\u8bc1\u8bad\u7ec3\u6548\u679c\uff0c\u9632\u6b62\u8fc7\u62df\u5408<\/td>\n<td align=\"left\">\u5212\u5206\u8bad\u7ec3\/\u9a8c\u8bc1\u96c6<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u4fdd\u5b58\/\u52a0\u8f7d\u6a21\u578b<\/td>\n<td align=\"left\">\u8bad\u7ec3\u597d\u7684\u53c2\u6570\u53ef\u4ee5\u4fdd\u5b58\u5230\u6587\u4ef6<\/td>\n<td align=\"left\">\u5c06\u6743\u91cd\u5199\u5165\u6587\u4ef6<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h3>12.2 \u6570\u5b66\u5c42\u9762\u7684\u6539\u8fdb<\/h3>\n<ul>\n<li><strong>ReLU \u6fc0\u6d3b\u51fd\u6570<\/strong>\uff1a\u7528 <code>max(0, x)<\/code> \u66ff\u4ee3 sigmoid\uff0c\u8ba1\u7b97\u66f4\u5feb\uff0c\u7f13\u89e3\u68af\u5ea6\u6d88\u5931<\/li>\n<li><strong>\u4ea4\u53c9\u71b5\u635f\u5931<\/strong>\uff1a\u5bf9\u4e8e\u5206\u7c7b\u95ee\u9898\uff0c\u4ea4\u53c9\u71b5\u901a\u5e38\u6bd4 MSE \u6548\u679c\u66f4\u597d<\/li>\n<li><strong>Softmax \u8f93\u51fa<\/strong>\uff1a\u591a\u5206\u7c7b\u65f6\u4f7f\u7528\uff0c\u5c06\u8f93\u51fa\u8f6c\u6362\u4e3a\u6982\u7387\u5206\u5e03<\/li>\n<li><strong>Batch Normalization<\/strong>\uff1a\u6bcf\u5c42\u8f93\u51fa\u505a\u6807\u51c6\u5316\uff0c\u52a0\u901f\u8bad\u7ec3<\/li>\n<li><strong>Dropout<\/strong>\uff1a\u8bad\u7ec3\u65f6\u968f\u673a\u4e22\u5f03\u90e8\u5206\u795e\u7ecf\u5143\uff0c\u9632\u6b62\u8fc7\u62df\u5408<\/li>\n<\/ul>\n<h3>12.3 \u66f4\u4e00\u822c\u7684\u795e\u7ecf\u7f51\u7edc\u6a21\u5f0f<\/h3>\n<p>\u672c\u7a0b\u5e8f\u7684\u7f51\u7edc\u7ed3\u6784\u53ef\u4ee5\u63a8\u5e7f\u5230\u4efb\u610f\u5c42\u6570\u3001\u4efb\u610f\u5bbd\u5ea6\uff1a<\/p>\n<pre><code class=\"c\">\/\/ \u901a\u7528\u5c42\u7ed3\u6784\uff08\u793a\u610f\uff09\ntypedef struct {\n    int input_size;\n    int output_size;\n    double *weights;    \/\/ \u4e00\u7ef4\u5b58\u50a8\u6743\u91cd\u77e9\u9635\n    double *bias;\n    double *input;      \/\/ \u672c\u5c42\u7684\u8f93\u5165\uff08\u4e0a\u4e00\u5c42\u7684\u6fc0\u6d3b\u503c\uff09\n    double *z;          \/\/ \u51c0\u8f93\u5165\n    double *output;     \/\/ \u6fc0\u6d3b\u503c\n    double *delta;      \/\/ \u8bef\u5dee\u9879\n} Layer;\n\ntypedef struct {\n    Layer *layers;\n    int num_layers;\n    double (*activation)(double);\n    double (*activation_deriv)(double);\n} DeepNet;<\/code><\/pre>\n<h3>12.4 \u4ece C \u5230\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6<\/h3>\n<p>\u7406\u89e3\u4e86\u8fd9\u4e2a C \u8bed\u8a00\u5b9e\u73b0\uff0c\u4f60\u5c31\u7406\u89e3\u4e86\u6240\u6709\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6\uff08TensorFlow\u3001PyTorch\u3001JAX\uff09\u7684\u6838\u5fc3\uff1a<\/p>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u6982\u5ff5<\/th>\n<th align=\"left\">C \u4ee3\u7801<\/th>\n<th align=\"left\">PyTorch \u7b49\u4ef7<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">\u5b9a\u4e49\u7f51\u7edc<\/td>\n<td align=\"left\"><code>NeuralNet<\/code> \u7ed3\u6784\u4f53<\/td>\n<td align=\"left\"><code>nn.Module<\/code> \u5b50\u7c7b<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u7ebf\u6027\u53d8\u6362<\/td>\n<td align=\"left\"><code>hidden_weights*x + hidden_biases<\/code><\/td>\n<td align=\"left\"><code>nn.Linear(2, 4)<\/code><\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u6fc0\u6d3b\u51fd\u6570<\/td>\n<td align=\"left\"><code>sigmoid()<\/code><\/td>\n<td align=\"left\"><code>nn.Sigmoid()<\/code><\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u635f\u5931\u51fd\u6570<\/td>\n<td align=\"left\"><code>0.5*(\u0177-y)\u00b2<\/code><\/td>\n<td align=\"left\"><code>nn.MSELoss()<\/code><\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u53cd\u5411\u4f20\u64ad<\/td>\n<td align=\"left\">\u624b\u52a8\u5b9e\u73b0<\/td>\n<td align=\"left\"><code>loss.backward()<\/code>\uff08\u81ea\u52a8\u5fae\u5206\uff09<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u53c2\u6570\u66f4\u65b0<\/td>\n<td align=\"left\">\u624b\u52a8\u5b9e\u73b0<\/td>\n<td align=\"left\"><code>optimizer.step()<\/code><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>\u73b0\u4ee3\u6846\u67b6\u81ea\u52a8\u5904\u7406\u4e86<\/strong>\uff1a<\/p>\n<ul>\n<li>\u68af\u5ea6\u8ba1\u7b97\uff08\u81ea\u52a8\u5fae\u5206\uff09<\/li>\n<li>\u53c2\u6570\u66f4\u65b0\uff08\u5404\u79cd\u4f18\u5316\u5668\uff09<\/li>\n<li>\u6279\u91cf\u5904\u7406\uff08\u6570\u636e\u52a0\u8f7d\u5668\uff09<\/li>\n<li>GPU \u52a0\u901f\uff08CUDA\uff09<\/li>\n<\/ul>\n<p>\u4f46<strong>\u57fa\u672c\u539f\u7406\u5b8c\u5168\u4e00\u81f4<\/strong>\uff0c\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u7406\u89e3\u8fd9\u4e2a C \u8bed\u8a00\u5b9e\u73b0\u5982\u6b64\u91cd\u8981\u3002<\/p>\n<h3>12.5 \u4ece <code>NeuralNet<\/code> \u5230\u901a\u7528\u6846\u67b6\uff1a\u7ed3\u6784\u4f53\u6f14\u8fdb\u8def\u7ebf<\/h3>\n<p>\u5982\u679c\u4f60\u60f3\u628a\u672c\u4ee3\u7801\u6539\u9020\u6210\u80fd\u5904\u7406\u4efb\u610f\u5c42\u6570\u3001\u4efb\u610f\u5927\u5c0f\u7684\u9879\u76ee\uff0c\u53ef\u4ee5\u6cbf\u7740\u4e0b\u9762\u7684\u8def\u7ebf\u9010\u6b65\u91cd\u6784\uff1a<\/p>\n<h4>\u9636\u6bb5 1\uff1a\u628a\u786c\u7f16\u7801\u7ed3\u6784\u4f53\u6539\u6210\u6309\u5c42\u7ec4\u7ec7<\/h4>\n<p>\u4e0d\u8981\u4e00\u4e0b\u5b50\u8df3\u5230\u52a8\u6001\u5185\u5b58\u3002\u53ef\u4ee5\u5148\u4fdd\u7559\u9759\u6001\u6570\u7ec4\uff0c\u4f46\u7528\u5b8f\u5b9a\u4e49\u5c42\u6570\u548c\u6bcf\u5c42\u5927\u5c0f\uff1a<\/p>\n<pre><code class=\"c\">#define NUM_LAYERS 3          \/\/ \u8f93\u5165\u5c42\u4e0d\u8ba1\u5165\uff0c\u8fd9\u91cc\u662f\u9690\u85cf+\u8f93\u51fa\n#define LAYER_SIZES {4, 1}    \/\/ \u9690\u85cf\u5c42 4 \u4e2a\uff0c\u8f93\u51fa\u5c42 1 \u4e2a<\/code><\/pre>\n<p>\u7136\u540e\u628a <code>hidden_weights\/hidden_biases\/hidden_z\/hidden_a\/hidden_delta<\/code> \u548c <code>output_weights\/output_biases\/output_z\/output_a\/output_delta<\/code> \u5408\u5e76\u6210\u6570\u7ec4\uff1a<\/p>\n<pre><code class=\"c\">typedef struct {\n    double W[NUM_LAYERS-1][MAX_LAYER_SIZE][MAX_LAYER_SIZE];  \/\/ \u5c42\u95f4\u6743\u91cd\n    double b[NUM_LAYERS-1][MAX_LAYER_SIZE];                  \/\/ \u5c42\u95f4\u504f\u7f6e\n    double z[NUM_LAYERS-1][MAX_LAYER_SIZE];                  \/\/ \u51c0\u8f93\u5165\n    double a[NUM_LAYERS-1][MAX_LAYER_SIZE];                  \/\/ \u6fc0\u6d3b\u503c\n    double delta[NUM_LAYERS-1][MAX_LAYER_SIZE];              \/\/ \u8bef\u5dee\u9879\n} NeuralNetV2;<\/code><\/pre>\n<p>\u8fd9\u79cd\u5199\u6cd5\u4ecd\u7136\u4f7f\u7528\u9759\u6001\u6570\u7ec4\uff0c\u4f46\u5faa\u73af\u5df2\u7ecf\u6309&#8221;\u5c42\u7d22\u5f15&#8221;\u7edf\u4e00\uff0c\u4ee3\u7801\u590d\u7528\u6027\u66f4\u597d\u3002<\/p>\n<h4>\u9636\u6bb5 2\uff1a\u5f15\u5165\u52a8\u6001\u5c42\u7ed3\u6784<\/h4>\n<p>\u5f53\u5c42\u6570\u548c\u6bcf\u5c42\u5927\u5c0f\u9700\u8981\u8fd0\u884c\u65f6\u914d\u7f6e\u65f6\uff0c\u6539\u7528 <code>Layer<\/code> \u7ed3\u6784\u4f53\u548c <code>malloc<\/code>\uff1a<\/p>\n<pre><code class=\"c\">typedef struct {\n    int input_size, output_size;\n    double *weights;   \/\/ \u5c55\u5e73\u5b58\u50a8\uff1aweights[output_size * input_size]\n    double *bias;\n    double *z, *a, *delta;\n    double (*activation)(double);\n    double (*activation_deriv)(double activation_value);\n} Layer;\n\ntypedef struct {\n    Layer *layers;\n    int num_layers;\n} NeuralNetV3;<\/code><\/pre>\n<p>\u6ce8\u610f\uff1a<\/p>\n<ul>\n<li>\u6743\u91cd\u4ece\u4e8c\u7ef4\u6570\u7ec4\u53d8\u6210\u4e00\u7ef4\u5c55\u5e73\uff0c\u8bbf\u95ee\u65f6\u7528 <code>weights[i * input_size + j]<\/code> \u4ee3\u66ff <code>W[i][j]<\/code>\u3002<\/li>\n<li>\u4e00\u7ef4\u5b58\u50a8\u7684\u597d\u5904\u662f\u4fbf\u4e8e\u5e8f\u5217\u5316\u5230\u6587\u4ef6\u3001\u4e5f\u4fbf\u4e8e\u540e\u7eed\u7528 BLAS \/ GPU \u52a0\u901f\u3002<\/li>\n<li>\u9700\u8981\u5728 <code>init_net<\/code> \u91cc <code>malloc<\/code>\uff0c\u5728\u7a0b\u5e8f\u7ed3\u675f\u65f6 <code>free<\/code>\uff0c\u907f\u514d\u5185\u5b58\u6cc4\u6f0f\u3002<\/li>\n<\/ul>\n<h4>\u9636\u6bb5 3\uff1a\u5206\u79bb&#8221;\u6a21\u578b\u53c2\u6570&#8221;\u4e0e&#8221;\u8ba1\u7b97\u56fe\u72b6\u6001&#8221;<\/h4>\n<p>\u518d\u8fdb\u4e00\u6b65\uff0c\u53ef\u4ee5\u5b66\u4e60\u73b0\u4ee3\u6846\u67b6\u7684<code>Parameter \/ Tensor<\/code>\u601d\u60f3\uff1a<\/p>\n<pre><code class=\"c\">typedef struct {\n    double *data;   \/\/ \u53c2\u6570\u503c\uff08W \u6216 b\uff09\n    double *grad;   \/\/ \u68af\u5ea6\n    int size;\n} Parameter;\n\ntypedef struct {\n    Parameter *weights;  \/\/ \u6bcf\u5c42\u4e00\u4e2a\u6743\u91cd\n    Parameter *biases;   \/\/ \u6bcf\u5c42\u4e00\u4e2a\u504f\u7f6e\n} Model;\n\ntypedef struct {\n    double **inputs;  \/\/ \u6bcf\u5c42\u7684\u8f93\u5165\n    double **z;       \/\/ \u6bcf\u5c42\u7684\u51c0\u8f93\u5165\n    double **a;       \/\/ \u6bcf\u5c42\u7684\u6fc0\u6d3b\n    double **delta;   \/\/ \u6bcf\u5c42\u7684\u8bef\u5dee\n} Cache;<\/code><\/pre>\n<p>\u6b64\u65f6 <code>forward(Model *model, Cache *cache, double *input)<\/code> \u53ea\u8bfb\u6a21\u578b\u3001\u5199\u7f13\u5b58\uff1b<code>backward<\/code> \u4e5f\u53ea\u8bfb\u6a21\u578b\u3001\u5199\u7f13\u5b58\uff0c\u5e76\u7d2f\u52a0\u68af\u5ea6\u5230 <code>Parameter.grad<\/code>\u3002\u8fd9\u6837\u6a21\u578b\u548c\u7f13\u5b58\u53ef\u4ee5\u72ec\u7acb\u4fdd\u5b58\u3001\u52a0\u8f7d\u548c\u8c03\u8bd5\u3002<\/p>\n<h4>\u9636\u6bb5 4\uff1a\u589e\u52a0\u6a21\u5757\u5316\u548c\u529f\u80fd\u5f00\u5173<\/h4>\n<p>\u6700\u540e\u53ef\u4ee5\u52a0\u5165\uff1a<\/p>\n<ul>\n<li>\u53ef\u63d2\u62d4\u6fc0\u6d3b\u51fd\u6570\uff08sigmoid \/ ReLU \/ tanh \/ softmax\uff09<\/li>\n<li>\u53ef\u914d\u7f6e\u635f\u5931\u51fd\u6570\uff08MSE \/ \u4ea4\u53c9\u71b5\uff09<\/li>\n<li>\u4f18\u5316\u5668\u63a5\u53e3\uff08SGD \/ Momentum \/ Adam\uff09<\/li>\n<li>\u6570\u636e\u52a0\u8f7d\u5668\uff08batch\u3001shuffle\uff09<\/li>\n<li>\u6a21\u578b\u4fdd\u5b58 \/ \u52a0\u8f7d\uff08\u4e8c\u8fdb\u5236\u6216 JSON\uff09<\/li>\n<li>\u63a8\u7406\u6a21\u5f0f\uff08\u5173\u95ed dropout\u3001\u4e0d\u8ba1\u7b97\u68af\u5ea6\uff09<\/li>\n<\/ul>\n<p>\u8fd9\u4e9b\u6269\u5c55\u7684\u5173\u952e\u601d\u60f3\u90fd\u662f\uff1a<strong>\u5148\u533a\u5206&#8221;\u53ef\u5b66\u4e60\u53c2\u6570&#8221;\u3001&#8221;\u8ba1\u7b97\u72b6\u6001&#8221;\u3001&#8221;\u7b97\u6cd5\u914d\u7f6e&#8221;\u4e09\u7c7b\u6570\u636e\uff0c\u518d\u628a\u786c\u7f16\u7801\u7684\u5177\u4f53\u503c\u6539\u6210\u53ef\u914d\u7f6e\u7684\u6570\u636e\u7ed3\u6784<\/strong>\u3002\u672c\u6559\u7a0b\u4e2d\u7684 <code>NeuralNet<\/code> \u5176\u5b9e\u5df2\u7ecf\u6697\u542b\u4e86\u8fd9\u4e09\u7c7b\uff1a<\/p>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u7c7b\u578b<\/th>\n<th align=\"left\">\u672c\u4ee3\u7801\u5b57\u6bb5<\/th>\n<th align=\"left\">\u672a\u6765\u6846\u67b6\u5bf9\u5e94\u6982\u5ff5<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">\u53ef\u5b66\u4e60\u53c2\u6570<\/td>\n<td align=\"left\"><code>hidden_weights<\/code>, <code>hidden_biases<\/code>, <code>output_weights<\/code>, <code>output_biases<\/code><\/td>\n<td align=\"left\"><code>Parameter<\/code> \/ <code>nn.Parameter<\/code><\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u8ba1\u7b97\u72b6\u6001<\/td>\n<td align=\"left\"><code>hidden_z<\/code>, <code>hidden_a<\/code>, <code>output_z<\/code>, <code>output_a<\/code>, <code>hidden_delta<\/code>, <code>output_delta<\/code><\/td>\n<td align=\"left\"><code>Tensor<\/code> \/ \u8ba1\u7b97\u56fe\u7f13\u5b58<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u7b97\u6cd5\u914d\u7f6e<\/td>\n<td align=\"left\">\u5b8f <code>INPUT_SIZE<\/code>, <code>HIDDEN_SIZE<\/code>, <code>OUTPUT_SIZE<\/code>\uff0c\u5b66\u4e60\u7387\u7b49<\/td>\n<td align=\"left\">\u914d\u7f6e\u6587\u4ef6 \/ \u8d85\u53c2\u6570\u5bf9\u8c61<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\u7406\u89e3\u4e86\u8fd9\u79cd\u5206\u5c42\uff0c\u518d\u8bfb PyTorch \u7684 <code>nn.Module<\/code>\u3001TensorFlow \u7684 <code>Layer<\/code>\u3001JAX \u7684 <code>params<\/code> \u5b57\u5178\u5c31\u4f1a\u975e\u5e38\u81ea\u7136\u3002<\/p>\n<hr>\n<h2>\u9644\u5f55 A\uff1a\u6570\u5b66\u516c\u5f0f\u901f\u67e5\u8868<\/h2>\n<table>\n<thead>\n<tr>\n<th align=\"left\">\u6982\u5ff5<\/th>\n<th align=\"left\">\u516c\u5f0f<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td align=\"left\">\u795e\u7ecf\u5143\u51c0\u8f93\u5165<\/td>\n<td align=\"left\">$z = \/sum_i w_i x_i + b$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">Sigmoid \u6fc0\u6d3b<\/td>\n<td align=\"left\">$a = \/sigma(z) = \/frac{1}{1+e^{-z}}$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">Sigmoid \u5bfc\u6570<\/td>\n<td align=\"left\">$\/sigma'(z) = \/sigma(z)(1-\/sigma(z)) = a(1-a)$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u635f\u5931\u51fd\u6570<\/td>\n<td align=\"left\">$L = \/frac{1}{2}(\/hat{y} &#8211; y)^2$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u8f93\u51fa\u5c42\u8bef\u5dee\u9879<\/td>\n<td align=\"left\">$\/delta_2 = (a_2 &#8211; y) \/cdot a_2 \/cdot (1 &#8211; a_2)$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u9690\u85cf\u5c42\u8bef\u5dee\u9879<\/td>\n<td align=\"left\">$\/delta_{1,i} = \/delta_2 \/cdot W2_{1,i} \/cdot a_{1,i} \/cdot (1 &#8211; a_{1,i})$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u68af\u5ea6\u4e0b\u964d\u66f4\u65b0<\/td>\n<td align=\"left\">$W \/leftarrow W &#8211; lr \/cdot \/delta \/cdot \/text{input}$<\/td>\n<\/tr>\n<tr>\n<td align=\"left\">\u504f\u7f6e\u66f4\u65b0<\/td>\n<td align=\"left\">$b \/leftarrow b &#8211; lr \/cdot \/delta$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2>\u9644\u5f55 B\uff1a\u63a8\u8350\u9605\u8bfb<\/h2>\n<ol>\n<li><strong>\u300aNeural Networks and Deep Learning\u300b<\/strong> \u2014 Michael Nielsen\uff08\u514d\u8d39\u5728\u7ebf\u4e66\uff0c\u9002\u5408\u5165\u95e8\uff09<\/li>\n<li><strong>\u300aDeep Learning\u300b<\/strong> \u2014 Ian Goodfellow, Yoshua Bengio, Aaron Courville\uff08&#8221;\u82b1\u4e66&#8221;\uff0c\u6df1\u5ea6\u5b66\u4e60\u5723\u7ecf\uff09<\/li>\n<li><strong>3Blue1Brown \u7684\u795e\u7ecf\u7f51\u7edc\u7cfb\u5217\u89c6\u9891<\/strong> \u2014 \u89c6\u89c9\u5316\u7406\u89e3\u53cd\u5411\u4f20\u64ad\uff08YouTube\uff09<\/li>\n<li><strong>CS231n: Convolutional Neural Networks for Visual Recognition<\/strong> \u2014 Stanford \u8bfe\u7a0b<\/li>\n<li><strong>PyTorch \u5b98\u65b9\u6559\u7a0b<\/strong> \u2014 \u4ece\u5b9e\u8df5\u5165\u624b\u5b66\u4e60\u6df1\u5ea6\u5b66\u4e60\u6846\u67b6<\/li>\n<\/ol>\n<hr>\n<blockquote><p>\u4f60\u8fd8\u53ef\u4ee5\u5c1d\u8bd5\u4fee\u6539\u9690\u85cf\u5c42\u5927\u5c0f\u3001\u5b66\u4e60\u7387\u3001\u8bad\u7ec3\u8f6e\u6b21\u7b49\u53c2\u6570\uff0c\u89c2\u5bdf\u5b83\u4eec\u5bf9\u8bad\u7ec3\u7ed3\u679c\u7684\u5f71\u54cd\u3002<\/p><\/blockquote>\n<p><a href=\"https:\/\/segmentfault.com\/a\/1190000048084308\">https:\/\/segmentfault.com\/a\/1190000048084308<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u672c\u6587\u6863\u57fa\u4e8eXOR\u95ee\u9898\uff0c\u5faa\u5e8f\u6e10\u8fdb\u5730\u8bb2\u89e3\u795e\u7ecf\u7f51\u7edc\u7684\u57fa\u672c\u539f\u7406\u3001\u6570\u5b66\u63a8\u5bfc\u548c\u4ee3\u7801\u5b9e\u73b0\u3002\u9002\u5408\u6709\u4e00\u5b9a\u7f16\u7a0b\u57fa\u7840\uff0c\u4f46\u521d\u6b21\u63a5\u89e6\u795e\u7ecf [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[4],"tags":[],"class_list":["post-10021421","post","type-post","status-publish","format-standard","hentry","category-share"],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/itmax.cn\/index.php?rest_route=\/wp\/v2\/posts\/10021421","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/itmax.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/itmax.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/itmax.cn\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/itmax.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=10021421"}],"version-history":[{"count":0,"href":"https:\/\/itmax.cn\/index.php?rest_route=\/wp\/v2\/posts\/10021421\/revisions"}],"wp:attachment":[{"href":"https:\/\/itmax.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=10021421"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/itmax.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=10021421"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/itmax.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=10021421"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}