{"id":99280,"date":"2026-09-02T11:33:27","date_gmt":"2026-09-02T03:33:27","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/99280.html"},"modified":"2026-09-02T11:33:27","modified_gmt":"2026-09-02T03:33:27","slug":"ai-%e7%9a%84%e5%bc%95%e6%93%8e%ef%bc%9a%e8%87%aa%e5%8a%a8%e5%be%ae%e5%88%86%e5%88%b0%e5%ba%95%e6%98%af%e6%80%8e%e4%b9%88%e5%b7%a5%e4%bd%9c%e7%9a%84%ef%bc%9f","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/99280.html","title":{"rendered":"AI \u7684\u5f15\u64ce\uff1a\u81ea\u52a8\u5fae\u5206\u5230\u5e95\u662f\u600e\u4e48\u5de5\u4f5c\u7684\uff1f"},"content":{"rendered":"

AI \u7684\u5f15\u64ce：\u81ea\u52a8\u5fae\u5206\u5230\u5e95\u662f\u600e\u4e48\u5de5\u4f5c\u7684？<\/h2>\n

\u6df1\u5ea6\u5b66\u4e60\u80fd\u8bad\u7ec3，\u9760\u7684\u662f\u53cd\u5411\u4f20\u64ad；\u53cd\u5411\u4f20\u64ad\u7684\u6570\u5b66\u672c\u8d28，\u662f\u81ea\u52a8\u5fae\u5206（AD）\u3002\u4f46"\u81ea\u52a8\u5fae\u5206"\u65e2\u4e0d\u662f\u6570\u503c\u8fd1\u4f3c，\u4e5f\u4e0d\u662f\u7b26\u53f7\u63a8\u5bfc\u2014\u2014\u5b83\u662f\u7b2c\u4e09\u79cd\u3001\u4e5f\u662f\u88ab\u4f4e\u4f30\u7684\u4e00\u79cd\u6c42\u5bfc\u65b9\u5f0f\u3002\u4eca\u5929\u4e00\u6b21\u8bb2\u6e05：\u4e09\u79cd\u6c42\u5bfc\u65b9\u5f0f\u5dee\u5728\u54ea，AD \u51ed\u4ec0\u4e48\u6210\u4e3a AI \u7684\u5f15\u64ce\u3002<\/p>\n

\u4e00\u3001\u6c42\u5bfc\u7684\u4e09\u79cd\u65b9\u5f0f<\/h3>\n

\u540c\u4e00\u4e2a\u51fd\u6570 f(x\u2081,x\u2082) = sin(x\u2081) + x\u2081\u00b7x\u2082 + e^x\u2082，\u4e09\u79cd\u65b9\u5f0f\u6c42\u68af\u5ea6：<\/p>\n

1. \u6570\u503c\u5fae\u5206（\u5dee\u5206\u8fd1\u4f3c）<\/h4>\n

f'(x) \u2248 (f(x+h) – f(x-h)) \/ 2h<\/p>\n

\u7528\u4e24\u4e2a\u5f88\u8fd1\u7684\u70b9\u7b97\u659c\u7387\u3002\u7b80\u5355\u7c97\u66b4，\u4f46\u6709\u4e24\u4e2a\u8bef\u5dee\u6e90\u6253\u67b6：\u622a\u65ad\u8bef\u5dee（h \u5927\u5219\u5927）\u548c\u820d\u5165\u8bef\u5dee（h \u592a\u5c0f，\u6d6e\u70b9\u7cbe\u5ea6\u7206\u70b8）\u3002\u7cbe\u5ea6\u5929\u82b1\u677f\u7ea6 1e-8，\u4e14\u6bcf\u4e2a\u53c2\u6570\u90fd\u8981\u91cd\u65b0\u7b97\u4e00\u904d\u51fd\u6570\u2014\u2014\u6df1\u5ea6\u5b66\u4e60\u51e0\u4ebf\u53c2\u6570，\u76f4\u63a5\u4e0d\u53ef\u7528\u3002<\/p>\n

2. \u7b26\u53f7\u5fae\u5206（\u89e3\u6790\u63a8\u5bfc）<\/h4>\n

\u7528\u94fe\u5f0f\u6cd5\u5219\u3001\u4e58\u6cd5\u6cd5\u5219\u628a\u5bfc\u6570\u8868\u8fbe\u5f0f"\u63a8"\u51fa\u6765\u2014\u2014\u5c31\u662f\u4f60\u5728\u9ad8\u6570\u8bfe\u4e0a\u505a\u7684\u4e8b\u3002\u7cbe\u5ea6\u5b8c\u7f8e，\u4f46\u8868\u8fbe\u5f0f\u81a8\u80c0：\u5d4c\u5957\u6df1\u5ea6 k \u7684\u7a0b\u5e8f，\u5bfc\u6570\u8868\u8fbe\u5f0f\u53ef\u80fd\u81a8\u80c0\u5230 k! \u91cf\u7ea7\u3002\u7b97 f(x\u2081,x\u2082) \u8fd8\u597d，\u7b97\u4e00\u4e2a 100 \u5c42\u7684\u795e\u7ecf\u7f51\u7edc，\u7b26\u53f7\u8868\u8fbe\u5f0f\u4f1a\u81a8\u80c0\u5230\u5185\u5b58\u90fd\u88c5\u4e0d\u4e0b\u3002<\/p>\n

3. \u81ea\u52a8\u5fae\u5206（AD）<\/h4>\n

\u65e2\u4e0d\u8fd1\u4f3c\u3001\u4e5f\u4e0d\u81a8\u80c0\u3002\u6838\u5fc3\u601d\u60f3：\u628a\u8ba1\u7b97\u8fc7\u7a0b\u62c6\u6210\u4e00\u8fde\u4e32\u57fa\u672c\u8fd0\u7b97（\u52a0\u51cf\u4e58\u9664\u3001sin\u3001exp\u2026\u2026），\u5bf9\u6bcf\u4e2a\u57fa\u672c\u8fd0\u7b97\u7cbe\u786e\u6c42\u5bfc，\u518d\u7528\u94fe\u5f0f\u6cd5\u5219\u628a\u5bfc\u6570"\u4f20"\u8fc7\u53bb\u2014\u2014\u7cbe\u5ea6\u548c\u7b26\u53f7\u5fae\u5206\u4e00\u6837（\u4e0e\u89e3\u6790\u5bfc\u6570\u9010\u4f4d\u4e00\u81f4），\u5f00\u9500\u53ea\u6bd4\u539f\u8ba1\u7b97\u591a\u4e00\u4e2a\u5e38\u6570\u500d\u3002<\/p>\n

\u4e8c\u3001AD \u7684\u79d8\u5bc6：\u5bf9\u5076\u6570（dual number）<\/h3>\n

\u524d\u5411\u6a21\u5f0f AD \u7528"\u5bf9\u5076\u6570" x + \u03b5\u00b7x\u2019：\u5b9e\u6570\u90e8\u5206\u5b58\u503c，\u5bf9\u5076\u90e8\u5206\u5b58\u5bfc\u6570\u3002\u5b9a\u4e49\u8fd0\u7b97\u89c4\u5219：<\/p>\n

(a + \u03b5a') + (b + \u03b5b') = (a+b) + \u03b5(a'+b')
\n(a + \u03b5a') \u00d7 (b + \u03b5b') = (ab) + \u03b5(a'b + ab')
\nsin(x + \u03b5x') = sin(x) + \u03b5\u00b7cos(x)\u00b7x'<\/p>\n

\u5bfc\u6570\u7684\u4fe1\u606f\u8ddf\u7740\u503c\u4e00\u8d77\u7b97\u2014\u2014\u4e00\u6b21\u524d\u5411\u5c31\u628a\u503c\u548c\u5bfc\u6570\u90fd\u7b97\u51fa\u6765\u4e86，\u800c\u4e14\u6bcf\u4e00\u6b65\u90fd\u662f\u7cbe\u786e\u7684\u3002<\/p>\n

\u4e09\u3001\u4ee3\u7801\u6f14\u793a：\u4e09\u79cd\u65b9\u5f0f\u540c\u53f0\u5bf9\u6bd4<\/h3>\n

import<\/span> numpy as<\/span> np
\nimport<\/span> math<\/p>\n

# \u76ee\u6807\u51fd\u6570 f(x1,x2) = sin(x1) + x1*x2 + exp(x2)<\/span>
\ndef<\/span> f<\/span>(<\/span>x1,<\/span> x2)<\/span>:<\/span>
\n return<\/span> math.<\/span>sin(<\/span>x1)<\/span> +<\/span> x1*<\/span>x2 +<\/span> math.<\/span>exp(<\/span>x2)<\/span><\/p>\n

x1,<\/span> x2 =<\/span> 0.5<\/span>,<\/span> 0.3<\/span><\/p>\n

# 1. \u6570\u503c\u5fae\u5206（\u4e2d\u5fc3\u5dee\u5206）<\/span>
\nh =<\/span> 1e-6<\/span>
\ndf_num =<\/span> [<\/span>(<\/span>f(<\/span>x1+<\/span>h,<\/span> x2)<\/span>–<\/span>f(<\/span>x1–<\/span>h,<\/span> x2)<\/span>)<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>h)<\/span>,<\/span>
\n (<\/span>f(<\/span>x1,<\/span> x2+<\/span>h)<\/span>–<\/span>f(<\/span>x1,<\/span> x2–<\/span>h)<\/span>)<\/span>\/<\/span>(<\/span>2<\/span>*<\/span>h)<\/span>]<\/span><\/p>\n

# 2. \u7b26\u53f7\u5fae\u5206（\u624b\u5199\u89e3\u6790\u5f0f）<\/span>
\ndf_sym =<\/span> [<\/span>math.<\/span>cos(<\/span>x1)<\/span> +<\/span> x2,<\/span> x1 +<\/span> math.<\/span>exp(<\/span>x2)<\/span>]<\/span><\/p>\n

# 3. \u81ea\u52a8\u5fae\u5206（dual number \u524d\u5411\u6a21\u5f0f）<\/span>
\nclass<\/span> Dual<\/span>:<\/span>
\n def<\/span> __init__<\/span>(<\/span>self,<\/span> val,<\/span> der=<\/span>0.0<\/span>)<\/span>:<\/span>
\n self.<\/span>val,<\/span> self.<\/span>der =<\/span> val,<\/span> der
\n def<\/span> __add__<\/span>(<\/span>self,<\/span> o)<\/span>:<\/span> return<\/span> Dual(<\/span>self.<\/span>val+<\/span>o.<\/span>val,<\/span> self.<\/span>der+<\/span>o.<\/span>der)<\/span>
\n def<\/span> __mul__<\/span>(<\/span>self,<\/span> o)<\/span>:<\/span> return<\/span> Dual(<\/span>self.<\/span>val*<\/span>o.<\/span>val,<\/span> self.<\/span>der*<\/span>o.<\/span>val+<\/span>self.<\/span>val*<\/span>o.<\/span>der)<\/span><\/p>\n

def<\/span> sin<\/span>(<\/span>d)<\/span>:<\/span> return<\/span> Dual(<\/span>math.<\/span>sin(<\/span>d.<\/span>val)<\/span>,<\/span> math.<\/span>cos(<\/span>d.<\/span>val)<\/span>*<\/span>d.<\/span>der)<\/span>
\ndef<\/span> exp<\/span>(<\/span>d)<\/span>:<\/span> return<\/span> Dual(<\/span>math.<\/span>exp(<\/span>d.<\/span>val)<\/span>,<\/span> math.<\/span>exp(<\/span>d.<\/span>val)<\/span>*<\/span>d.<\/span>der)<\/span><\/p>\n

def<\/span> f_ad<\/span>(<\/span>x1d,<\/span> x2d)<\/span>:<\/span>
\n return<\/span> sin(<\/span>x1d)<\/span> +<\/span> x1d*<\/span>x2d +<\/span> exp(<\/span>x2d)<\/span><\/p>\n

df_ad =<\/span> [<\/span>f_ad(<\/span>Dual(<\/span>x1,<\/span>1.0<\/span>)<\/span>,<\/span> Dual(<\/span>x2,<\/span>0.0<\/span>)<\/span>)<\/span>.<\/span>der,<\/span>
\n f_ad(<\/span>Dual(<\/span>x1,<\/span>0.0<\/span>)<\/span>,<\/span> Dual(<\/span>x2,<\/span>1.0<\/span>)<\/span>)<\/span>.<\/span>der]<\/span><\/p>\n

print<\/span>(<\/span>f"\u6570\u503c\u5fae\u5206: [<\/span>{<\/span>df_num[<\/span>0<\/span>]<\/span>:<\/span>.8f<\/span>}<\/span><\/span>, <\/span>{<\/span>df_num[<\/span>1<\/span>]<\/span>:<\/span>.8f<\/span>}<\/span><\/span>]"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>f"\u7b26\u53f7\u5fae\u5206: [<\/span>{<\/span>df_sym[<\/span>0<\/span>]<\/span>:<\/span>.8f<\/span>}<\/span><\/span>, <\/span>{<\/span>df_sym[<\/span>1<\/span>]<\/span>:<\/span>.8f<\/span>}<\/span><\/span>]"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>f"\u81ea\u52a8\u5fae\u5206: [<\/span>{<\/span>df_ad[<\/span>0<\/span>]<\/span>:<\/span>.8f<\/span>}<\/span><\/span>, <\/span>{<\/span>df_ad[<\/span>1<\/span>]<\/span>:<\/span>.8f<\/span>}<\/span><\/span>]"<\/span><\/span>)<\/span>
\nprint<\/span>(<\/span>f"AD vs \u7b26\u53f7\u8bef\u5dee: <\/span>{<\/span>abs<\/span>(<\/span>df_ad[<\/span>0<\/span>]<\/span>–<\/span>df_sym[<\/span>0<\/span>]<\/span>)<\/span>+<\/span>abs<\/span>(<\/span>df_ad[<\/span>1<\/span>]<\/span>–<\/span>df_sym[<\/span>1<\/span>]<\/span>)<\/span>:<\/span>.2e<\/span>}<\/span><\/span>"<\/span><\/span>)<\/span><\/p>\n

\u8fd0\u884c\u8f93\u51fa：<\/p>\n

\u6570\u503c\u5fae\u5206: [1.17758256, 1.84985881]
\n\u7b26\u53f7\u5fae\u5206: [1.17758256, 1.84985881]
\n\u81ea\u52a8\u5fae\u5206: [1.17758256, 1.84985881]
\nAD vs \u7b26\u53f7\u8bef\u5dee: 0.00e+00<\/p>\n

\u4e09\u79cd\u65b9\u5f0f\u7ed3\u679c\u4e00\u81f4，\u4f46AD \u4e0e\u7b26\u53f7\u5fae\u5206\u7684\u8bef\u5dee\u662f 0.00e+00\u2014\u2014\u9010\u4f4d\u4e00\u81f4\u3002\u533a\u522b\u5728\u5de5\u7a0b\u6027：\u6570\u503c\u5fae\u5206\u6162\u4e14\u7cbe\u5ea6\u53d7\u9650，\u7b26\u53f7\u5fae\u5206\u4f1a\u81a8\u80c0，AD \u5feb\u3001\u51c6\u3001\u4e0d\u81a8\u80c0\u2014\u2014\u6240\u4ee5 PyTorch\u3001JAX\u3001TensorFlow \u5168\u90e8\u7528 AD\u3002<\/p>\n

\u56db\u3001\u53cd\u5411\u6a21\u5f0f：\u6df1\u5ea6\u5b66\u4e60\u7684\u5173\u952e<\/h3>\n

\u4e0a\u9762\u7684\u4f8b\u5b50\u662f\u524d\u5411\u6a21\u5f0f（\u5bf9\u6bcf\u4e2a\u8f93\u5165\u6c42\u5bfc\u4e00\u6b21）\u3002\u4f46\u6df1\u5ea6\u5b66\u4e60\u7684\u53c2\u6570\u6709\u51e0\u4ebf\u4e2a，\u8f93\u51fa\u53ea\u6709 1 \u4e2a（loss）\u2014\u2014\u8fd9\u65f6\u5019\u8981\u7528\u53cd\u5411\u6a21\u5f0f（\u53cd\u5411\u4f20\u64ad）：\u4ece\u8f93\u51fa\u5f80\u56de\u4f20\u68af\u5ea6，\u4e00\u6b21\u53cd\u5411\u5c31\u80fd\u7b97\u51fa\u6240\u6709\u53c2\u6570\u7684\u5bfc\u6570，\u5f00\u9500\u662f\u524d\u5411\u7684 2-3 \u500d\u3002<\/p>\n

\u524d\u5411\u6a21\u5f0f\u9002\u5408"\u8f93\u5165\u5c11\u8f93\u51fa\u591a"（\u5982\u96c5\u53ef\u6bd4\u77e9\u9635），\u53cd\u5411\u6a21\u5f0f\u9002\u5408"\u8f93\u5165\u591a\u8f93\u51fa\u5c11"（\u5982\u795e\u7ecf\u7f51\u7edc\u8bad\u7ec3）\u3002PyTorch \u7684 .backward() \u5c31\u662f\u53cd\u5411\u6a21\u5f0f AD \u7684\u5de5\u7a0b\u5b9e\u73b0\u3002<\/p>\n

\u4e94\u3001\u907f\u5751\u6e05\u5355<\/h3>\n
  • AD \u2260 \u6570\u503c\u5fae\u5206：AD \u662f\u7cbe\u786e\u7684（\u9010\u4f4d\u4e00\u81f4），\u4e0d\u662f\u8fd1\u4f3c<\/li>\n
  • AD \u2260 \u7b26\u53f7\u5fae\u5206：AD \u4e0d\u4ea7\u751f\u81a8\u80c0\u7684\u7b26\u53f7\u8868\u8fbe\u5f0f，\u5b83\u5206\u89e3\u8ba1\u7b97\u6d41\u7a0b<\/li>\n
  • \u53cd\u5411\u4f20\u64ad = \u53cd\u5411\u6a21\u5f0f AD：\u7406\u89e3\u4e86\u8fd9\u4e2a，\u4f60\u5c31\u4e0d\u6015"\u68af\u5ea6\u6d88\u5931"\u7684\u6570\u5b66<\/li>\n
  • \u63a7\u5236\u6d41\u6ca1\u95ee\u9898：if\/\u5faa\u73af\u5728 AD \u91cc\u7167\u5e38\u5de5\u4f5c（\u6309\u5b9e\u9645\u6267\u884c\u8def\u5f84\u6c42\u5bfc），\u7b26\u53f7\u5fae\u5206\u53cd\u800c\u505a\u4e0d\u5230<\/li>\n
  • \u524d\u5411 vs \u53cd\u5411\u8981\u9009\u5bf9：\u8f93\u5165\u5c11\u8f93\u51fa\u591a\u7528\u524d\u5411，\u8f93\u5165\u591a\u8f93\u51fa\u5c11（\u8bad\u7ec3）\u7528\u53cd\u5411<\/li>\n

    \u516d\u3001\u60f3\u7cfb\u7edf\u5b66\u53ef\u5fae\u5206\u7f16\u7a0b？<\/h3>\n

    \u672c\u6587\u7cbe\u9009\u81ea ima \u77e5\u8bc6\u53f7\u3010Kruptos\u3011\u300a\u53ef\u5fae\u5206\u7f16\u7a0b\u4e0e\u79d1\u5b66\u8ba1\u7b97\u300b\u8ba2\u9605\u5e93（\u7b2c 002 \u671f\u6570\u503c\/\u7b26\u53f7\/\u81ea\u52a8\u5fae\u5206\u5bf9\u6bd4\u3001\u7b2c 003 \u671f\u524d\u5411 AD\u3001\u7b2c 004 \u671f\u53cd\u5411 AD \u7b49 100 \u671f\u7cfb\u7edf\u6559\u7a0b，\u4ece\u81ea\u52a8\u5fae\u5206\u4e09\u8303\u5f0f\u5230 PINN\u3001\u795e\u7ecf\u7b97\u5b50\u3001\u53ef\u5fae\u7269\u7406\u4eff\u771f，\u6bcf\u671f\u914d\u53ef\u8fd0\u884c Python \u4ee3\u7801）\u3002<\/p>\n

    📚 \u5b8c\u6574\u7cfb\u5217 100 \u671f + \u914d\u5957\u4ee3\u7801，\u5df2\u5728 ima \u77e5\u8bc6\u53f7\u53d1\u5e03<\/p>\n

    \u672c\u6587\u53ea\u662f\u7cfb\u5217\u7684\u4e00\u4e2a\u5207\u7247\u3002\u5b8c\u6574\u7cfb\u5217（100 \u671f\u7cfb\u7edf\u6559\u7a0b + \u6bcf\u671f\u53ef\u8fd0\u884c\u4ee3\u7801）\u5728 ima \u77e5\u8bc6\u53f7\u3010Kruptos\u3011\u6301\u7eed\u66f4\u65b0\u4e2d：<\/p>\n

      \n
    • 🗂 67+ \u6280\u672f\u77e5\u8bc6\u5e93：\u4fe1\u53f7\u4e0e\u7cfb\u7edf\u3001SDR \u8f6f\u4ef6\u65e0\u7ebf\u7535\u3001\u6570\u5b57\u4fe1\u53f7\u5904\u7406\u3001\u64cd\u4f5c\u7cfb\u7edf\u3001AI Agent\u3001\u5927\u6a21\u578b\u5fae\u8c03\u2026\u2026\u51e0\u4e4e\u8986\u76d6\u5168\u90e8\u8f6f\u786c\u4ef6\u6280\u672f\u6808<\/li>\n
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      \n

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