{"id":17806,"date":"2024-12-20T15:03:41","date_gmt":"2024-12-20T07:03:41","guid":{"rendered":"http:\/\/139.9.1.231\/?p=17806"},"modified":"2025-01-06T20:17:30","modified_gmt":"2025-01-06T12:17:30","slug":"the-flow-matching-model","status":"publish","type":"post","link":"http:\/\/139.9.1.231\/index.php\/2024\/12\/20\/the-flow-matching-model\/","title":{"rendered":"The flow matching model:\u6d41\u5339\u914d\u6a21\u578b"},"content":{"rendered":"\n<p class=\"has-light-pink-background-color has-background\"><strong>\u8bba\u6587\uff1a<a one-link-mark=\"yes\" href=\"https:\/\/arxiv.org\/abs\/2210.02747\"><em>https:\/\/arxiv.org\/abs\/2210.02747<\/em><\/a><\/strong><\/p>\n\n\n\n<p><strong>\u535a\u5ba2\uff1a<\/strong><a href=\"https:\/\/lilianweng.github.io\/posts\/2018-10-13-flow-models\/\"><em>https:\/\/lilianweng.github.io\/posts\/2018-10-13-flow-models\/<\/em><\/a><\/p>\n\n\n\n\n\n<p>\u6700\u8fd1\uff0c Flux.ai\u7684<strong> Flux \u7cfb\u5217<\/strong>\u6a21\u578b\u56e0\u5404\u79cd\u539f\u56e0\u800c\u53d7\u5230\u79d1\u6280\u754c\u548c\u975e\u79d1\u6280\u754c\u7684\u5e7f\u6cdb\u5173\u6ce8\uff0c\u6a21\u578b\u901f\u5ea6\u5feb\u3001\u6613\u4e8e\u4f7f\u7528\uff08\u8fd9\u8981\u5f52\u529f\u4e8e\u6269\u6563\u5668(diffusers)\uff09\u5e76\u4e14\u6613\u4e8e\u8c03\u6574\uff08\u518d\u6b21\u611f\u8c22\u6269\u6563\u5668\uff09\u3002\u4f46\u662f\uff0cFlux \u51fa\u8272\u7684\u56fe\u50cf\u6587\u672c\u5bf9\u9f50\u548c\u9ad8\u8d28\u91cf\u751f\u6210\u80cc\u540e\u7684\u539f\u56e0\u662f\u4e00\u79cd<strong>\u8d85\u8d8a\u6807\u51c6\u6269\u6563\u8fc7\u7a0b\u7684\u65b0\u65b9\u6cd5\uff0c\u79f0\u4e3a\u201c\u6d41\u5339\u914d\uff08Flow matching\uff09\u201d\u3002<\/strong><\/p>\n\n\n\n<p>\u6bcf\u4e2a\u751f\u6210\u6a21\u578b(generative model)\u7406\u60f3\u60c5\u51b5\u4e0b\u90fd\u662f\u5bc6\u5ea6\u4f30\u8ba1(density estimaor)\uff1b\u56e0\u6b64\u6a21\u62df\u6982\u7387\u5bc6\u5ea6\uff0c\u6700\u7ec8\u662f JPD\uff0c\u5177\u6709\u4e24\u4e2a\u9884\u671f\u7279\u5f81\uff0c\u5373\u91c7\u6837\u548c\u538b\u7f29\uff0c\u538b\u7f29\u57fa\u672c\u4e0a\u662f\u5c06\u6570\u636e\u63a8\u9001\u5230\u4fe1\u606f\u7a7a\u95f4\uff0c\u8fd9\u4f3c\u4e4e\u662f\u8f83\u4f4e\u7ef4\u7684\uff0c\u800c\u91c7\u6837\u662f\u4ece\u4efb\u4f55\u7279\u5f81\u5206\u5e03\uff08z\uff09\u5f00\u59cb\u751f\u6210 P(x|z) \u7684\u80fd\u529b\uff0c\u53ef\u4ee5\u662f\u6b63\u6001\u5206\u5e03\uff08\u5982 VAE \u7684\u60c5\u51b5\uff09\uff0c\u56e0\u6b64\uff0c\u5728\u975e\u5e38\u9ad8\u7684\u5c42\u6b21\u4e0a\uff0c\u6211\u4eec\u8bd5\u56fe\u627e\u5230\u5c06 z \u6620\u5c04\u5230 x \u4ee5\u53ca\u5c06 x \u6620\u5c04\u5230 z\uff08\u91c7\u6837\u548c\u538b\u7f29\uff09\u7684\u6620\u5c04\/\u51fd\u6570\u3002<\/p>\n\n\n\n<h3>\u4ec0\u4e48\u662f Flow Matching\uff1f<\/h3>\n\n\n\n<p>Flow Matching \u662f\u4e00\u79cd\u7528\u4e8e\u8bad\u7ec3\u751f\u6210\u6a21\u578b\u7684\u65b9\u6cd5\uff0c\u5b83\u57fa\u4e8e\u5bf9\u6d41\uff08flow\uff09\u7684\u6982\u5ff5\u3002\u5728\u751f\u6210\u6a21\u578b\u4e2d\uff0c\u201cflow\u201d\u901a\u5e38\u6307\u7684\u662f\u901a\u8fc7\u53ef\u9006\u7684\u53d8\u6362\uff08reversible transformation\uff09\u9010\u6b65\u5c06\u590d\u6742\u7684\u5206\u5e03\u6620\u5c04\u4e3a\u7b80\u5355\u7684\u5206\u5e03\uff08\u5982\u9ad8\u65af\u5206\u5e03\uff09\uff0c\u4ece\u800c\u4f7f\u751f\u6210\u6a21\u578b\u80fd\u591f\u66f4\u5bb9\u6613\u5730\u5bf9\u590d\u6742\u6570\u636e\u8fdb\u884c\u5efa\u6a21\u3002Flow Matching \u7684\u76ee\u6807\u662f\u901a\u8fc7\u6700\u5c0f\u5316\u6a21\u578b\u751f\u6210\u7684\u6d41\u4e0e\u771f\u5b9e\u6570\u636e\u5206\u5e03\u7684\u6d41\u4e4b\u95f4\u7684\u5dee\u5f02\uff0c\u6765\u5b66\u4e60\u4e00\u4e2a\u80fd\u591f\u751f\u6210\u9ad8\u8d28\u91cf\u6837\u672c\u7684\u6a21\u578b\u3002<\/p>\n\n\n\n<p>\u5728 Flow Matching \u4e2d\uff0c\u901a\u8fc7\u5bf9\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u6bcf\u4e00\u6b65\u7684\u5339\u914d\u8fdb\u884c\u5efa\u6a21\uff0c\u6a21\u578b\u9010\u6e10\u5b66\u4e60\u5982\u4f55\u4ece\u7b80\u5355\u7684\u566a\u58f0\u5206\u5e03\u4e2d\u751f\u6210\u590d\u6742\u7684\u76ee\u6807\u6570\u636e\u3002\u8fd9\u79cd\u65b9\u6cd5\u53ef\u4ee5\u7528\u4e8e\u89e3\u51b3\u4f20\u7edf\u751f\u6210\u6a21\u578b\u4e2d\u4e00\u4e9b\u96be\u4ee5\u5904\u7406\u7684\u95ee\u9898\uff0c\u5982\u6a21\u5f0f\u5d29\u6e83\uff08mode collapse\uff09\u7b49\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" src=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/08\/image-86-1024x498.png\" alt=\"\" class=\"wp-image-17847\" width=\"538\" height=\"261\" srcset=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/08\/image-86-1024x498.png 1024w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/08\/image-86-300x146.png 300w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/08\/image-86-768x374.png 768w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/08\/image-86.png 1104w\" sizes=\"(max-width: 538px) 100vw, 538px\" \/><\/figure><\/div>\n\n\n\n<p>\u5047\u8bbe\u4e24\u4e2a Normalizing Flows\uff0c\u4e00\u4e2a\u8868\u793a\u4e3a z\uff08\u6f5c\u5728\u6216\u53ef\u5904\u7406\u5206\u5e03\uff09\uff0c\u53e6\u4e00\u4e2a\u8868\u793a\u4e3a X\uff08\u6570\u636e\u5206\u5e03\uff09\uff0c\u56e0\u4e3a\u6211\u4eec\u60f3\u8981\u627e\u5230\u4e00\u4e2a\u53ef\u4ee5\u5c06 z \u6620\u5c04\u5230 x \u7684\u51fd\u6570\uff0c\u6211\u4eec\u4f1a\u5f97\u5230 X \u548c Z \u5bc6\u5ea6\u4e4b\u95f4\u7684\u5173\u7cfb\uff0c\u8fd9\u5fc5\u7136\u6307\u51fa\u5047\u8bbe X \u548c Z \u662f\u5171\u8f6d\u5206\u5e03\uff08\u53d8\u6362\u524d\u540e\u540c\u4e00\u5bb6\u65cf z \u7684\u5206\u5e03\uff09\uff0cX \u548c Z \u7684\u53d8\u5316\u5e94\u8be5\u662f\u76f8\u5bf9\u7684\uff0c\u56e0\u6b64\uff0cX \u7684\u53d8\u5316\u662f Z \u7684\u67d0\u4e2a\u51fd\u6570\uff0c\u53cd\u4e4b\u4ea6\u7136\u3002\u4f46\u662f\uff0c\u6309\u67d0\u4e2a\u91cf\u7f29\u653e\u3002\u8fd9\u4e2a\u91cf\u7531\u96c5\u53ef\u6bd4\u77e9\u9635\u7ed9\u51fa\u7684 z \u548c x \u4e4b\u95f4\u6bcf\u4e2a\u7ef4\u5ea6\u7684\u53d8\u5316\u8868\u793a\uff0c\u5728\u975e\u5e38\u7b80\u5355\u7684\u5c3a\u5ea6\u4e0a\uff0c\u5b83\u57fa\u672c\u4e0a\u662f Z \u548c X \u4e4b\u95f4\u53d8\u91cf\u7684\u53d8\u5316\u3002\u4f46\u662f\uff0c\u5b83\u4e0d\u662f\u90a3\u4e48\u7b80\u5355\uff0c\u56e0\u4e3a X \u548c Z \u5b9e\u9645\u4e0a\u5e76\u4e0d\u5171\u8f6d\uff0c\u56e0\u6b64\uff0c\u6211\u4eec\u53ea\u5269\u4e0b\u8fed\u4ee3\u91c7\u6837\u548c\u8fd1\u4f3c\u65b9\u6cd5\uff0c\u6bd4\u5982\u6700\u4f73\u4f20\u8f93\u6216\u5409\u5e03\u65af\u91c7\u6837\uff08\u7528\u4e8e RBM\uff09\u3002\u9274\u4e8e\u8fd9\u4e9b\u9650\u5236\uff0c\u5927\u591a\u6570\u65b9\u6cd5\u90fd\u7ed5\u9053\u53bb\u6a21\u62df\u5206\u5e03\u5e76\u8fd1\u4f3c\u975e\u7cbe\u786e\u6620\u5c04\uff0c\u800c\u50cf Normalizing flow \u8fd9\u6837\u7684\u65b9\u6cd5\u5219\u505a\u51fa\u7b80\u5316\u5047\u8bbe\uff0c\u4f7f\u8ba1\u7b97\u548c\u516c\u5f0f\u6613\u4e8e\u5904\u7406\uff0c\u5f62\u5f0f\u4e3a p(x)dx = p(z)dz\uff0c\u53ef\u4ee5\u5c06\u5176\u91cd\u65b0\u8868\u8ff0\u4e3a\u4e24\u4e2a\u9879\uff0c\u7b2c\u4e00\u4e2a\u662f MLE \u9879\uff0c\u7b2c\u4e8c\u4e2a\u662f\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" src=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-69.png\" alt=\"\" class=\"wp-image-23256\" width=\"317\" height=\"203\" srcset=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-69.png 532w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-69-300x192.png 300w\" sizes=\"(max-width: 317px) 100vw, 317px\" \/><\/figure><\/div>\n\n\n\n<p>\uff08\u6807\u51c6\u5316\u662f\u56e0\u4e3a\u53d8\u91cf\u7684\u53d8\u5316\u603b\u662f\u7ed9\u51fa\u4e00\u4e2a\u6807\u51c6\u5316\u7684\u5bc6\u5ea6\u51fd\u6570\uff0c\u6d41\u52a8\u662f\u56e0\u4e3a\u5b83\u8fed\u4ee3\u5730\u6a21\u62df\u4ece\u6e90\u5230\u76ee\u6807\u7684\u8f68\u8ff9\/\u6d41\u52a8\uff09<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" src=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-70.png\" alt=\"\" class=\"wp-image-23258\" width=\"529\" height=\"165\" srcset=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-70.png 769w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-70-300x94.png 300w\" sizes=\"(max-width: 529px) 100vw, 529px\" \/><\/figure><\/div>\n\n\n\n<p>\u95ee\u9898\u4ece\u8fd9\u91cc\u5f00\u59cb\uff0c\u8fd9\u6837\u7684\u51fd\u6570\u5b58\u5728\u9700\u8981\u4e24\u4e2a\u6761\u4ef6\uff0c<\/p>\n\n\n\n<p>1. p(x,z) \u7684 MLE \u516c\u5f0f\u5fc5\u987b\u662f\u53cc\u5c04\u7684\u3002<\/p>\n\n\n\n<p>2. \u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u884c\u5217\u5f0f\u662f\u53ef\u6709\u6548\u8ba1\u7b97\u7684\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" src=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-71.png\" alt=\"\" class=\"wp-image-23260\" width=\"370\" height=\"120\" srcset=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-71.png 715w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-71-300x98.png 300w\" sizes=\"(max-width: 370px) 100vw, 370px\" \/><\/figure><\/div>\n\n\n\n<p>\u8981\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0c\u6211\u4eec\u9700\u8981\u5047\u8bbe z \u548c X \u4e4b\u95f4\u7684\u72b6\u6001\u4f9d\u8d56\u6027\uff0c\u4f7f\u5f97\u5b83\u662f\u53cc\u5c04\u7684\u5e76\u4e14\u884c\u5217\u5f0f\u53ef\u4ee5\u6709\u6548\u8ba1\u7b97\uff0c\u6709\u4e09\u79cd\u4e3b\u8981\u65b9\u6cd5<\/p>\n\n\n\n<p>1. \u8026\u5408\u5757\uff1a\u57fa\u672c\u4e0a\u4f60\u5c06 z \u5206\u6210\u4e24\u5757\uff0c\u53ea\u6709\u6700\u540e k \u503c\u9884\u6d4b X \u7684\u6700\u540e k \u503c\uff08\u901a\u8fc7\u57fa\u4e8e\u5747\u503c\/\u65b9\u5dee\u7684\u91c7\u6837\uff09\uff0cX \u7684\u5176\u4ed6\u90e8\u5206\u57fa\u672c\u4e0a\u662f z \u7684\u76f4\u63a5\u590d\u5236\uff0c\u8fd9\u6709\u4f55\u5e2e\u52a9\uff1f\u7531\u4e8e\u8fd9\u79cd\u65b9\u6cd5\uff0c\u96c5\u53ef\u6bd4\u77e9\u9635\u53d8\u6210\u4e86\u5bf9\u89d2\u77e9\u9635\uff0c\u5de6\u4e0a\u89d2\uff08&lt;k\uff09\u90e8\u5206\u662f\u6052\u7b49\u77e9\u9635\uff0c\u53f3\u4e0b\u89d2\uff08&gt;=k\uff09\u53d8\u6210\u5143\u7d20\u4e58\u79ef\uff0c\u53f3\u4e0a\u89d2\u53d8\u6210 0\uff0c\u56e0\u4e3a z\uff08&lt;k\uff09\u548c x\uff08&gt;=k\uff09\u4e4b\u95f4\u6ca1\u6709\u4f9d\u8d56\u5173\u7cfb\uff0c\u56e0\u6b64\uff0c\u96c5\u53ef\u6bd4\u884c\u5217\u5f0f\u7684\u8ba1\u7b97\u662f\u6709\u6548\u7684\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" src=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-72.png\" alt=\"\" class=\"wp-image-23262\" width=\"266\" height=\"233\"\/><\/figure><\/div>\n\n\n\n<p>2. AR \u6d41\u6216\u81ea\u56de\u5f52\u6d41\u662f\u4e0b\u4e00\u4e2a\u5408\u4e4e\u903b\u8f91\u7684\u6269\u5c55\uff0c\u4e0e\u5176\u5236\u4f5c\u5927 k \u5757\uff0c\u4e3a\u4ec0\u4e48\u4e0d\u5c06\u6bcf\u4e2a\u72b6\u6001\/\u7279\u5f81\u89c6\u4e3a\u9a6c\u5c14\u53ef\u592b\u94fe\u7684\u4e00\u90e8\u5206\uff0c\u4ece\u800c\u6d88\u9664\u989d\u5916\u7684\u4f9d\u8d56\u5173\u7cfb\uff0c\u8fd9\u4f1a\u5bfc\u81f4\u96c5\u53ef\u6bd4\u77e9\u9635\u7684\u4e0b\u4e09\u89d2\u77e9\u9635\uff0c\u8fd9\u4e5f\u5f88\u5bb9\u6613\u8ba1\u7b97\u3002\u4f46\u662f\uff0c\u8fd9\u79cd\u65b9\u6cd5\u4fdd\u7559\u4e86\u66f4\u591a\u7684\u7279\u5f81\uff0c\u5e76\u4e14\u4e0d\u6613\u53d7\u5230\u6211\u4eec\u5728\u8026\u5408\u5c42\u4e2d\u4e3a\u4fdd\u7559\u7279\u5f81\u800c\u8fdb\u884c\u7684\u7f6e\u6362\u64cd\u4f5c\u7684\u5f71\u54cd\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" src=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-73.png\" alt=\"\" class=\"wp-image-23263\" width=\"516\" height=\"259\" srcset=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-73.png 751w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-73-300x151.png 300w\" sizes=\"(max-width: 516px) 100vw, 516px\" \/><figcaption>\u56fe MAF \u548c IAF \u7684\u6bd4\u8f83\u3002\u5177\u6709\u5df2\u77e5\u5bc6\u5ea6\u7684\u53d8\u91cf\u4e3a\u7eff\u8272\uff0c\u800c\u672a\u77e5\u7684\u53d8\u91cf\u4e3a\u7ea2\u8272\u3002<\/figcaption><\/figure><\/div>\n\n\n\n<p>3. \u6700\u540e\uff0c\u6b8b\u5dee\u6d41\uff0c\u6211\u4eec\u4fdd\u7559\u6574\u4e2a\u7279\u5f81\u7a7a\u95f4\uff0c\u800c\u4e0d\u727a\u7272\u8ba1\u7b97\u3002\u8fd9\u4e2a\u60f3\u6cd5\u5f88\u7b80\u5355\uff0c\u4f46\u5374\u6709\u975e\u5e38\u590d\u6742\u7684\u6570\u5b66\u652f\u6301\u3002\u516c\u5f0f\u662f\u6b8b\u5dee\u5f62\u5f0f x = z + f(z)\uff0c\u4f46\u8fd9\u4e0d\u662f\u53cc\u5c04\uff0c\u56e0\u4e3a f \u662f\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u3002\u6709\u8da3\u7684\u662f\uff0c\u591a\u4e8f\u4e86 Banach \u548c\u4ed6\u7684\u6536\u7f29\u6620\u5c04\uff0c\u5728\u7406\u60f3\u60c5\u51b5\u4e0b\uff0c\u5b58\u5728\u4e00\u4e2a\u552f\u4e00\u7684 z*\uff0c\u5b83\u603b\u662f\u6620\u5c04\u5230\u76f8\u540c\u7684 x\uff08\u7a33\u5b9a\u72b6\u6001 z\uff09\uff0c\u56e0\u6b64\uff0c\u5b83\u4e5f\u53d8\u6210\u4e86\u53cc\u5c04\uff0c\u5f62\u5f0f\u4e3a x = z* + f(z*)\uff0c\u5176\u4e2d f \u662f\u4e00\u4e2a\u6536\u7f29\u6620\u5c04\uff08\u51fd\u6570\u53d7 Lipschitz \u5c0f\u4e8e 1 \u7684\u9650\u5236\uff0c\u56e0\u6b64\uff0cz \u7684\u53d8\u5316\u53d7 X \u7684\u53d8\u5316\u7684\u9650\u5236\uff09\uff0c\u8be5\u5f62\u5f0f\u8fd8\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u79cd\u5728\u7ed9\u5b9a\u5148\u524d z(k) \u7684\u60c5\u51b5\u4e0b\u8868\u793a z(k+1) \u7684\u65b9\u6cd5\uff0c\u8fd9\u6709\u52a9\u4e8e\u8fed\u4ee3\u8fd1\u4f3c X\uff0c\u800c\u4e0d\u662f\u5355\u6b21\u6846\u67b6\u3002\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u76f8\u540c\u7684\u516c\u5f0f\u4ece z(0) \u8f6c\u5230 z(t)\uff0c\u4e5f\u53ef\u4ee5\u6062\u590d\u56de\u6765\uff0c\u542c\u8d77\u6765\u5f88\u719f\u6089\uff0c\u8fd9\u5927\u81f4\u5c31\u662f\u6269\u6563\u3002\u90a3\u4e48\u884c\u5217\u5f0f\u5462\uff0c\u8fed\u4ee3\u53d8\u6362\u5bfc\u81f4\u96c5\u53ef\u6bd4\u77e9\u9635\u8ff9\u7684\u65e0\u7a77\u9879\u4e4b\u548c\uff0c\u8fd9\u5bf9\u4e8e\u6ee1\u79e9\u96c5\u53ef\u6bd4\u77e9\u9635\u6765\u8bf4\u662f\u53ef\u6015\u7684\uff0c\u4f46\u53ef\u4ee5\u901a\u8fc7\u7c7b\u4f3c\u7684\u77e9\u9635\u516c\u5f0f\u4f7f\u7528\u54c8\u94a6\u68ee\u65b9\u6cd5\u8fdb\u884c\u8ff9\u4f30\u8ba1\u6765\u7b80\u5355\u5730\u8ba1\u7b97\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" width=\"781\" height=\"295\" src=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-74.png\" alt=\"\" class=\"wp-image-23264\" srcset=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-74.png 781w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-74-300x113.png 300w, http:\/\/139.9.1.231\/wp-content\/uploads\/2024\/12\/image-74-768x290.png 768w\" sizes=\"(max-width: 781px) 100vw, 781px\" \/><figcaption>Fig. 4. Three substeps in one step of flow in Glow.<\/figcaption><\/figure>\n\n\n\n<p>\u5230\u76ee\u524d\u4e3a\u6b62\uff0c\u6211\u4eec\u8ba8\u8bba\u7684\u6240\u6709\u5185\u5bb9\u90fd\u5bf9\u6837\u672c\u548c\u8f68\u8ff9\u72b6\u6001\uff08z=&gt;x\uff09\u505a\u51fa\u4e86\u79bb\u6563\u5047\u8bbe\uff0c\u4e3a\u4ec0\u4e48\u4e0d\u4f7f\u5176\u8fde\u7eed\u6216\u57fa\u672c\u4e0a\u6210\u4e3a\u8fde\u7eed\u7684\u6b8b\u5dee\u6d41\u5462\uff1f<\/p>\n\n\n\n<p>\u6b8b\u5dee\u5f62\u5f0f x(k+1) = x(k) + af(x(k)) \u53ef\u4ee5\u5199\u6210\u4e00\u4e2a\u5fae\u5206\uff0c\u5176\u4e2d k \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\uff0c\u8fd9\u770b\u8d77\u6765\u5f88\u719f\u6089\u5417\uff1f\u662f\u7684\uff0c\u8fd9\u73b0\u5728\u662f\u4e00\u4e2a\u795e\u7ecf ODE\uff0c\u4f46\u662f\uff0c\u6211\u4eec\u6b63\u5728\u5c1d\u8bd5\u5bf9\u6982\u7387\u5bc6\u5ea6\u5728 t \u65f6\u7684\u72b6\u6001\u8fdb\u884c\u5efa\u6a21\uff0c\u8be5\u72b6\u6001\u4fdd\u6301\u4e0d\u53d8\u3002\u5bc6\u5ea6\u7684\u72b6\u6001\u53d8\u5316\u901a\u8fc7\u8fde\u7eed\u6027\u6216\u4f20\u8f93\u65b9\u7a0b\u5efa\u6a21\u3002\u5177\u4f53\u800c\u8a00\uff0c\u4ece\u5bc6\u5ea6\u51fd\u6570\u7684\u4e00\u90e8\u5206\u79fb\u52a8\u5230\u53e6\u4e00\u90e8\u5206\u7684\u8d28\u91cf\u53ef\u4ee5\u770b\u4f5c\u662f\u539f\u59cb\u8d28\u91cf\u4e0e\u5f53\u524d\/\u79fb\u52a8\u540e\u8d28\u91cf\u4e4b\u95f4\u7684\u53d1\u6563\u3002\u8fd9\u79cd\u53d1\u6563\u662f\u6574\u4e2a\u8f68\u8ff9\u4e0a\u7684\u8fde\u7eed\u51fd\u6570\uff0c\u5fc5\u987b\u8fdb\u884c\u79ef\u5206\uff0c\u56e0\u6b64\u5fc5\u987b\u901a\u8fc7\u6570\u503c ODE \u6c42\u89e3\uff0c\u8fd9\u4f7f\u5f97\u5b83\u4e0d\u53ef\u6269\u5c55\u3002<\/p>\n\n\n\n<p>\u8fd9\u5c31\u50cf\u72b6\u6001\u7684\u6bd4\u8f83\uff0c\u4e3a\u4ec0\u4e48\u4e0d\u6bd4\u8f83\u8def\u5f84\u5462\uff1f<\/p>\n\n\n\n<p>\u8fd9\u5f15\u51fa\u4e86 Flux \u80cc\u540e\u7684\u65b9\u6cd5\uff0c\u5373\u6d41\u5339\u914d\u3002\u60f3\u6cd5\u662f\u8fd9\u6837\u7684\u2026\u2026\u6211\u4eec\u4ece\u4e00\u4e2a\u975e\u5e38\u7b80\u5355\u7684\u5206\u5e03\u5f00\u59cb\uff0c\u5e76\u5c06\u5176\u79fb\u5411\u9884\u671f\u7684\u5206\u5e03\uff0c\u4f46\u662f\uff0c\u7531\u4e8e\u6211\u4eec\u4e0d\u77e5\u9053\u9884\u671f\u7684\u5206\u5e03\uff0c\u6211\u4eec\u901a\u8fc7\u8fed\u4ee3\u6270\u52a8\u6765\u8c03\u8282\u5b83\uff0c\u5e76\u4f30\u8ba1\u5df2\u77e5\u7684\u9644\u52a0\u566a\u58f0\uff0c\u6700\u7ec8\u6a21\u62df\u5e95\u5c42\u5206\u5e03\uff0c\u8fd9\u4e2a\u8fc7\u7a0b\u7b80\u79f0\u4e3a\u6761\u4ef6\u6d41\u5339\u914d\uff0c\u7406\u8bba\u4e0a\u5df2\u7ecf\u8bc1\u660e\uff0c\u5728\u7406\u60f3\u6761\u4ef6\u4e0b\uff0c\u6761\u4ef6\u76ee\u6807\u548c\u65e0\u6761\u4ef6\u76ee\u6807\u662f\u5b8c\u5168\u76f8\u540c\u7684\uff0c\u56e0\u6b64\uff0c\u901a\u8fc7\u4f18\u5316 CFM\u3010\u6761\u4ef6\u6d41\u5339\u914d\u3011\uff0c\u6211\u4eec\u503e\u5411\u4e8e\u4f18\u5316\u6d41\u5339\u914d\u80cc\u540e\u7684\u4e3b\u8981\u76ee\u6807\u3002<\/p>\n\n\n\n<div class=\"wp-block-image\"><figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" src=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2025\/01\/image-1024x331.png\" alt=\"\" class=\"wp-image-23619\" width=\"448\" height=\"145\" srcset=\"http:\/\/139.9.1.231\/wp-content\/uploads\/2025\/01\/image-1024x331.png 1024w, http:\/\/139.9.1.231\/wp-content\/uploads\/2025\/01\/image-300x97.png 300w, http:\/\/139.9.1.231\/wp-content\/uploads\/2025\/01\/image-768x249.png 768w, http:\/\/139.9.1.231\/wp-content\/uploads\/2025\/01\/image-1536x497.png 1536w, http:\/\/139.9.1.231\/wp-content\/uploads\/2025\/01\/image.png 1560w\" sizes=\"(max-width: 448px) 100vw, 448px\" \/><\/figure><\/div>\n\n\n\n<p>\u8fd9\u6b63\u662f\u6269\u6563\u8fc7\u7a0b\uff0c\u4f46\u662f\uff0c\u4e3b\u8981\u533a\u522b\u5728\u4e8e\u8fb9\u754c\u6761\u4ef6\u6216\u521d\u59cb\u548c\u6700\u7ec8\u72b6\u6001\u7684\u5b9a\u4e49\uff0c\u6269\u6563\u8fc7\u7a0b\u5047\u8bbe\u7eaf\u566a\u58f0\u4e3a z(t)\uff0c\u6570\u636e\u4e3a z(0)\uff0c\u4f46\u662f\uff0c\u8fd9\u662f\u5728 t \u8d8b\u5411\u4e8e\u65e0\u7a77\u5927\u65f6\u7684\u5047\u8bbe\u3002\u4f46\u662f\uff0c\u8fd9\u5728\u7ecf\u9a8c\u4e0a\u662f\u4e0d\u53ef\u884c\u7684\uff0c\u76f8\u53cd\uff0c\u6211\u4eec\u4f1a\u5728\u8db3\u591f\u7684\u65f6\u95f4\u6233\u5185\u6267\u884c\u6b64\u64cd\u4f5c\uff0c\u56e0\u6b64\uff0c\u6211\u4eec\u4fdd\u6301\u5728\u66f4\u6d45\u7684\u6d41\u5f62\u4e2d\u5e76\u4e14\u901f\u5ea6\u4e5f\u66f4\u6162\uff08\u7a0d\u540e\u7531 LCM \u5904\u7406\uff09\uff0c\u8fd9\u5c31\u50cf\u5728\u4f4e\u5206\u8fa8\u7387\u5730\u56fe\u4e2d\u5bfb\u627e\u65b9\u5f0f\/\u65b9\u5411\uff0c\u800c\u5728\u6d41\u5339\u914d\u4e2d\uff0c\u7eaf\u566a\u58f0\u548c\u6570\u636e\u7a7a\u95f4\u88ab\u5efa\u6a21\u4e3a lerp\uff08\u7ebf\u6027\u63d2\u503c\uff09\uff0c\u5f62\u5f0f\u4e3a x(t) = t*x(t-1) + (1-t)*x(0)\uff0c\u56e0\u6b64\uff0c\u5728 t=0 \u65f6\u6211\u4eec\u662f\u7eaf\u6570\u636e\u6837\u672c\uff0c\u800c\u5728 t=t \u65f6\u6211\u4eec\u662f\u7eaf\u566a\u58f0\uff0c\u8fd9\u4e3a\u6a21\u578b\u63d0\u4f9b\u4e86\u66f4\u7cbe\u7ec6\u7684\u72b6\u6001\/\u6d41\u5f62\uff0c\u56e0\u6b64\uff0c\u66f4\u5177\u4ee3\u8868\u6027\u7684\u9ad8\u5206\u8fa8\u7387\u5730\u56fe\uff0c\u4e5f\u79f0\u4e3a\u6761\u4ef6\u6d41\u5339\u914d\uff08\u901a\u8fc7\u566a\u58f0\u5206\u5e03\u8fdb\u884c\u6761\u4ef6\u8c03\u8282\uff09\u3002<\/p>\n\n\n\n<h3>Flow Matching \u5728\u8bed\u97f3\u751f\u6210\u4e2d\u7684\u5e94\u7528<\/h3>\n\n\n\n<p>\u5728\u8bed\u97f3\u751f\u6210\u4efb\u52a1\u4e2d\uff0cFlow Matching \u53ef\u4ee5\u7528\u4e8e\u5efa\u6a21\u4ece\u8bed\u97f3\u7279\u5f81\uff08\u5982\u6885\u5c14\u9891\u8c31\u56fe\u3001F0\u7b49\uff09\u5230\u6ce2\u5f62\u7684\u751f\u6210\u8fc7\u7a0b\u3002\u7531\u4e8e\u8bed\u97f3\u4fe1\u53f7\u662f\u9ad8\u5ea6\u590d\u6742\u4e14\u5177\u6709\u975e\u7ebf\u6027\u7684\u65f6\u5e8f\u6570\u636e\uff0cFlow Matching \u53ef\u4ee5\u6709\u6548\u5730\u6355\u6349\u8bed\u97f3\u6570\u636e\u4e2d\u7684\u590d\u6742\u5206\u5e03\uff0c\u5e76\u751f\u6210\u9ad8\u8d28\u91cf\u7684\u8bed\u97f3\u6837\u672c\u3002<\/p>\n\n\n\n<h4>\u6b65\u9aa4\u6982\u8ff0\uff1a<\/h4>\n\n\n\n<ol><li><strong>\u8f93\u5165\u7279\u5f81\u63d0\u53d6<\/strong>\uff1a\u9996\u5148\uff0c\u5c06\u8bed\u97f3\u6570\u636e\u8f6c\u6362\u4e3a\u7279\u5f81\u8868\u793a\uff0c\u5982\u6885\u5c14\u9891\u8c31\u56fe\u6216\u5176\u5b83\u58f0\u5b66\u7279\u5f81\u3002<\/li><li><strong>Flow Model \u6784\u5efa<\/strong>\uff1a\u5efa\u7acb\u4e00\u4e2a\u57fa\u4e8e\u6d41\u7684\u751f\u6210\u6a21\u578b\uff0c\u901a\u8fc7\u4e00\u7cfb\u5217\u53ef\u9006\u53d8\u6362\u5c06\u7b80\u5355\u7684\u5206\u5e03\uff08\u5982\u6807\u51c6\u9ad8\u65af\u5206\u5e03\uff09\u6620\u5c04\u5230\u8bed\u97f3\u6570\u636e\u7684\u5206\u5e03\u3002<\/li><li><strong>Flow Matching \u8bad\u7ec3<\/strong>\uff1a\u901a\u8fc7 Flow Matching \u6280\u672f\uff0c\u6700\u5c0f\u5316\u6a21\u578b\u751f\u6210\u7684\u8bed\u97f3\u5206\u5e03\u4e0e\u771f\u5b9e\u8bed\u97f3\u6570\u636e\u5206\u5e03\u4e4b\u95f4\u7684\u5dee\u5f02\uff0c\u9010\u6b65\u5b66\u4e60\u751f\u6210\u9ad8\u8d28\u91cf\u8bed\u97f3\u4fe1\u53f7\u7684\u80fd\u529b\u3002<\/li><li><strong>\u8bed\u97f3\u751f\u6210<\/strong>\uff1a\u8bad\u7ec3\u5b8c\u6210\u540e\uff0c\u4f7f\u7528\u6a21\u578b\u751f\u6210\u8bed\u97f3\u4fe1\u53f7\uff0c\u4ece\u8f93\u5165\u7279\u5f81\u6620\u5c04\u5230\u8bed\u97f3\u6ce2\u5f62\u3002<\/li><\/ol>\n\n\n\n<h3>\u4f18\u70b9<\/h3>\n\n\n\n<ul><li><strong>\u9ad8\u8d28\u91cf\u751f\u6210<\/strong>\uff1a\u7531\u4e8e Flow Matching \u80fd\u591f\u7cbe\u786e\u5730\u5efa\u6a21\u590d\u6742\u7684\u5206\u5e03\uff0c\u751f\u6210\u7684\u8bed\u97f3\u5f80\u5f80\u66f4\u81ea\u7136\u3001\u66f4\u63a5\u8fd1\u771f\u5b9e\u8bed\u97f3\u3002<\/li><li><strong>\u7a33\u5b9a\u6027<\/strong>\uff1a\u76f8\u6bd4\u4e00\u4e9b\u5176\u4ed6\u751f\u6210\u65b9\u6cd5\uff08\u5982 GANs\uff09\uff0cFlow Matching \u66f4\u52a0\u7a33\u5b9a\uff0c\u907f\u514d\u4e86\u6a21\u5f0f\u5d29\u6e83\u7b49\u95ee\u9898\u3002<\/li><li><strong>\u53ef\u9006\u6027<\/strong>\uff1aFlow \u6a21\u578b\u7684\u53ef\u9006\u6027\u4f7f\u5f97\u5bf9\u751f\u6210\u8fc7\u7a0b\u7684\u63a7\u5236\u66f4\u52a0\u7075\u6d3b\u548c\u7cbe\u786e\u3002<\/li><\/ul>\n\n\n\n<h3>\u5e94\u7528\u573a\u666f<\/h3>\n\n\n\n<p>Flow Matching for Speech Generation \u5728\u6587\u672c\u5230\u8bed\u97f3\uff08TTS\uff09\u7cfb\u7edf\u3001\u8bed\u97f3\u8f6c\u6362\uff08Voice Conversion\uff09\u3001\u8bed\u97f3\u589e\u5f3a\uff08Speech Enhancement\uff09\u7b49\u9886\u57df\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u524d\u666f\u3002<\/p>\n\n\n\n<p>\u901a\u8fc7 Flow Matching\uff0c\u8bed\u97f3\u751f\u6210\u6a21\u578b\u53ef\u4ee5\u66f4\u6709\u6548\u5730\u5904\u7406\u590d\u6742\u7684\u8bed\u97f3\u4fe1\u53f7\uff0c\u63d0\u4f9b\u66f4\u9ad8\u8d28\u91cf\u7684\u8f93\u51fa\u3002\u8fd9\u79cd\u65b9\u6cd5\u6b63\u9010\u6b65\u6210\u4e3a\u8bed\u97f3\u751f\u6210\u6280\u672f\u53d1\u5c55\u7684\u91cd\u8981\u65b9\u5411\u4e4b\u4e00\u3002<\/p>\n\n\n\n<p class=\"has-light-pink-background-color has-background\"><a href=\"https:\/\/blog.csdn.net\/weixin_44966641\/article\/details\/139842872\">https:\/\/blog.csdn.net\/weixin_44966641\/article\/details\/139842872<\/a><\/p>\n\n\n\n<p><strong>\u6761\u4ef6\u6d41\u5339\u914d<\/strong>\uff08Conditional Flow Matching, CFM\uff09<\/p>\n\n\n\n<p>\u6761\u4ef6\u6d41\u5339\u914d\u662f\u4e00\u79cd\u751f\u6210\u6a21\u578b\u65b9\u6cd5\uff0c\u65e8\u5728\u5c06\u6761\u4ef6\u4fe1\u606f\uff08\u5982\u6807\u7b7e\u3001\u7279\u5f81\u7b49\uff09\u548c\u751f\u6210\u7684\u6570\u636e\u5206\u5e03\u5339\u914d\u8d77\u6765\u3002\u5b83\u901a\u8fc7\u6761\u4ef6\u4fe1\u606f\u6765\u6307\u5bfc\u751f\u6210\u8fc7\u7a0b\uff0c\u4f7f\u5f97\u751f\u6210\u6a21\u578b\u80fd\u591f\u751f\u6210\u4e0e\u7ed9\u5b9a\u6761\u4ef6\u4e00\u81f4\u7684\u6570\u636e\u3002<\/p>\n\n\n\n<p><strong>\u6700\u4f18\u4f20\u8f93\u6761\u4ef6\u6d41\u5339\u914d\u6a21\u578b\uff08OT-CFM\uff09<\/strong><\/p>\n\n\n\n<p>OT-CFM \u7ed3\u5408\u4e86\u6700\u4f18\u4f20\u8f93\u548c\u6761\u4ef6\u6d41\u5339\u914d\u7684\u4f18\u52bf\uff0c\u901a\u8fc7\u6700\u4f18\u4f20\u8f93\u7406\u8bba\u63d0\u4f9b\u7684\u8ddd\u79bb\u5ea6\u91cf\uff08\u5982 Wasserstein \u8ddd\u79bb\uff09\uff0c\u4ee5\u6307\u5bfc\u751f\u6210\u6a21\u578b\u5728\u590d\u6742\u6761\u4ef6\u4e0b\u5339\u914d\u76ee\u6807\u5206\u5e03\u3002\u5177\u4f53\u6765\u8bf4\uff0cOT-CFM \u7684\u5de5\u4f5c\u65b9\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n\n<ol><li><strong>\u5efa\u6a21\u76ee\u6807<\/strong>\uff1aOT-CFM \u7684\u76ee\u6807\u662f\u5b66\u4e60\u4e00\u4e2a\u751f\u6210\u6a21\u578b\uff0c\u4f7f\u5f97\u5728\u7ed9\u5b9a\u6761\u4ef6\u4e0b\uff0c\u751f\u6210\u7684\u6570\u636e\u5206\u5e03\u4e0e\u76ee\u6807\u5206\u5e03\u4e4b\u95f4\u7684\u6700\u4f18\u4f20\u8f93\u8ddd\u79bb\u6700\u5c0f\u5316\u3002\u8fd9\u6837\u53ef\u4ee5\u4fdd\u8bc1\u751f\u6210\u7684\u6570\u636e\u4e0d\u4ec5\u771f\u5b9e\uff0c\u800c\u4e14\u6ee1\u8db3\u6761\u4ef6\u8981\u6c42\u3002<\/li><li><strong>\u6d41\u5339\u914d<\/strong>\uff1a\u6a21\u578b\u901a\u8fc7\u6761\u4ef6\u6d41\u5339\u914d\u6280\u672f\uff0c\u786e\u4fdd\u751f\u6210\u8fc7\u7a0b\u53d7\u6761\u4ef6\u4fe1\u606f\u7684\u63a7\u5236\uff0c\u9010\u6b65\u8c03\u6574\u751f\u6210\u8fc7\u7a0b\uff0c\u4f7f\u5176\u66f4\u8d34\u8fd1\u76ee\u6807\u5206\u5e03\u3002<\/li><li><strong>\u4f18\u5316\u8fc7\u7a0b<\/strong>\uff1a\u901a\u8fc7\u6700\u4f18\u4f20\u8f93\u7684\u8ddd\u79bb\u5ea6\u91cf\uff08\u5982 Wasserstein \u8ddd\u79bb\uff09\u6765\u4f18\u5316\u751f\u6210\u6a21\u578b\uff0c\u4f7f\u5f97\u6a21\u578b\u751f\u6210\u7684\u6570\u636e\u5206\u5e03\u4e0e\u76ee\u6807\u6570\u636e\u5206\u5e03\u4e4b\u95f4\u7684\u4f20\u8f93\u6210\u672c\u6700\u5c0f\u5316\u3002<\/li><\/ol>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8bba\u6587\uff1ahttps:\/\/arxiv.org\/abs\/2210.02747 \u535a\u5ba2\uff1ahttps:\/\/lilianwe &hellip; <a href=\"http:\/\/139.9.1.231\/index.php\/2024\/12\/20\/the-flow-matching-model\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">The flow matching model:\u6d41\u5339\u914d\u6a21\u578b<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4,38,34],"tags":[],"_links":{"self":[{"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/posts\/17806"}],"collection":[{"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/comments?post=17806"}],"version-history":[{"count":34,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/posts\/17806\/revisions"}],"predecessor-version":[{"id":23772,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/posts\/17806\/revisions\/23772"}],"wp:attachment":[{"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/media?parent=17806"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/categories?post=17806"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/139.9.1.231\/index.php\/wp-json\/wp\/v2\/tags?post=17806"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}