{"id":5532,"date":"2025-01-10T12:07:10","date_gmt":"2025-01-10T04:07:10","guid":{"rendered":"https:\/\/nullthought.net\/?p=5532"},"modified":"2025-01-10T12:07:11","modified_gmt":"2025-01-10T04:07:11","slug":"pcnde%e6%a1%86%e6%9e%b6%ef%bc%88physics-constrained-coupled-neural-differential-equations%ef%bc%8c%e7%89%a9%e7%90%86%e7%ba%a6%e6%9d%9f%e7%a5%9e%e7%bb%8f%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b%ef%bc%89","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=5532","title":{"rendered":"PCNDE\u6846\u67b6\uff08Physics-constrained coupled neural differential equations\uff0c\u7269\u7406\u7ea6\u675f\u795e\u7ecf\u5fae\u5206\u65b9\u7a0b\uff09\uff1a\u5b9e\u73b0\u8f83\u9ad8\u7cbe\u5ea6\u548c\u8ba1\u7b97\u6548\u7387\u76841D\u8840\u6d41\u6a21\u578b"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2411.05631\" target=\"_blank\" rel=\"noreferrer noopener\">Physics-constrained coupled neural differential equations for one dimensional blood flow modeling<\/a><\/strong>\u63d0\u51fa\u4e86PCNDE\u6846\u67b6\uff08Physics-constrained coupled neural differential equations\uff0c\u7269\u7406\u7ea6\u675f\u795e\u7ecf\u5fae\u5206\u65b9\u7a0b\u6846\u67b6\uff09\uff0c\u8be5\u6846\u67b6\u5728\u63d0\u9ad81D\u8840\u6d41\u6a21\u578b\u7cbe\u5ea6\u7684\u540c\u65f6\uff0c\u4fdd\u6301\u4e86\u8f83\u9ad8\u8ba1\u7b97\u6548\u7387\uff0c\u6210\u529f\u5f25\u8865\u4e86\u4f20\u7edf1D FEM\u6a21\u578b\u5728\u590d\u6742\u8840\u6d41\u60c5\u51b5\u4e0b\u7684\u4e0d\u8db3\u3002\u901a\u8fc7\u7a7a\u95f4\u57df\u7684\u91cd\u6784\u548c\u5085\u91cc\u53f6\u5468\u671f\u6027\u7ea6\u675f\uff0cPCNDE\u5728\u7cbe\u5ea6\u548c\u7a33\u5b9a\u6027\u65b9\u9762\u5747\u8868\u73b0\u51fa\u4e86\u8f83\u4f73\u6027\u80fd\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aHunor Csala, Arvind Mohan, Daniel Livescu, Amirhossein Arzani\uff0c\u6765\u81eaUniversity of Utah\u548cLos Alamos National Laboratory\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"881\" height=\"549\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-10.png\" alt=\"\" class=\"wp-image-5533\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-10.png 881w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-10-300x187.png 300w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-10-768x479.png 768w\" sizes=\"auto, (max-width: 881px) 100vw, 881px\" \/><figcaption class=\"wp-element-caption\"><strong><a href=\"https:\/\/arxiv.org\/abs\/2411.05631\" target=\"_blank\" rel=\"noreferrer noopener\">Physics-constrained coupled neural differential equations for one dimensional blood flow modeling<\/a><\/strong><\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e00\u3001\u7814\u7a76\u80cc\u666f\u4e0e\u76ee\u6807<\/h4>\n\n\n\n<p>\u5fc3\u8840\u7ba1\u75be\u75c5\u662f\u5168\u7403\u8303\u56f4\u5185\u4e3b\u8981\u7684\u6b7b\u4ea1\u539f\u56e0\u4e4b\u4e00\uff0c\u7406\u89e3\u8840\u6db2\u6d41\u52a8\u7684\u52a8\u529b\u5b66\u7279\u6027\u5bf9\u4e8e\u4e34\u5e8a\u8bca\u65ad\u4e0e\u6cbb\u7597\u81f3\u5173\u91cd\u8981\u3002\u4f20\u7edf\u7684\u8840\u6d41\u52a8\u529b\u5b66\u5efa\u6a21\u65b9\u6cd5\uff0c\u5305\u62ec\u8be6\u7ec6\u7684\u4e09\u7ef4\u6d41\u4f53-\u7ed3\u6784\u8026\u5408\uff08FSI\uff09\u6a21\u578b\uff0c\u80fd\u591f\u63d0\u4f9b\u51c6\u786e\u7684\u8840\u6d41\u7ec6\u8282\uff0c\u4f46\u8ba1\u7b97\u4ee3\u4ef7\u6781\u9ad8\uff0c\u9650\u5236\u4e86\u5176\u5728\u4e34\u5e8a\u4e2d\u7684\u5e7f\u6cdb\u5e94\u7528\u3002\u4e3a\u4e86\u63d0\u9ad8\u8ba1\u7b97\u6548\u7387\u5e76\u5728\u4fdd\u6301\u4e00\u5b9a\u51c6\u786e\u6027\u7684\u540c\u65f6\u4f7f\u6a21\u578b\u9002\u7528\u4e8e\u5b9e\u65f6\u5206\u6790\uff0c\u7b80\u5316\u6a21\u578b\u6210\u4e3a\u4e00\u4e2a\u5fc5\u8981\u65b9\u5411\u3002<\/p>\n\n\n\n<p>1D\u8840\u6d41\u6a21\u578b\u901a\u8fc7\u5bf9Navier-Stokes\u65b9\u7a0b\u8fdb\u884c\u622a\u9762\u5e73\u5747\u5316\u7b80\u5316\u4e86\u590d\u6742\u7684\u4e09\u7ef4\u65b9\u7a0b\u7cfb\u7edf\uff0c\u901a\u5e38\u7528\u4e8e\u6a21\u62df\u8840\u7ba1\u5185\u7684\u8840\u6db2\u6d41\u52a8\u3002\u5c3d\u7ba11D\u6a21\u578b\u8ba1\u7b97\u4e0a\u66f4\u52a0\u9ad8\u6548\uff0c\u4f46\u5176\u7cbe\u5ea6\u76f8\u8f83\u4e8e3D\u6a21\u578b\u8f83\u4f4e\uff0c\u5c24\u5176\u5728\u590d\u6742\u51e0\u4f55\u7ed3\u6784\uff08\u5982\u52a8\u8109\u72ed\u7a84\uff09\u4e0b\uff0c\u4f20\u7edf\u7684\u6709\u9650\u5143\u65b9\u6cd5\uff08FEM\uff091D\u6a21\u578b\u5f80\u5f80\u65e0\u6cd5\u6709\u6548\u6355\u6349\u8fd9\u4e9b\u590d\u6742\u6d41\u52a8\u60c5\u51b5\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\uff0c\u672c\u6587\u63d0\u51fa\u4e86\u4e00\u79cd\u65b0\u578b\u7684\u7269\u7406\u7ea6\u675f\u795e\u7ecf\u5fae\u5206\u65b9\u7a0b\uff08PCNDE\uff09\u6846\u67b6\uff0c\u65e8\u5728\u901a\u8fc7\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\u63d0\u9ad81D\u8840\u6d41\u6a21\u578b\u7684\u7cbe\u5ea6\uff0c\u540c\u65f6\u4fdd\u6301\u5176\u8ba1\u7b97\u6548\u7387\u3002\u8be5\u65b9\u6cd5\u7ed3\u5408\u4e86\u7269\u7406\u6a21\u578b\u4e0e\u6570\u636e\u9a71\u52a8\u65b9\u6cd5\uff0c\u521b\u65b0\u6027\u5730\u5c06\u8840\u6d41\u52a8\u529b\u5b66\u7684\u7269\u7406\u7ea6\u675f\u4e0e\u795e\u7ecf\u7f51\u7edc\u76f8\u7ed3\u5408\uff0c\u4ece\u800c\u5728\u8ba1\u7b97\u901f\u5ea6\u548c\u51c6\u786e\u5ea6\u4e4b\u95f4\u5b9e\u73b0\u5e73\u8861\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e8c\u3001\u65b9\u6cd5\u4e0e\u6846\u67b6<\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">1. \u7269\u7406\u6a21\u578b\u4e0e\u6570\u636e\u751f\u6210<\/h5>\n\n\n\n<p>\u4f20\u7edf\u76841D\u8840\u6d41\u6a21\u578b\u901a\u8fc7\u5bf93D Navier-Stokes\u65b9\u7a0b\u8fdb\u884c\u622a\u9762\u5e73\u5747\u7b80\u5316\uff0c\u5f97\u5230\u5982\u4e0b\u4e24\u6761\u4e3b\u8981\u7684\u65b9\u7a0b\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u8fde\u7eed\u6027\u65b9\u7a0b<\/strong>\uff1a\u63cf\u8ff0\u4e86\u8840\u6d41\u901f\u7387\uff08Q\uff09\u4e0e\u8840\u7ba1\u622a\u9762\u79ef\uff08S\uff09\u4e4b\u95f4\u7684\u5173\u7cfb\uff1a<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"169\" height=\"77\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-11.png\" alt=\"\" class=\"wp-image-5534\"\/><\/figure>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u52a8\u91cf\u65b9\u7a0b<\/strong>\uff1a\u901a\u8fc7\u52a8\u91cf\u5b88\u6052\u8ba1\u7b97\u538b\u529b\u53d8\u5316\uff1a<\/li>\n<\/ul>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"547\" height=\"70\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-12.png\" alt=\"\" class=\"wp-image-5535\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-12.png 547w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/01\/image-12-300x38.png 300w\" sizes=\"auto, (max-width: 547px) 100vw, 547px\" \/><\/figure>\n\n\n\n<p>\u5176\u4e2d\uff0cQ\u662f\u6d41\u91cf\uff0cS\u662f\u8840\u7ba1\u622a\u9762\u79ef\uff0c\u03c1\u662f\u8840\u6db2\u5bc6\u5ea6\uff0c\u03bd\u662f\u8840\u6db2\u7684\u8fd0\u52a8\u7c98\u5ea6\uff0cp\u662f\u538b\u529b\uff0ct\u662f\u65f6\u95f4\uff0cz\u662f\u6cbf\u8840\u7ba1\u8f74\u5411\u7684\u5750\u6807\u3002<\/p>\n\n\n\n<p>\u4e3a\u8bad\u7ec3PCNDE\u6a21\u578b\uff0c\u672c\u6587\u4f7f\u7528SimVascular\u8fdb\u884c100\u4e2a\u4e0d\u540c3D FSI\u6a21\u62df\uff0c\u6db5\u76d6\u4e8610\u79cd\u4e0d\u540c\u7684\u52a8\u8109\u72ed\u7a84\u6bd4\u4f8b\uff08\u03b2\uff09\u548c10\u79cd\u4e0d\u540c\u7684\u6d41\u5165\u8fb9\u754c\u6761\u4ef6\u3002\u901a\u8fc7\u8fd9\u4e9b\u6a21\u62df\u6570\u636e\u751f\u6210\u8bad\u7ec3\u96c6\uff0c\u4e3a\u540e\u7eed\u7684PCNDE\u6a21\u578b\u63d0\u4f9b\u57fa\u7840\u6570\u636e\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">2. PCNDE\u6846\u67b6<\/h5>\n\n\n\n<p>PCNDE\u6846\u67b6\u7684\u6838\u5fc3\u521b\u65b0\u5728\u4e8e\u5c06\u4f20\u7edf\u7684\u65f6\u95f4\u57df\u795e\u7ecf\u5fae\u5206\u65b9\u7a0b\uff08NDE\uff09\u91cd\u65b0\u8868\u8ff0\u4e3a\u7a7a\u95f4\u57df\u7684\u795e\u7ecf\u5fae\u5206\u65b9\u7a0b\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u672c\u6587\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u5b9e\u73b0\u4e86PCNDE\u7684\u8ba1\u7b97\u6846\u67b6\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u7a7a\u95f4\u57df\u52a8\u91cf\u65b9\u7a0b<\/strong>\uff1a\u901a\u8fc7\u795e\u7ecf\u7f51\u7edc\u62df\u5408\u52a8\u91cf\u65b9\u7a0b\u4e2d\u7684\u672a\u77e5\u9879\uff0c\u5e76\u5c06\u65b9\u7a0b\u8f6c\u6362\u4e3a\u7a7a\u95f4\u5f62\u5f0f\u8fdb\u884c\u6c42\u89e3\uff0c\u89e3\u51b3\u4e86\u4f20\u7edf\u65f6\u95f4\u57df\u65b9\u6cd5\u4e2d\u884d\u751f\u4e0d\u5e73\u6ed1\u7684\u95ee\u9898\u3002<\/li>\n\n\n\n<li><strong>\u5085\u91cc\u53f6\u7ea7\u6570\u7ea6\u675f<\/strong>\uff1a\u4e3a\u786e\u4fdd\u8840\u7ba1\u622a\u9762\u79ef\u7684\u65f6\u95f4\u5468\u671f\u6027\uff0c\u4f7f\u7528\u5085\u91cc\u53f6\u7ea7\u6570\u5bf9\u9762\u79ef\u7684\u9884\u6d4b\u503c\u8fdb\u884c\u4fee\u6b63\u3002\u8fd9\u6837\u53ef\u4ee5\u5f3a\u5236\u6267\u884c\u8840\u6d41\u5468\u671f\u7684\u5468\u671f\u6027\u7279\u5f81\u3002<\/li>\n\n\n\n<li><strong>\u538b\u529b\u6a21\u578b<\/strong>\uff1a\u538b\u529b\u65b9\u7a0b\u91c7\u7528\u4e24\u9636\u6bb5\u6a21\u578b\uff1a\u7b2c\u4e00\u9636\u6bb5\u4f7f\u7528\u795e\u7ecf\u7f51\u7edc\u62df\u5408\u538b\u529b\u7684\u65f6\u95f4\u53d8\u5316\uff0c\u7b2c\u4e8c\u9636\u6bb5\u5219\u91c7\u7528\u89e3\u6790\u6a21\u578b\u6355\u6349\u7a7a\u95f4\u4e0a\u7684\u538b\u529b\u53d8\u5316\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u8fd9\u4e00\u6846\u67b6\u7684\u76ee\u6807\u662f\u901a\u8fc7\u7ed3\u5408\u7269\u7406\u77e5\u8bc6\u548c\u6570\u636e\u9a71\u52a8\u7684\u5b66\u4e60\uff0c\u63d0\u9ad81D\u6a21\u578b\u7684\u7cbe\u5ea6\uff0c\u540c\u65f6\u4fdd\u63011D\u6a21\u578b\u5728\u8ba1\u7b97\u4e0a\u7684\u4f18\u52bf\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e09\u3001\u5b9e\u9a8c\u7ed3\u679c\u4e0e\u5206\u6790<\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">1. \u6d41\u91cf\u3001\u622a\u9762\u79ef\u548c\u538b\u529b\u7684\u9884\u6d4b\u7ed3\u679c<\/h5>\n\n\n\n<p>PCNDE\u6a21\u578b\u5728\u6d41\u91cf\u3001\u622a\u9762\u79ef\u548c\u538b\u529b\u9884\u6d4b\u65b9\u9762\u8868\u73b0\u51fa\u663e\u8457\u4f18\u4e8e\u4f20\u7edf1D FEM\u6a21\u578b\u7684\u7cbe\u5ea6\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u6d41\u91cf\u9884\u6d4b<\/strong>\uff1aPCNDE\u6a21\u578b\u5728\u6d4b\u8bd5\u96c6\u4e0a\u7684\u8bef\u5dee\u76f8\u6bd4\u4f20\u7edf1D FEM\u5c0f3\u81f35\u500d\uff0c\u80fd\u8f83\u597d\u5730\u6355\u6349\u8840\u6d41\u7684\u65f6\u95f4\u548c\u7a7a\u95f4\u53d8\u5316\u3002<\/li>\n\n\n\n<li><strong>\u622a\u9762\u79ef\u9884\u6d4b<\/strong>\uff1aPCNDE\u80fd\u591f\u8f83\u597d\u5730\u9884\u6d4b\u8840\u7ba1\u622a\u9762\u79ef\u7684\u53d8\u5316\uff0c\u5c24\u5176\u662f\u5728\u52a8\u8109\u72ed\u7a84\u5904\uff0c\u4f46\u5728\u5468\u671f\u7684\u5f00\u59cb\u548c\u7ed3\u675f\u9636\u6bb5\u6709\u4e00\u5b9a\u7684\u8fc7\u5ea6\u4f30\u8ba1\u3002<\/li>\n\n\n\n<li><strong>\u538b\u529b\u9884\u6d4b<\/strong>\uff1aPCNDE\u80fd\u591f\u66f4\u51c6\u786e\u5730\u6a21\u62df\u538b\u529b\u53d8\u5316\uff0c\u5c24\u5176\u662f\u5728\u8f83\u9ad8\u538b\u529b\u533a\uff0c\u4f20\u7edf1D FEM\u6a21\u578b\u5728\u6b64\u8868\u73b0\u4e0d\u4f73\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u6b64\u5916\uff0cPCNDE\u6a21\u578b\u7684\u9884\u6d4b\u7ed3\u679c\u4e0e3D\u6a21\u578b\u7684\u5e73\u5747\u6570\u636e\u76f8\u6bd4\uff0c\u6700\u5927\u8bef\u5dee\u4e0d\u8d85\u8fc71.2%\uff0c\u8868\u660e\u5176\u5728\u5904\u7406\u771f\u5b9e\u4e16\u754c\u4e2d\u590d\u6742\u7684\u8840\u6d41\u6a21\u578b\u65f6\u5177\u6709\u8f83\u9ad8\u7684\u9c81\u68d2\u6027\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">2. \u7a33\u5b9a\u6027\u4e0e\u9c81\u68d2\u6027\u5206\u6790<\/h5>\n\n\n\n<p>\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0cPCNDE\u6a21\u578b\u5c55\u793a\u4e86\u826f\u597d\u7684\u7a33\u5b9a\u6027\uff0c\u5c24\u5176\u662f\u5728\u901a\u8fc7\u7a7a\u95f4\u57df\u7684\u8f6c\u6362\u5904\u7406\u52a8\u91cf\u65b9\u7a0b\u540e\uff0c\u6a21\u578b\u5728\u8026\u5408\u7a33\u5b9a\u6027\u548c\u5149\u6ed1\u6027\u65b9\u9762\u663e\u8457\u6539\u8fdb\u3002\u5c3d\u7ba1\u521d\u671f\u7684\u65f6\u95f4\u57df\u6a21\u578b\u5728\u6d41\u901f\u7684\u7a7a\u95f4\u5bfc\u6570\u4e0a\u5b58\u5728\u4e0d\u5e73\u6ed1\u95ee\u9898\uff0c\u4f46\u8fd9\u79cd\u95ee\u9898\u901a\u8fc7\u7a7a\u95f4\u57df\u7684\u8f6c\u6362\u5f97\u5230\u4e86\u6709\u6548\u89e3\u51b3\u3002\u6a21\u578b\u5bf9\u8f93\u5165\u6570\u636e\u7684\u566a\u58f0\u4e5f\u8868\u73b0\u51fa\u4e86\u8f83\u5f3a\u7684\u9c81\u68d2\u6027\uff0c\u901a\u8fc7\u5728\u8f93\u5165\u8fb9\u754c\u6761\u4ef6\u4e2d\u6dfb\u52a0\u566a\u58f0\uff0cPCNDE\u7684\u9884\u6d4b\u7ed3\u679c\u4ecd\u7136\u80fd\u591f\u4fdd\u6301\u4e0e\u771f\u5b9e\u6570\u636e\u76f8\u4f3c\u7684\u7cbe\u5ea6\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3. \u8bef\u5dee\u5206\u6790<\/h5>\n\n\n\n<p>PCNDE\u6a21\u578b\u5728\u6d4b\u8bd5\u96c6\u4e2d\u7684\u76f8\u5bf9\u8bef\u5dee\u660e\u663e\u4f4e\u4e8e1D FEM\u6a21\u578b\u3002\u5c24\u5176\u662f\u5728\u6d41\u91cf\u548c\u538b\u529b\u9884\u6d4b\u65b9\u9762\uff0cPCNDE\u6a21\u578b\u7684\u8bef\u5dee\u5747\u4f4e\u4e8e1%\uff0c\u800c\u4f20\u7edf\u76841D FEM\u6a21\u578b\u5219\u5b58\u5728\u66f4\u5927\u8bef\u5dee\u3002\u56fe\u8868\u4e2d\u7684\u8bef\u5dee\u5206\u5e03\u663e\u793a\uff0c\u6700\u5927\u8bef\u5dee\u4e3b\u8981\u96c6\u4e2d\u5728\u52a8\u8109\u72ed\u7a84\u533a\u57df\u548c\u7279\u6b8a\u7684\u6d41\u5165\u6ce2\u5f62\u6761\u4ef6\u4e0b\uff0c\u8fd9\u8868\u660ePCNDE\u6a21\u578b\u80fd\u591f\u5728\u8fd9\u4e9b\u590d\u6742\u60c5\u51b5\u4e0b\u63d0\u4f9b\u66f4\u7cbe\u786e\u7684\u9884\u6d4b\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u56db\u3001\u8ba8\u8bba\u4e0e\u672a\u6765\u5c55\u671b<\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">1. \u65b9\u6cd5\u7684\u4f18\u52bf<\/h5>\n\n\n\n<p>PCNDE\u6846\u67b6\u7ed3\u5408\u4e86\u7269\u7406\u7ea6\u675f\u548c\u6570\u636e\u9a71\u52a8\u65b9\u6cd5\uff0c\u6210\u529f\u5730\u5f25\u8865\u4e86\u4f20\u7edf1D\u6a21\u578b\u5728\u7cbe\u5ea6\u4e0a\u7684\u4e0d\u8db3\u3002\u901a\u8fc7\u91c7\u7528\u5085\u91cc\u53f6\u7ea7\u6570\u5f3a\u5236\u65f6\u95f4\u5468\u671f\u6027\uff0cPCNDE\u4e0d\u4ec5\u63d0\u5347\u4e86\u6d41\u91cf\u548c\u538b\u529b\u7684\u9884\u6d4b\u51c6\u786e\u6027\uff0c\u8fd8\u5728\u8ba1\u7b97\u6548\u7387\u4e0a\u4fdd\u6301\u4e861D\u6a21\u578b\u7684\u4f18\u52bf\u3002\u8be5\u65b9\u6cd5\u7279\u522b\u9002\u7528\u4e8e\u9700\u8981\u5feb\u901f\u9884\u6d4b\u7684\u4e34\u5e8a\u5e94\u7528\uff0c\u80fd\u591f\u5728\u4e0d\u727a\u7272\u7cbe\u5ea6\u7684\u60c5\u51b5\u4e0b\u63d0\u4f9b\u5b9e\u65f6\u6a21\u62df\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">2. \u5f53\u524d\u6311\u6218\u4e0e\u4e0d\u8db3<\/h5>\n\n\n\n<p>\u5c3d\u7ba1PCNDE\u5728\u5927\u591a\u6570\u6d4b\u8bd5\u60c5\u51b5\u4e0b\u8868\u73b0\u826f\u597d\uff0c\u4f46\u4ecd\u9762\u4e34\u4e00\u4e9b\u6311\u6218\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u590d\u6742\u51e0\u4f55\u7ed3\u6784<\/strong>\uff1a\u6a21\u578b\u4e3b\u8981\u9488\u5bf9\u7406\u60f3\u5316\u7684\u52a8\u8109\u51e0\u4f55\uff0c\u5728\u5904\u7406\u5177\u6709\u66f4\u590d\u6742\u5f2f\u66f2\u6216\u591a\u5206\u652f\u7ed3\u6784\u65f6\uff0c\u53ef\u80fd\u9700\u8981\u8fdb\u4e00\u6b65\u8c03\u6574\u548c\u4f18\u5316\u3002<\/li>\n\n\n\n<li><strong>\u591a\u5c3a\u5ea6\u5efa\u6a21\u95ee\u9898<\/strong>\uff1a\u8840\u6d41\u52a8\u529b\u5b66\u662f\u4e00\u4e2a\u5178\u578b\u7684\u591a\u5c3a\u5ea6\u95ee\u9898\uff0c\u65f6\u95f4\u548c\u7a7a\u95f4\u7684\u53d8\u5316\u5e45\u5ea6\u76f8\u5dee\u60ac\u6b8a\u3002\u5728\u5904\u7406\u5982\u52a8\u8109\u72ed\u7a84\u7b49\u95ee\u9898\u65f6\uff0c\u5982\u4f55\u5728\u7a7a\u95f4\u548c\u65f6\u95f4\u5c3a\u5ea6\u4e0a\u66f4\u7cbe\u786e\u5730\u62df\u5408\u4ecd\u662f\u4e00\u4e2a\u6311\u6218\u3002<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">3. \u672a\u6765\u53d1\u5c55\u65b9\u5411<\/h5>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u60a3\u8005\u7279\u5f02\u6027\u51e0\u4f55\u7ed3\u6784\u7684\u5904\u7406<\/strong>\uff1a\u672a\u6765\u7684\u7814\u7a76\u5c06\u8fdb\u4e00\u6b65\u6269\u5c55PCNDE\u6846\u67b6\uff0c\u8003\u8651\u60a3\u8005\u4e2a\u4f53\u7684\u590d\u6742\u52a8\u8109\u51e0\u4f55\u548c\u53d8\u5f62\uff0c\u8fdb\u800c\u63d0\u9ad8\u5176\u5728\u4e34\u5e8a\u4e2d\u7684\u5b9e\u9645\u5e94\u7528\u4ef7\u503c\u3002<\/li>\n\n\n\n<li><strong>\u66f4\u5927\u89c4\u6a21\u7684\u8bad\u7ec3\u6570\u636e\u96c6<\/strong>\uff1a\u4e3a\u4e86\u63d0\u9ad8\u6a21\u578b\u7684\u6cdb\u5316\u80fd\u529b\uff0c\u672a\u6765\u9700\u8981\u6784\u5efa\u4e00\u4e2a\u6db5\u76d6\u66f4\u591a\u590d\u6742\u51e0\u4f55\u548c\u4e0d\u540c\u8fb9\u754c\u6761\u4ef6\u7684\u5927\u89c4\u6a21\u8bad\u7ec3\u6570\u636e\u96c6\u3002<\/li>\n\n\n\n<li><strong>\u4e0e\u81ea\u52a8\u5316\u68c0\u6d4b\u5de5\u5177\u7ed3\u5408<\/strong>\uff1a\u7ed3\u5408\u81ea\u52a8\u5316\u5de5\u5177\uff08\u5982\u57fa\u4e8e\u56fe\u5f62\u7684\u5b66\u4e60\u65b9\u6cd5\uff09\u6765\u66f4\u6709\u6548\u5730\u5904\u7406\u591a\u5206\u652f\u52a8\u8109\u7684\u5efa\u6a21\u95ee\u9898\uff0c\u63d0\u5347\u6a21\u578b\u7684\u9002\u5e94\u80fd\u529b\u548c\u666e\u9002\u6027\u3002<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>\u5bf9\u4e8e\u662f\u5426\u91c7\u7528\u4e86<a href=\"https:\/\/nullthought.net\/?p=4287\" target=\"_blank\" rel=\"noreferrer noopener\">\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edcPINNs\uff08Physics-Informed Neural Networks\uff09<\/a>\uff0c\u8bba\u6587\u4f5c\u8005<a href=\"https:\/\/www.linkedin.com\/in\/csalahunor\/\" target=\"_blank\" rel=\"noreferrer noopener\">Hunor Csala<\/a>\u6709\u5982\u4e0b<a href=\"https:\/\/www.linkedin.com\/feed\/update\/urn:li:activity:7282627628178579463?commentUrn=urn%3Ali%3Acomment%3A%28activity%3A7282627628178579463%2C7282847552939458560%29&amp;replyUrn=urn%3Ali%3Acomment%3A%28activity%3A7282627628178579463%2C7282870673151074304%29&amp;dashCommentUrn=urn%3Ali%3Afsd_comment%3A%287282847552939458560%2Curn%3Ali%3Aactivity%3A7282627628178579463%29&amp;dashReplyUrn=urn%3Ali%3Afsd_comment%3A%287282870673151074304%2Curn%3Ali%3Aactivity%3A7282627628178579463%29\" target=\"_blank\" rel=\"noreferrer noopener\">\u89e3\u91ca<\/a>\uff1a<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p class=\"has-small-font-size\">\u6211\u4eec\u4f7f\u7528\u4e86\u666e\u9002\u5fae\u5206\u65b9\u7a0b\uff08Rackauckas\u7b49\uff0c2020\uff09\u6846\u67b6\uff0c\u5c06\u795e\u7ecf\u7f51\u7edc\u548c\u5176\u4ed6\u53ef\u4f18\u5316\u7684\u5411\u91cf\u4f5c\u4e3a\u5fae\u5206\u65b9\u7a0b\u4e2d\u7684\u9879\u3002PDE\uff08\u504f\u5fae\u5206\u65b9\u7a0b\uff09\u901a\u8fc7\u7ecf\u5178\u7684\u6570\u503c\u65b9\u6cd5\u6c42\u89e3\uff0c\u53ea\u662f\u91c7\u7528\u53ef\u5fae\u7684\u65b9\u5f0f\u4ee5\u4fbf\u5141\u8bb8\u53cd\u5411\u4f20\u64ad\u3002\u5728\u635f\u5931\u51fd\u6570\u4e2d\uff0c\u7269\u7406\u7ea6\u675f\u5e76\u4e0d\u50cf\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINN\uff09\u4e2d\u90a3\u6837\u88ab\u5f3a\u5236\u6267\u884c\uff1b\u635f\u5931\u4ec5\u4ec5\u662f\u9884\u6d4b\u4e0e\u6570\u636e\u4e4b\u95f4\u7684\u5747\u65b9\u8bef\u5dee\uff08MSE\uff09\u3002\u6b64\u5916\uff0c\u795e\u7ecf\u7f51\u7edc\u7684\u8f93\u5165\u662f\u611f\u5174\u8da3\u7684\u53d8\u91cf\uff08\u4f8b\u5982\u6d41\u91cf\uff09\uff0c\u800c\u4e0d\u662f\u7a7a\u95f4\u548c\u65f6\u95f4\u3002\u56e0\u6b64\uff0c\u8fd9\u79cd\u65b9\u6cd5\u4e0ePINN\u6709\u5f88\u5927\u7684\u4e0d\u540c\u3002<\/p>\n<\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>\u8bba\u6587Physics-constrained coupled neural differential equat [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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equat&hellip;","_links":{"self":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5532","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5532"}],"version-history":[{"count":1,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5532\/revisions"}],"predecessor-version":[{"id":5536,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5532\/revisions\/5536"}],"wp:attachment":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5532"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5532"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5532"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}