{"id":5623,"date":"2025-02-09T15:11:28","date_gmt":"2025-02-09T07:11:28","guid":{"rendered":"https:\/\/nullthought.net\/?p=5623"},"modified":"2025-02-09T15:11:30","modified_gmt":"2025-02-09T07:11:30","slug":"%e5%88%a9%e7%94%a8%e6%b7%b7%e5%90%88%e6%95%b4%e6%95%b0%e7%ba%bf%e6%80%a7%e4%bc%98%e5%8c%96%ef%bc%88mixed-integer-linear-programming-milp%ef%bc%89%e6%94%b9%e8%bf%9b-pinns%e5%88%9d%e5%a7%8b%e6%9d%83","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=5623","title":{"rendered":"\u5229\u7528\u6df7\u5408\u6574\u6570\u7ebf\u6027\u4f18\u5316\uff08Mixed-Integer Linear Programming, MILP\uff09\u6539\u8fdb PINNs\u521d\u59cb\u6743\u91cd\uff0c\u63d0\u9ad8\u8bad\u7ec3\u6536\u655b\u901f\u5ea6\u548c\u7cbe\u5ea6"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><strong><a href=\"https:\/\/nullthought.net\/?p=4287\" target=\"_blank\" rel=\"noreferrer noopener\">\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08Physics-Informed Neural Networks\uff0cPINNs\uff09<\/a><\/strong>\u662f\u4e00\u79cd\u65b0\u5174\u7684\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\uff0c\u5b83\u7ed3\u5408\u4e86\u524d\u9988\u795e\u7ecf\u7f51\u7edc\uff08Feedforward Neural Network\uff0cFNN\uff09\u4e0e\u7528\u5e38\u5fae\u5206\u65b9\u7a0b\uff08ODE\uff09\u6216\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u8868\u793a\u7684\u7cfb\u7edf\u5148\u9a8c\u77e5\u8bc6\u3002\u8fd9\u79cd\u65b9\u6cd5\u5728\u6d41\u4f53\u7269\u7406\u3001\u52a8\u529b\u7cfb\u7edf\u3001\u80fd\u6e90\u7cfb\u7edf\u3001\u70ed\u6269\u6563\u7b49\u591a\u4e2a\u79d1\u5b66\u9886\u57df\u5f97\u5230\u4e86\u5e7f\u6cdb\u5e94\u7528\u3002PINNs \u7684\u4f18\u52bf\u5728\u4e8e\u80fd\u591f\u7ed3\u5408\u7269\u7406\u89c4\u5f8b\u8fdb\u884c\u5efa\u6a21\uff0c\u4f7f\u5176\u80fd\u591f\u5e94\u7528\u4e8e\u6c42\u89e3\u590d\u6742\u7684\u7269\u7406\u95ee\u9898\u3002\u7136\u800c\uff0cPINNs \u4ecd\u7136\u5b58\u5728\u8bad\u7ec3\u6536\u655b\u901f\u5ea6\u8f83\u6162\u7684\u95ee\u9898\uff0c\u5c24\u5176\u662f\u5728\u6d89\u53ca\u73b0\u5b9e\u590d\u6742\u7cfb\u7edf\u7684\u60c5\u51b5\u4e0b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5728 PINNs \u7684\u6807\u51c6\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\uff0c\u7f51\u7edc\u6743\u91cd\u7684\u521d\u59cb\u5316\u901a\u5e38\u91c7\u7528\u968f\u673a\u5206\u5e03\uff0c\u5982 Glorot\uff08Xavier\uff09\u521d\u59cb\u5316\u65b9\u6cd5\u3002\u7136\u800c\uff0c\u8fd9\u79cd\u968f\u673a\u521d\u59cb\u5316\u53ef\u80fd\u5bfc\u81f4\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u7684\u6536\u655b\u74f6\u9888\uff0c\u4f7f\u5f97\u7f51\u7edc\u96be\u4ee5\u6709\u6548\u6536\u655b\u751a\u81f3\u65e0\u6cd5\u6536\u655b\u3002\u56e0\u6b64\uff0c\u7814\u7a76\u8005\u4eec\u5f00\u59cb\u63a2\u7d22\u66f4\u4f18\u7684\u521d\u59cb\u5316\u65b9\u6cd5\uff0c\u4ee5\u63d0\u9ad8 PINNs \u7684\u6536\u655b\u901f\u5ea6\u548c\u7a33\u5b9a\u6027\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2501.16153\" target=\"_blank\" rel=\"noreferrer noopener\">MILP initialization for solving parabolic PDEs with PINNs<\/a><\/strong>\u63d0\u51fa\u4e86\u4e00\u79cd\u5229\u7528<strong>\u6df7\u5408\u6574\u6570\u7ebf\u6027\u89c4\u5212\uff08Mixed-Integer Linear Programming, MILP\uff09\u4f18\u5316<\/strong>\u7684\u65b9\u6cd5\u6765\u6539\u8fdb PINNs \u7684\u521d\u59cb\u6743\u91cd\u3002\u4f5c\u8005\u7814\u7a76\u4e86\u4e24\u79cd\u4e0d\u540c\u7684\u4f18\u5316\u7b56\u7565\u4f5c\u4e3a\u7f51\u7edc\u8bad\u7ec3\u7684\u9884\u8bad\u7ec3\u6b65\u9aa4\uff1a<strong>\uff081\uff09\u4ec5\u5305\u542b\u8fb9\u754c\u6761\u4ef6\u7684\u4f18\u5316\uff1b\uff082\uff09\u7ed3\u5408\u7269\u7406\u7ea6\u675f\u7684\u4f18\u5316<\/strong>\u3002\u4f18\u5316\u4e3b\u8981\u9488\u5bf9 PINNs \u7684\u7b2c\u4e00\u5c42\u6743\u91cd\uff0c\u800c\u5176\u4ed6\u5c42\u4ecd\u91c7\u7528\u968f\u673a\u521d\u59cb\u5316\u3002\u8bba\u6587\u91c7\u7528<strong>\u70ed\u6269\u6563\u65b9\u7a0b<\/strong>\u4f5c\u4e3a\u6d4b\u8bd5\u6848\u4f8b\uff0c\u6a21\u62df\u7535\u529b\u53d8\u538b\u5668\u7684\u6e29\u5ea6\u5206\u5e03\uff0c\u5e76\u5206\u6790\u4e86\u4e0d\u540c\u521d\u59cb\u5316\u7b56\u7565\u5bf9 PINNs \u8bad\u7ec3\u6536\u655b\u901f\u5ea6\u7684\u5f71\u54cd\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8bba\u6587\u4f5c\u8005\u4e3aSirui Li, Federica Bragone, Matthieu Barreau, Kateryna Morozovska\uff0c\u6765\u81eaKTH Royal Institute of Technology\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"849\" height=\"466\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/02\/image-6.png\" alt=\"\" class=\"wp-image-5625\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/02\/image-6.png 849w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/02\/image-6-300x165.png 300w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/02\/image-6-768x422.png 768w\" sizes=\"auto, (max-width: 849px) 100vw, 849px\" \/><figcaption class=\"wp-element-caption\"><strong><a href=\"https:\/\/arxiv.org\/abs\/2501.16153\" target=\"_blank\" rel=\"noreferrer noopener\">MILP initialization for solving parabolic PDEs with PINNs<\/a><\/strong><\/figcaption><\/figure>\n\n\n\n<h5 class=\"wp-block-heading\"><strong>\u4e00\u3001\u8bba\u6587\u65b9\u6cd5<\/strong><\/h5>\n\n\n\n<h6 class=\"wp-block-heading\"><strong>1. \u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINNs\uff09<\/strong><\/h6>\n\n\n\n<p class=\"wp-block-paragraph\">PINNs \u901a\u8fc7\u5728\u795e\u7ecf\u7f51\u7edc\u7684\u635f\u5931\u51fd\u6570\u4e2d\u663e\u5f0f\u5730\u878d\u5165\u7269\u7406\u65b9\u7a0b\uff0c\u4f7f\u7f51\u7edc\u80fd\u591f\u5b66\u4e60\u7cfb\u7edf\u7684\u52a8\u529b\u5b66\u7279\u6027\u3002\u672c\u6587\u4f7f\u7528\u4e86\u4e00\u4e2a\u5177\u6709\u5355\u4e2a\u9690\u85cf\u5c42\u7684 PINN\uff0c\u5176\u6570\u5b66\u8868\u793a\u5982\u4e0b\uff1a\u00fb(X)=W<sub>2<\/sub>\u03c3(W<sub>1<\/sub>X+b<sub>1<\/sub>)+b<sub>2<\/sub>\uff0c\u5176\u4e2d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03c3\u4e3a\u9690\u85cf\u5c42\u7684\u6fc0\u6d3b\u51fd\u6570\uff08\u5982 tanh\u2061\uff09<\/li>\n\n\n\n<li>W<sub>1<\/sub>,W<sub>2<\/sub>\u4e3a\u6743\u91cd\u77e9\u9635<\/li>\n\n\n\n<li>b<sub>1<\/sub>,b<sub>2<\/sub>\u200b\u4e3a\u504f\u7f6e<\/li>\n\n\n\n<li>X\u4e3a\u8f93\u5165\u5411\u91cf<\/li>\n\n\n\n<li>\u00fb\u4e3a\u7f51\u7edc\u8f93\u51fa<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">PINNs \u7684\u635f\u5931\u51fd\u6570\u7531<strong>\u8fb9\u754c\u6761\u4ef6\u8bef\u5dee<\/strong>\uff08MSE<sub>u<\/sub>\uff09\u548c<strong>\u65b9\u7a0b\u6b8b\u5dee\u8bef\u5dee<\/strong>\uff08MSE<sub>f<\/sub>\u200b\uff09\u7ec4\u6210\uff1a<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">L=\u03bb<sub>u<\/sub>MSE(\u00fb(X<sub>u<\/sub>),u)+\u03bb<sub>f<\/sub>MSE(f[\u00fb](X<sub>f<\/sub>),0)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5176\u4e2d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>X<sub>u<\/sub>\u548cu\u5206\u522b\u8868\u793a\u8fb9\u754c\u70b9\u7684\u8f93\u5165\u4e0e\u76ee\u6807\u8f93\u51fa<\/li>\n\n\n\n<li>X<sub>f<\/sub>\u4e3a\u65b9\u7a0b\u6c42\u89e3\u70b9\uff08collocation points\uff09\uff0cPINNs \u5728\u8fd9\u4e9b\u70b9\u4e0a\u6ee1\u8db3\u504f\u5fae\u5206\u65b9\u7a0b<\/li>\n<\/ul>\n\n\n\n<h6 class=\"wp-block-heading\"><strong>2. \u70ed\u6269\u6563\u65b9\u7a0b<\/strong><\/h6>\n\n\n\n<p class=\"wp-block-paragraph\">\u672c\u6587\u7684\u5b9e\u9a8c\u6d4b\u8bd5\u4f7f\u7528 <strong>\u4e00\u7ef4\u70ed\u6269\u6563\u65b9\u7a0b<\/strong> \u6765\u6a21\u62df\u7535\u529b\u53d8\u538b\u5668\u7684\u6e29\u5ea6\u5206\u5e03\uff1a<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">\u03c1c<sub>p<\/sub>u<sub>t<\/sub>=ku<sub>xx<\/sub>+q<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5176\u4e2d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03c1\u4e3a\u5bc6\u5ea6<\/li>\n\n\n\n<li>c<sub>p<\/sub>\u4e3a\u6bd4\u70ed\u5bb9<\/li>\n\n\n\n<li>k\u4e3a\u70ed\u5bfc\u7387<\/li>\n\n\n\n<li>q\u4e3a\u70ed\u6e90\u9879\uff0c\u5b9a\u4e49\u4e3a\uff1a<\/li>\n<\/ul>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">q=P<sub>0<\/sub>+P<sub>K<\/sub>(x,t)\u2212h(u\u2212T<sub>a<\/sub>)<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5176\u4e2d\uff1aP<sub>0<\/sub>\u4e3a\u7a7a\u8f7d\u635f\u8017\uff1bP<sub>K<\/sub>(x,t)\u4e3a\u8d1f\u8f7d\u635f\u8017\uff0c\u4f9d\u8d56\u4e8e\u8d1f\u8f7d\u56e0\u5b50 K(t)\uff1bhhh \u4e3a\u5bf9\u6d41\u6362\u70ed\u7cfb\u6570\uff1bT<sub>a<\/sub>\u4e3a\u73af\u5883\u6e29\u5ea6\uff1b<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u53d8\u538b\u5668\u7684\u6e29\u5ea6\u8fb9\u754c\u6761\u4ef6\u4e3a\uff1a<\/p>\n\n\n\n<p class=\"has-text-align-center wp-block-paragraph\">u(x<sub>0<\/sub>,t)=Ta, u(x<sub>end<\/sub>,t)=T<sub>o<\/sub><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5176\u4e2d T<sub>o<\/sub>\u4e3a\u53d8\u538b\u5668\u9876\u90e8\u6cb9\u6e29\u3002<\/p>\n\n\n\n<h6 class=\"wp-block-heading\"><strong>3. \u4f18\u5316\u6a21\u578b<\/strong><\/h6>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3a\u4e86\u89e3\u51b3 PINNs \u968f\u673a\u521d\u59cb\u5316\u5bfc\u81f4\u7684\u8bad\u7ec3\u74f6\u9888\u95ee\u9898\uff0c\u672c\u6587\u91c7\u7528 <strong>\u6df7\u5408\u6574\u6570\u7ebf\u6027\u4f18\u5316\uff08MILP\uff09<\/strong> \u6765\u6539\u8fdb\u6743\u91cd\u521d\u59cb\u5316\u3002\u4e3b\u8981\u4f18\u5316\u76ee\u6807\u662f\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u8fb9\u754c\u9884\u8bad\u7ec3\uff08Boundary Pre-training\uff09\uff1a<\/strong> \u4ec5\u5728\u8fb9\u754c\u70b9\u4e0a\u62df\u5408\u6e29\u5ea6\u3002<\/li>\n\n\n\n<li><strong>\u5168\u57df\u9884\u8bad\u7ec3\uff08Full Pre-training\uff09\uff1a<\/strong> \u5728\u8fb9\u754c\u70b9\u548c\u7269\u7406\u65b9\u7a0b\u6b8b\u5dee\u70b9\u4e0a\u62df\u5408\u6e29\u5ea6\u3002<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">MILP \u901a\u8fc7\u5c06\u6fc0\u6d3b\u51fd\u6570 tanh\u2061(x)\u8fd1\u4f3c\u4e3a <strong>\u9971\u548c\u51fd\u6570\uff08Saturation function\uff09<\/strong> \u6765\u7b80\u5316\u4f18\u5316\u95ee\u9898\uff1a<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"449\" height=\"121\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/02\/image-5.png\" alt=\"\" class=\"wp-image-5624\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2025\/02\/image-5.png 449w, https:\/\/nullthought.net\/wp-content\/uploads\/2025\/02\/image-5-300x81.png 300w\" sizes=\"auto, (max-width: 449px) 100vw, 449px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u5229\u7528\u8be5\u8fd1\u4f3c\u65b9\u6cd5\uff0cMILP \u901a\u8fc7\u6c42\u89e3\u7ebf\u6027\u89c4\u5212\u95ee\u9898\u6765\u786e\u5b9a\u6700\u4f18\u7684\u521d\u59cb\u6743\u91cd\uff0c\u5e76\u4f7f\u7528\u8be5\u6743\u91cd\u8fdb\u884c PINNs \u8bad\u7ec3\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\"><strong>\u4e09\u3001\u7ed3\u679c\u4e0e\u8ba8\u8bba<\/strong><\/h5>\n\n\n\n<h6 class=\"wp-block-heading\"><strong>1. \u9884\u8bad\u7ec3\u521d\u59cb\u5316\u6548\u679c<\/strong><\/h6>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u91c7\u7528 MILP \u9884\u8bad\u7ec3\u7684\u521d\u59cb\u6743\u91cd\u5728<strong>\u6e29\u5ea6\u9884\u6d4b\u8303\u56f4<\/strong>\u4e0a\u66f4\u63a5\u8fd1\u771f\u5b9e\u89e3\uff08\u7531 COMSOL \u6a21\u62df\u63d0\u4f9b\uff09\u3002<\/li>\n\n\n\n<li><strong>\u8fb9\u754c\u9884\u8bad\u7ec3<\/strong>\u548c<strong>\u5168\u57df\u9884\u8bad\u7ec3<\/strong>\u76f8\u6bd4<strong>\u968f\u673a\u521d\u59cb\u5316\u7684 PINN<\/strong>\u5728\u521d\u59cb\u9884\u6d4b\u65f6\u5177\u6709\u66f4\u5408\u7406\u7684\u6e29\u5ea6\u5206\u5e03\u3002<\/li>\n<\/ul>\n\n\n\n<h6 class=\"wp-block-heading\"><strong>2. \u6536\u655b\u901f\u5ea6\u5206\u6790<\/strong><\/h6>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u8fb9\u754c\u9884\u8bad\u7ec3 PINN \u7684\u6536\u655b\u901f\u5ea6\u6700\u5feb<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>32 \u4e2a\u9690\u85cf\u795e\u7ecf\u5143\u65f6\uff0c\u8fb9\u754c\u9884\u8bad\u7ec3\u7684 MSE \u4e0b\u964d\u81f3 0.0184\uff0c\u76f8\u6bd4\u666e\u901a PINN\uff080.0579\uff09\u6709\u663e\u8457\u63d0\u5347\u3002<\/li>\n\n\n\n<li>60 \u4e2a\u9690\u85cf\u795e\u7ecf\u5143\u65f6\uff0c\u8fb9\u754c\u9884\u8bad\u7ec3\u7684 MSE \u4e0b\u964d\u81f3 0.0059\uff0c\u800c\u666e\u901a PINN \u4e3a 0.0395\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u5168\u57df\u9884\u8bad\u7ec3\u7684\u6536\u655b\u901f\u5ea6\u672a\u663e\u8457\u63d0\u5347<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u7531\u4e8e\u7269\u7406\u635f\u5931\u51fd\u6570\u7684\u975e\u7ebf\u6027\u903c\u8fd1\u5b58\u5728\u8bef\u5dee\uff0c\u5168\u57df\u9884\u8bad\u7ec3\u7684\u6548\u679c\u53cd\u800c\u4e0d\u5982\u8fb9\u754c\u9884\u8bad\u7ec3\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u8bad\u7ec3\u65f6\u95f4\u6bd4\u8f83<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li><strong>\u8fb9\u754c\u9884\u8bad\u7ec3 PINN \u8bad\u7ec3\u65f6\u95f4\u663e\u8457\u7f29\u77ed<\/strong>\uff08\u7ea6 20 \u5206\u949f\uff09\u3002<\/li>\n\n\n\n<li>\u666e\u901a PINN \u548c\u5168\u57df\u9884\u8bad\u7ec3 PINN \u9700\u8981\u7ea6 30-35 \u5206\u949f\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<h6 class=\"wp-block-heading\"><strong>3. \u9884\u6d4b\u7cbe\u5ea6\u8bc4\u4f30<\/strong><\/h6>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u7ecf\u8fc7 10000 \u8f6e\u8bad\u7ec3\u540e\uff0c\u6240\u6709 PINN \u6a21\u578b\u90fd\u80fd\u51c6\u786e\u9884\u6d4b\u6e29\u5ea6\u5206\u5e03<\/strong>\uff0c\u4f46\u8fb9\u754c\u9884\u8bad\u7ec3 PINN \u4ecd\u7136\u5728\u8bef\u5dee\u4e0a\u66f4\u5c0f\u3002<\/li>\n\n\n\n<li><strong>\u4e0d\u540c\u65f6\u523b\u7684\u6e29\u5ea6\u5206\u5e03\u66f2\u7ebf\u4e0e COMSOL \u7ed3\u679c\u4e00\u81f4<\/strong>\uff0c\u8868\u660e\u4f18\u5316\u540e\u7684 PINN \u8bad\u7ec3\u80fd\u591f\u6709\u6548\u6355\u6349\u70ed\u6269\u6563\u8fc7\u7a0b\u3002<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\"><strong>\u56db\u3001\u7ed3\u8bba<\/strong><\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">\u672c\u6587\u7814\u7a76\u4e86\u4f7f\u7528<strong>\u6df7\u5408\u6574\u6570\u7ebf\u6027\u4f18\u5316\uff08MILP\uff09<\/strong>\u6765\u6539\u8fdb PINNs \u7684\u521d\u59cb\u6743\u91cd\uff0c\u4ee5\u63d0\u9ad8\u8bad\u7ec3\u6536\u655b\u901f\u5ea6\u548c\u7cbe\u5ea6\u3002\u4e3b\u8981\u7ed3\u8bba\u5982\u4e0b\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u8fb9\u754c\u9884\u8bad\u7ec3\u6bd4\u968f\u673a\u521d\u59cb\u5316\u663e\u8457\u63d0\u5347\u6536\u655b\u901f\u5ea6\u548c\u9884\u6d4b\u7cbe\u5ea6\u3002<\/strong><\/li>\n\n\n\n<li><strong>\u5168\u57df\u9884\u8bad\u7ec3\u672a\u80fd\u660e\u663e\u6539\u5584 PINN \u8bad\u7ec3\uff0c\u53ef\u80fd\u53d7\u5230\u975e\u7ebf\u6027\u8fd1\u4f3c\u8bef\u5dee\u7684\u5f71\u54cd\u3002<\/strong><\/li>\n\n\n\n<li><strong>\u5728\u7535\u529b\u53d8\u538b\u5668\u70ed\u6269\u6563\u95ee\u9898\u4e0a\uff0c\u4f18\u5316\u540e\u7684 PINN \u8bad\u7ec3\u65f6\u95f4\u7f29\u77ed\u7ea6 40%\uff0c\u6536\u655b\u8bef\u5dee\u964d\u4f4e 3-5 \u500d\u3002<\/strong><\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u672a\u6765\u7814\u7a76\u65b9\u5411\u5305\u62ec\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u4f18\u5316\u6fc0\u6d3b\u51fd\u6570\u8fd1\u4f3c\u65b9\u6cd5<\/strong>\uff0c\u63d0\u5347\u7269\u7406\u7ea6\u675f\u9884\u8bad\u7ec3\u7684\u7cbe\u5ea6\u3002<\/li>\n\n\n\n<li><strong>\u5c06\u8be5\u65b9\u6cd5\u63a8\u5e7f\u81f3\u66f4\u590d\u6742\u7684 PDE \u95ee\u9898<\/strong>\uff0c\u5982\u6e4d\u6d41\u6a21\u62df\u548c\u6d41\u56fa\u8026\u5408\u95ee\u9898\u3002<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08Physics-Informed Neural Networks\uff0cPINNs\uff09\u662f\u4e00\u79cd\u65b0\u5174\u7684\u6df1 [&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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