{"id":4287,"date":"2024-08-10T12:00:49","date_gmt":"2024-08-10T04:00:49","guid":{"rendered":"https:\/\/nullthought.net\/?p=4287"},"modified":"2025-12-01T15:44:31","modified_gmt":"2025-12-01T07:44:31","slug":"%e7%89%a9%e7%90%86%e4%bf%a1%e6%81%af%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9cpinns%ef%bc%88physics-informed-neural-networks%ef%bc%89","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=4287","title":{"rendered":"\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edcPINNs\uff08Physics-Informed Neural Networks\uff09"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\"><strong>Physics-Informed Neural Networks (PINNs)<\/strong> \u662f\u4e00\u79cd\u5c06\u7269\u7406\u77e5\u8bc6\u878d\u5165\u795e\u7ecf\u7f51\u7edc\u7684\u6280\u672f\uff0c\u4e3b\u8981\u7528\u4e8e\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDEs\uff09\u548c\u5176\u4ed6\u53d7\u7269\u7406\u5b9a\u5f8b\u7ea6\u675f\u7684\u95ee\u9898\u3002\u4f20\u7edf\u7684\u795e\u7ecf\u7f51\u7edc\u901a\u5e38\u9700\u8981\u5927\u91cf\u7684\u6807\u6ce8\u6570\u636e\u6765\u8fdb\u884c\u8bad\u7ec3\uff0c\u4f46\u5728\u7269\u7406\u95ee\u9898\u4e2d\uff0c\u83b7\u5f97\u5927\u89c4\u6a21\u7684\u9ad8\u8d28\u91cf\u6570\u636e\u53ef\u80fd\u975e\u5e38\u56f0\u96be\u3002PINNs\u901a\u8fc7\u5c06\u5df2\u77e5\u7684\u7269\u7406\u89c4\u5f8b\uff08\u5982\u5b88\u6052\u5b9a\u5f8b\u3001\u8fb9\u754c\u6761\u4ef6\u7b49\uff09\u5f15\u5165\u635f\u5931\u51fd\u6570\u4e2d\uff0c\u51cf\u5c11\u4e86\u5bf9\u5927\u89c4\u6a21\u6570\u636e\u7684\u4f9d\u8d56\uff0c<strong>\u63d0\u9ad8\u4e86\u6a21\u578b\u7684\u6cdb\u5316\u80fd\u529b<\/strong>\u3002<\/p>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08Physics-Informed Neural Networks, PINNs\uff09\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/3SC22VAGxvo?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">PINNs\u53ef\u7ffb\u8bd1\u6210<strong>\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc<\/strong>\uff0c\u6216\u8005<strong>\u7269\u7406\u5f15\u5bfc\u795e\u7ecf\u7f51\u7edc<\/strong>\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u4e00\u3001PINNs \u7684\u57fa\u672c\u539f\u7406<\/h5>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u7269\u7406\u7ea6\u675f<\/strong>\uff1a<strong>\u5728PINNs\u4e2d\uff0c\u7269\u7406\u7ea6\u675f\u901a\u5e38\u901a\u8fc7\u635f\u5931\u51fd\u6570\u7684\u5f62\u5f0f\u88ab\u663e\u5f0f\u5730\u52a0\u5165\u6a21\u578b\u4e2d<\/strong>\u3002\u4f8b\u5982\uff0c\u6c42\u89e3\u6d41\u4f53\u529b\u5b66\u4e2d\u7684Navier-Stokes\u65b9\u7a0b\u65f6\uff0c\u53ef\u4ee5\u5c06\u65b9\u7a0b\u7684\u6b8b\u5dee\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\u7684\u4e00\u90e8\u5206\uff0c\u4f7f\u7f51\u7edc\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u81ea\u52a8\u6ee1\u8db3\u7269\u7406\u65b9\u7a0b\u3002<\/li>\n\n\n\n<li><strong>\u81ea\u9002\u5e94\u6743\u91cd<\/strong>\uff1a\u4e3a\u4e86\u5e73\u8861\u6570\u636e\u9a71\u52a8\u7684\u635f\u5931\u548c\u7269\u7406\u7ea6\u675f\u7684\u635f\u5931\uff0cPINNs\u5f15\u5165\u4e86\u81ea\u9002\u5e94\u6743\u91cd\u673a\u5236\u3002\u8fd9\u79cd\u673a\u5236\u80fd\u591f\u52a8\u6001\u8c03\u6574\u7269\u7406\u635f\u5931\u548c\u6570\u636e\u635f\u5931\u4e4b\u95f4\u7684\u6bd4\u91cd\uff0c\u4ece\u800c\u5728\u590d\u6742\u95ee\u9898\u4e0a\u53d6\u5f97\u66f4\u597d\u7684\u6548\u679c\u3002<\/li>\n\n\n\n<li><strong>\u68af\u5ea6\u4f20\u64ad<\/strong>\uff1aPINNs\u4f7f\u7528\u81ea\u52a8\u5fae\u5206\u6765\u8ba1\u7b97\u7269\u7406\u65b9\u7a0b\u7684\u5bfc\u6570\uff0c\u786e\u4fdd\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u80fd\u591f\u6709\u6548\u5730\u4f18\u5316\u795e\u7ecf\u7f51\u7edc\u53c2\u6570\u3002\u8fd9\u79cd\u65b9\u6cd5\u53ef\u4ee5\u5728\u4e0d\u663e\u5f0f\u6c42\u89e3\u65b9\u7a0b\u7684\u60c5\u51b5\u4e0b\uff0c\u83b7\u5f97\u4e0e\u7269\u7406\u6a21\u578b\u4e00\u81f4\u7684\u89e3\u3002<\/li>\n<\/ol>\n\n\n\n<h5 class=\"wp-block-heading\">\u4e8c\u3001\u6700\u65b0\u7814\u7a76\u8fdb\u5c55<\/h5>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u9ad8\u6548\u6c42\u89e3\u590d\u6742PDEs<\/strong>\uff1a\u6700\u65b0\u7684\u7814\u7a76\u8868\u660e\uff0cPINNs\u5728\u6c42\u89e3\u590d\u6742\u7684\u591a\u5c3a\u5ea6\u3001\u591a\u7269\u7406\u573a\u95ee\u9898\u4e0a\u8868\u73b0\u51fa\u8272\u3002\u4f8b\u5982\uff0c\u5728\u91cf\u5b50\u529b\u5b66\u3001\u6c14\u5019\u5efa\u6a21\u3001\u6750\u6599\u79d1\u5b66\u7b49\u9886\u57df\uff0cPINNs\u5df2\u88ab\u6210\u529f\u5e94\u7528\uff0c\u5e76\u5c55\u793a\u4e86\u5176\u5728\u9ad8\u7ef4\u95ee\u9898\u4e0a\u7684\u4f18\u52bf\u3002<\/li>\n\n\n\n<li><strong>\u6269\u5c55\u81f3\u591a\u4efb\u52a1\u5b66\u4e60<\/strong>\uff1a\u8fd1\u671f\u7684\u7814\u7a76\u63a2\u7d22\u4e86\u5982\u4f55\u5c06PINNs\u6269\u5c55\u5230\u591a\u4efb\u52a1\u5b66\u4e60\u4e2d\uff0c\u4f7f\u5176\u80fd\u591f\u540c\u65f6\u6c42\u89e3\u591a\u4e2a\u76f8\u5173\u8054\u7684\u7269\u7406\u95ee\u9898\u3002\u8fd9\u79cd\u65b9\u6cd5\u4e0d\u4ec5\u63d0\u9ad8\u4e86\u8ba1\u7b97\u6548\u7387\uff0c\u8fd8\u63d0\u5347\u4e86\u6a21\u578b\u7684\u6574\u4f53\u51c6\u786e\u6027\u3002<\/li>\n\n\n\n<li><strong>\u6df7\u5408\u7269\u7406\u6a21\u578b\u548c\u6570\u636e\u9a71\u52a8\u6a21\u578b<\/strong>\uff1a\u4e3a\u4e86\u8fdb\u4e00\u6b65\u589e\u5f3a\u6a21\u578b\u7684\u7cbe\u5ea6\uff0c\u7814\u7a76\u8005\u4eec\u6b63\u5728\u63a2\u7d22\u5c06PINNs\u4e0e\u4f20\u7edf\u7684\u7269\u7406\u6a21\u62df\u548c\u6570\u636e\u9a71\u52a8\u6a21\u578b\u76f8\u7ed3\u5408\u3002\u8fd9\u79cd\u6df7\u5408\u6a21\u578b\u80fd\u591f\u5728\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u514b\u670d\u5355\u4e00\u65b9\u6cd5\u7684\u5c40\u9650\u6027\uff0c\u5b9e\u73b0\u66f4\u9ad8\u7684\u9884\u6d4b\u7cbe\u5ea6\u548c\u66f4\u5e7f\u6cdb\u7684\u5e94\u7528\u8303\u56f4\u3002<\/li>\n\n\n\n<li><strong>\u81ea\u9002\u5e94\u91c7\u6837\u7b56\u7565<\/strong>\uff1a\u968f\u7740PINNs\u7684\u5e94\u7528\u8303\u56f4\u6269\u5927\uff0c\u5982\u4f55\u6709\u6548\u5730\u9009\u62e9\u8bad\u7ec3\u6837\u672c\u6210\u4e3a\u4e00\u4e2a\u5173\u952e\u95ee\u9898\u3002\u6700\u65b0\u7684\u7814\u7a76\u63d0\u51fa\u4e86\u4e00\u4e9b\u81ea\u9002\u5e94\u91c7\u6837\u7b56\u7565\uff0c\u80fd\u591f\u6839\u636e\u95ee\u9898\u7684\u7279\u6027\u52a8\u6001\u9009\u62e9\u6700\u6709\u4fe1\u606f\u91cf\u7684\u6837\u672c\u70b9\u8fdb\u884c\u8bad\u7ec3\uff0c\u4ece\u800c\u63d0\u9ad8\u8bad\u7ec3\u6548\u7387\u5e76\u51cf\u5c11\u8ba1\u7b97\u6210\u672c\u3002<\/li>\n\n\n\n<li><strong>\u8ba1\u7b97\u6548\u7387\u7684\u63d0\u5347<\/strong>\uff1a\u7531\u4e8ePINNs\u901a\u5e38\u9700\u8981\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\uff0c\u7814\u7a76\u8005\u4eec\u6b63\u5728\u5f00\u53d1\u65b0\u7684\u4f18\u5316\u7b97\u6cd5\u548c\u5e76\u884c\u8ba1\u7b97\u6280\u672f\uff0c\u4ee5\u63d0\u5347PINNs\u7684\u8ba1\u7b97\u6548\u7387\u3002\u8fd9\u4e9b\u8fdb\u5c55\u4f7f\u5f97PINNs\u5728\u5de5\u4e1a\u754c\u548c\u5b66\u672f\u754c\u7684\u5e94\u7528\u53d8\u5f97\u66f4\u52a0\u5b9e\u9645\u53ef\u884c\u3002<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u603b\u4f53\u800c\u8a00\uff0cPhysics-Informed Neural Networks (PINNs) \u662f\u4e00\u79cd\u5177\u6709\u5de8\u5927\u6f5c\u529b\u7684\u6280\u672f\uff0c\u6b63\u5728\u5404\u4e2a\u9886\u57df\u4e2d\u5c55\u793a\u5176\u4f18\u52bf\u3002\u968f\u7740\u8ba1\u7b97\u8d44\u6e90\u7684\u63d0\u5347\u548c\u7b97\u6cd5\u7684\u6539\u8fdb\uff0cPINNs\u6709\u671b\u5728\u672a\u6765\u7684\u79d1\u5b66\u8ba1\u7b97\u548c\u5de5\u7a0b\u5e94\u7528\u4e2d\u53d1\u6325\u66f4\u52a0\u91cd\u8981\u7684\u4f5c\u7528\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">\u4e09\u3001\u5224\u65ad\u662f\u5426\u9002\u5408\u4f7f\u7528 PINNs \u7684\u4e00\u822c\u6807\u51c6<\/h5>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u2705 \u6709\u660e\u786e\u7269\u7406\u65b9\u7a0b\uff08ODE\/PDE \u7b49\uff09<\/li>\n\n\n\n<li>\u2705 \u6570\u636e\u6709\u9650\uff08\u7279\u522b\u662f\u8fb9\u754c\/\u521d\u59cb\u70b9\uff09<\/li>\n\n\n\n<li>\u2705 \u9700\u4ece\u6570\u636e\u4e2d\u63a8\u7406\u6216\u53cd\u6f14\u672a\u77e5\u53c2\u6570\/\u8fb9\u754c\u6761\u4ef6<\/li>\n\n\n\n<li>\u2705 \u4f20\u7edf\u6570\u503c\u89e3\u592a\u6162\uff0c\u9700 <a href=\"https:\/\/nullthought.net\/?p=4426\" target=\"_blank\" rel=\"noreferrer noopener\">surrogate model<\/a><\/li>\n\n\n\n<li>\u2705 \u62e5\u6709\u4e00\u5b9a\u7b97\u529b\u8d44\u6e90\u652f\u6301\u8bad\u7ec3<\/li>\n\n\n\n<li>\u2705 \u9700\u8981\u89e3\u91ca\u6027\u66f4\u5f3a\u7684\u9884\u6d4b\u6a21\u578b\uff08\u76f8\u6bd4\u9ed1\u76d2\u795e\u7ecf\u7f51\u7edc\uff09<\/li>\n<\/ol>\n\n\n\n<h5 class=\"wp-block-heading\">\u56db\u3001PINN\u7684\u9ad8\u7ef4\u975e\u7ebf\u6027\u5efa\u6a21\u7b56\u7565\u4e0e\u53d8\u79cd\u6bd4\u8f83<\/h5>\n\n\n\n<p class=\"wp-block-paragraph\">\u5de5\u4e1a\u573a\u666f\u4e2d\u7684\u6545\u969c\u8bca\u65ad\u95ee\u9898\u5e38\u6d89\u53ca<strong>\u9ad8\u7ef4\u5ea6<\/strong>\uff08\u7a7a\u95f4\u7ef4\u5ea6\u591a\u3001\u53c2\u6570\u7ef4\u5ea6\u9ad8\uff09\u3001<strong>\u5f3a\u975e\u7ebf\u6027<\/strong>\u548c<strong>\u9006\u95ee\u9898<\/strong>\u6c42\u89e3\u9700\u6c42\u3002\u4e3a\u6b64\uff0c\u6807\u51c6PINN\u6846\u67b6\u884d\u751f\u51fa\u591a\u79cd\u6539\u8fdb\u7b56\u7565\u548c\u53d8\u79cd\u6a21\u578b\u6765\u63d0\u5347\u9002\u5e94\u6027\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>4.1 \u9ad8\u7ef4\u4e0e\u591a\u5c3a\u5ea6\u95ee\u9898\uff1a\u9ad8\u7ef4\u4e3b\u8981\u6307\u7a7a\u95f4\u7ef4\u5ea6\u5347\u9ad8\uff08\u4f8b\u59823D\u7ed3\u6784\uff09\u6216\u53c2\u6570\u7ef4\u5ea6\u589e\u52a0\uff08\u540c\u65f6\u4f30\u8ba1\u591a\u4e2a\u672a\u77e5\u53c2\u6570\uff09\u3002\u6b64\u7c7b\u95ee\u9898\u4f1a\u5bfc\u81f4PINN\u8bad\u7ec3\u56f0\u96be\uff0c\u8868\u73b0\u4e3a\u6536\u655b\u53d8\u6162\u6216\u7cbe\u5ea6\u4e0b\u964d\u3002XPINN\uff08Extended PINN\uff09\u901a\u8fc7\u7a7a\u95f4-\u65f6\u95f4\u57df\u5206\u89e3\u6765\u7f13\u89e3\u9ad8\u7ef4\u538b\u529b<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=independent%20computational%20unit,the%20solution%20of%20nonlinear%20PDE\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=on%20arbitrary%20complex%20geometries,along%20with%20its%20flexible%20domain\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=on%20arbitrary%20complex%20geometries,along%20with%20its%20flexible%20domain\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u5176\u505a\u6cd5\u662f\u5c06\u6c42\u89e3\u57df\u5212\u5206\u4e3a\u82e5\u5e72\u5b50\u57df\uff0c\u6bcf\u4e2a\u5b50\u57df\u4f7f\u7528\u72ec\u7acb\u7684\u7f51\u7edc\u8bad\u7ec3\uff0c\u5e76\u5728\u63a5\u53e3\u5904\u4ea4\u6362\u4fe1\u606f<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=strengths%20of%20both%20classical%20and,the%20solution%20of%20nonlinear%20PDE\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u6bcf\u4e2a\u7f51\u7edc\u53ea\u9700\u5b66\u4e60\u5168\u5c40\u95ee\u9898\u7684\u4e00\u90e8\u5206\uff0c\u964d\u4f4e\u4e86\u590d\u6742\u5ea6\uff0c\u800c\u4e14\u5b50\u7f51\u7edc\u53ef\u4ee5\u5e76\u884c\u8bad\u7ec3<\/strong>\uff0c\u52a0\u901f\u8ba1\u7b97<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=on%20arbitrary%20complex%20geometries,along%20with%20its%20flexible%20domain\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=characteristics,for%20complex%20scientific%20computing%20problems\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=characteristics,for%20complex%20scientific%20computing%20problems\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002XPINN\u5141\u8bb8\u9488\u5bf9\u4e0d\u540c\u5b50\u57df\u9009\u7528\u4e0d\u540c\u7f51\u7edc\u7ed3\u6784\u548c\u8d85\u53c2\u6570\uff0c\u7075\u6d3b\u6027\u66f4\u5f3a<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=independent%20computational%20unit,the%20solution%20of%20nonlinear%20PDE\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u4f8b\u5982\uff0c\u5bf9\u4e8e\u4e00\u4e2a\u5305\u542b\u56fa\u4f53\u548c\u6d41\u4f53\u533a\u57df\u7684\u8026\u5408\u95ee\u9898\uff0c\u53ef\u4ee5\u7528\u4e24\u4e2aPINN\u5b50\u7f51\u5206\u522b\u5904\u7406\u56fa\u4f53PDE\u548c\u6d41\u4f53PDE\uff0c\u5728\u4ea4\u754c\u9762\u6ee1\u8db3\u8fde\u7eed\u6761\u4ef6<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=extended%20PINNs%20%28XPINNs%29%C2%A0%28,interfaces%2C%20cPINNs%20ensure%20the%20conservative\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=characteristics,for%20complex%20scientific%20computing%20problems\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=characteristics,for%20complex%20scientific%20computing%20problems\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u53c8\u5982\u5bf9\u4e8e\u65f6\u95f4\u957f\u533a\u95f4\u7684\u95ee\u9898\uff0c\u53ef\u6309\u65f6\u95f4\u5206\u6bb5\uff0c\u7528XPINN\u9010\u6bb5\u6c42\u89e3\u4ee5\u4fdd\u6301\u7cbe\u5ea6\u3002\u9664\u4e86XPINN\uff0c<strong>\u81ea\u9002\u5e94\u91c7\u6837<\/strong>\u4e5f\u662f\u91cd\u8981\u7b56\u7565<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=The%20architecture%20and%20training%20strategy,adaptive%20learning%20rate%20schedules%2C%20including\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\uff1a\u5728\u9ad8\u7ef4\u60c5\u51b5\u4e0b\uff0c\u8bef\u5dee\u5f80\u5f80\u96c6\u4e2d\u4e8e\u5c40\u90e8\u533a\u57df\uff08\u4f8b\u5982\u8fb9\u754c\u5c42\u5904\uff09\u3002Residual-based adaptive refinement (RAR)\u65b9\u6cd5\u4f1a\u52a8\u6001\u76d1\u6d4bPINN\u7684\u6b8b\u5dee\u5206\u5e03\uff0c\u5c06\u66f4\u591a\u8bad\u7ec3\u70b9\u6295\u653e\u5728\u9ad8\u6b8b\u5dee\u533a\u57df<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=The%20architecture%20and%20training%20strategy,adaptive%20learning%20rate%20schedules%2C%20including\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u8fd9\u6837\u7f51\u7edc\u91cd\u70b9\u5b66\u4e60\u56f0\u96be\u533a\u57df\uff0c\u53ef\u66f4\u5feb\u903c\u8fd1\u590d\u6742\u89e3\u3002\u6b64\u5916\uff0c\u9ad8\u7ef4\u60c5\u51b5\u4e0b\u7f51\u7edc\u53c2\u6570\u591a\uff0c\u5bb9\u6613\u9677\u5165\u5c40\u90e8\u6781\u5c0f\u503c\uff0c\u81ea\u9002\u5e94\u8c03\u6574\u5b66\u4e60\u7387\uff08\u5982<strong>\u5b66\u4e60\u7387\u4f59\u5f26\u9000\u706b<\/strong>\u3001<strong>warm-up<\/strong>\u7b49\uff09\u6709\u52a9\u4e8e\u66f4\u6709\u6548\u5730\u63a2\u7d22\u635f\u5931\u9762<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=ensure%20better%20coverage%20of%20the,getting%20trapped%20in%20local%20minima\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=turbulence,66\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=turbulence,66\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u603b\u7684\u6765\u8bf4\uff0c\u9ad8\u7ef4\u591a\u5c3a\u5ea6\u95ee\u9898\u9700\u8981\u7ed3\u5408<strong>\u533a\u57df\u5212\u5206\u3001\u91cd\u70b9\u91c7\u6837\u548c\u4f18\u5316\u8c03\u5ea6<\/strong>\u7b49\u7efc\u5408\u624b\u6bb5\u6765\u4fdd\u8bc1\u8bad\u7ec3\u6548\u679c\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">4<strong>.2 \u975e\u7ebf\u6027\u4e0e\u521a\u6027\u95ee\u9898\uff1a\u975e\u7ebf\u6027PDE\uff08\u5982Navier-Stokes\u3001\u975e\u7ebf\u6027\u5f39\u6027\uff09\u4f1a\u5bfc\u81f4PINN\u7684\u635f\u5931\u51fd\u6570\u9ad8\u5ea6\u975e\u51f8\uff0c\u8bad\u7ec3\u96be\u5ea6\u4e0a\u5347\u3002\u800c\u521a\u6027<\/strong>\u95ee\u9898\uff08stiff\uff0c\u7528\u4e8e\u63cf\u8ff0\u5b58\u5728\u591a\u65f6\u95f4\u5c3a\u5ea6\u7684\u7cfb\u7edf\uff09\u5219\u53ef\u80fd\u51fa\u73b0\u7f51\u7edc\u96be\u4ee5\u540c\u65f6\u8fd1\u4f3c\u5feb\u6162\u53d8\u91cf\u7684\u95ee\u9898\u3002\u9488\u5bf9\u975e\u7ebf\u6027\uff0c\u5e38\u7528\u7b56\u7565\u5305\u62ec\uff1a1\uff09<strong>\u8fc1\u79fb\u5b66\u4e60\/\u9010\u6b65\u8bad\u7ec3<\/strong>\uff1a\u5148\u8bad\u7ec3\u7ebf\u6027\u7b80\u5316\u7248\u672c\uff0c\u518d\u5c06\u7f51\u7edc\u53c2\u6570\u7528\u4e8e\u975e\u7ebf\u6027\u7248\u672c\u4f5c\u521d\u59cb<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=interactions%20in%20a%20bluff,flows%2C%20it%20exhibited%20limitations%20in\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u4f8b\u5982\u71c3\u70e7\u4e0d\u7a33\u5b9a\u95ee\u9898\u4e2d\uff0c\u5148\u4e0d\u8003\u8651\u975e\u7ebf\u6027\u6e90\u9879\u8bad\u7ec3\u7ebf\u6027\u58f0\u5b66PINN\uff0c\u518d\u9010\u6b65\u52a0\u5165\u975e\u7ebf\u6027\u9879\u30022\uff09<strong>\u591a\u4efb\u52a1\u6743\u91cd\u5e73\u8861<\/strong>\uff1a\u975e\u7ebf\u6027PDE\u901a\u5e38\u6e90\u4e8e\u591a\u4e2a\u7269\u7406\u89c4\u5f8b\u53e0\u52a0\uff0c\u5bf9\u5e94\u635f\u5931\u7684\u4e0d\u540c\u90e8\u5206\u3002\u53ef\u91c7\u7528<strong>\u81ea\u9002\u5e94\u6743\u91cdPINN<\/strong>\uff08Self-Adaptive PINN, SA-PINN\uff09\uff0c\u4ee4\u635f\u5931\u6743\u91cd\u4f5c\u4e3a\u53ef\u4ee5\u8bad\u7ec3\u7684\u53c2\u6570\uff0c\u7f51\u7edc\u81ea\u52a8\u8c03\u8282\u6743\u91cd\u5927\u5c0f\uff0c\u4f7f\u5404\u90e8\u5206\u6b8b\u5dee\u68af\u5ea6\u8d8b\u4e8e\u5747\u8861<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=observational%20data%20and%20and%20enforce,can%20be%20computed%20as%20follows\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u5b9e\u8df5\u8bc1\u660e\u8fd9\u6709\u52a9\u4e8e\u5e94\u5bf9\u975e\u7ebf\u6027\u9879\u4e3b\u5bfc\u6216\u91cf\u7ea7\u5dee\u5f02\u5927\u7684\u60c5\u51b5\uff0c\u52a0\u5feb\u6536\u655b\u5e76\u907f\u514d\u67d0\u90e8\u5206\u8bef\u5dee\u59cb\u7ec8\u5c45\u9ad8\u4e0d\u4e0b<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=compressor%20cascade%20using%20PINNs,dimensional%20subsonic%20linear%20compressor%20cascade\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=Compressor%20Cascade%20Flow%20%20with,Surrogate%20Modeling%2C%20%20Turbulence%20Modeling\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=Compressor%20Cascade%20Flow%20%20with,Surrogate%20Modeling%2C%20%20Turbulence%20Modeling\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u30023\uff09<strong>\u6fc0\u6d3b\u51fd\u6570\u8c03\u63a7<\/strong>\uff1a\u5bf9\u4e8e\u521a\u6027\u95ee\u9898\uff0c\u53ef\u91c7\u7528\u5206\u6bb5\u7ebf\u6027\u7684ReLU\u6216\u7269\u7406\u610f\u4e49\u66f4\u5f3a\u7684\u7279\u6b8a\u6fc0\u6d3b\uff0c\u4ee5\u66f4\u597d\u8868\u8fbe\u9661\u5ced\u53d8\u5316\u3002\u8fd8\u6709<strong>\u591a\u57df\u65f6\u95f4\u6b65\u8fdb<\/strong>\uff08XPINN\u65f6\u95f4\u5b50\u57df\u5212\u5206\uff09\u7b49\u65b9\u6cd5\u3002\u9700\u8981\u5f3a\u8c03\u7684\u662f\uff0cPINN\u89e3\u51b3\u521a\u6027\u95ee\u9898\u4ecd\u5177\u6311\u6218\u6027\uff0c\u662f\u5f53\u524d\u7814\u7a76\u70ed\u70b9\u4e4b\u4e00\uff0c\u4e00\u4e9b\u53d8\u4f53\uff08\u5982<strong>\u57fa\u4e8eRunge-Kutta\u7684PINN<\/strong>\u3001<strong>\u591a\u5c3a\u5ea6PINN<\/strong>\uff09\u6b63\u5728\u63a2\u7d22\u4e4b\u4e2d\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">4<strong>.3 \u9006\u95ee\u9898\u6c42\u89e3\uff1a\u9006\u95ee\u9898\u662fPINN\u7684\u4e00\u5927\u5f3a\u9879\uff0c\u5373\u5229\u7528\u90e8\u5206\u53ef\u89c2\u6d4b\u6570\u636e\u63a8\u65ad\u672a\u77e5\u53c2\u6570\u6216\u573a\u5206\u5e03\u3002\u5728PINN\u4e2d\uff0c\u8fd9\u901a\u8fc7\u5c06\u672a\u77e5\u91cf\u89c6\u4e3a\u6269\u5c55\u53c2\u6570\u5171\u540c\u8bad\u7ec3\u6765\u5b9e\u73b0<a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC10781337\/#:~:text=The%20proposed%20framework%20hinges%20upon,26\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov<\/a><a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC10781337\/#:~:text=where%20%20and%20%20denote,each%20training%20iteration%2C%20the%20PINN\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC10781337\/#:~:text=where%20%20and%20%20denote,each%20training%20iteration%2C%20the%20PINN\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov<\/a>\u3002\u4f8b\u5982\u672a\u77e5\u53c2\u6570\u4f5c\u4e3a\u7f51\u7edc\u989d\u5916\u8f93\u51fa\u8282\u70b9\u6216\u76f4\u63a5\u4f5c\u4e3a\u5f85\u4f18\u5316\u53d8\u91cf\u63d2\u5165\u635f\u5931\u51fd\u6570\uff0c\u5229\u7528\u7269\u7406\u6b8b\u5dee\u548c\u6570\u636e\u8bef\u5dee\u7684\u5171\u540c\u4f5c\u7528\uff0c\u5c06\u5176\u6536\u655b\u5230\u6b63\u786e\u503c\u3002PINN\u6c42\u89e3\u9006\u95ee\u9898\u76f8\u6bd4\u4f20\u7edf\u65b9\u6cd5\uff08\u5982\u6700\u5c0f\u4e8c\u4e58\u4f18\u5316\u6709\u9650\u5143\uff09\u6709\u4ee5\u4e0b\u4f18\u52bf\uff1a\u4e00\u662f\u4e00\u6b21\u8bad\u7ec3\u540c\u65f6\u83b7\u5f97\u53c2\u6570\u548c\u573a<\/strong>\uff0c\u907f\u514d\u4e86\u6bcf\u5047\u8bbe\u53c2\u6570\u4e00\u6b21\u4eff\u771f\u7684\u53cd\u590d\u8fed\u4ee3<a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC10781337\/#:~:text=where%20%20and%20%20denote,each%20training%20iteration%2C%20the%20PINN\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov<\/a>\u3002\u4e8c\u662f\u7269\u7406\u65b9\u7a0b\u63d0\u4f9b\u6b63\u5219\uff0c\u4f7f\u89e3\u5373\u4fbf\u5728\u89c2\u6d4b\u6570\u636e\u7a00\u758f\u65f6\u4e5f\u6709\u826f\u597d\u7ea6\u675f<a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC12111269\/#:~:text=into%20the%20loss%20function%2C%20thereby,However%2C%20traditional%20PINNs\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov<\/a><a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC12111269\/#:~:text=this%20approach%20for%20inverse%20problems%2C,integration%20with%20generative%20models%20to\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC12111269\/#:~:text=this%20approach%20for%20inverse%20problems%2C,integration%20with%20generative%20models%20to\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov<\/a>\u3002\u4e8b\u5b9e\u4e0a\uff0cRaissi\u7b49\u4eba\u5f00\u521bPINN\u65f6\u91cd\u70b9\u9a8c\u8bc1\u4e86\u5176\u5728<strong>\u53c2\u6570\u8fa8\u8bc6<\/strong>\u548c<strong>\u9690\u542b\u51fd\u6570\u53d1\u73b0<\/strong>\u7b49\u9006\u95ee\u9898\u4e0a\u7684\u5353\u8d8a\u6027\u80fd<a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC12111269\/#:~:text=into%20the%20loss%20function%2C%20thereby,However%2C%20traditional%20PINNs\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov<\/a>\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u9006\u95ee\u9898\u5f80\u5f80\u75c5\u6001\u6216\u4e0d\u9002\u5b9a\uff0c\u4e3a\u63d0\u9ad8\u7a33\u5065\u6027\uff0c\u73b0\u4ee3PINN\u5f15\u5165\u4e86<strong>\u6b63\u5219\u5316<\/strong>\uff08\u5982Tikhonov\u6b63\u5219\uff09\u6216<strong>\u8d1d\u53f6\u65afPINN<\/strong>\uff08Bayesian PINN, \u5728\u8bad\u7ec3\u540e\u8fdb\u884c\u53c2\u6570\u540e\u9a8c\u5206\u5e03\u63a8\u65ad\uff09\u6765\u91cf\u5316\u4e0d\u786e\u5b9a\u6027\u3002\u540e\u8005\u5c5e\u4e8ePINN\u7684Bayesian\u53d8\u4f53\uff0c\u53ef\u5728\u7ed9\u51fa\u53c2\u6570\u4f30\u8ba1\u503c\u7684\u540c\u65f6\u7ed9\u51fa\u53ef\u4fe1\u533a\u95f4\uff0c\u63d0\u9ad8\u8bca\u65ad\u51b3\u7b56\u7684\u53ef\u9760\u6027<a href=\"https:\/\/ww2.mathworks.cn\/discovery\/physics-informed-neural-networks.html#:~:text=PINN%20%E4%BA%8E2017%20%E5%B9%B4%E9%A6%96%E6%AC%A1%E6%8E%A8%E5%87%BA%EF%BC%8C%E7%8E%B0%E5%9C%A8%E6%9C%89%E8%AE%B8%E5%A4%9A%E5%8F%98%E4%BD%93%EF%BC%8C%E5%85%B6%E4%B8%AD%E5%8C%85%E6%8B%AC%EF%BC%9A,VPINN%29%EF%BC%8C%E5%AE%83%E4%BB%AC%E5%B0%86PDE%20%E7%9A%84%E5%BC%B1%E5%BD%A2%E5%BC%8F%E7%BA%B3%E5%85%A5%E6%8D%9F%E5%A4%B1%E5%87%BD%E6%95%B0%E4%B8%AD\" target=\"_blank\" rel=\"noreferrer noopener\">ww2.mathworks.cn<\/a>\u3002\u5728\u5de5\u7a0b\u6545\u969c\u8bca\u65ad\u4e2d\uff0cInverse PINN\u6982\u5ff5\u5e38\u88ab\u63d0\u53ca\uff0c\u610f\u6307\u4e13\u95e8\u7528\u4e8e\u53cd\u6f14\u7cfb\u7edf\u5065\u5eb7\u72b6\u6001\u7684PINN\u6a21\u578b<a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC12111269\/#:~:text=2,temperature%2C%20pressure\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov<\/a>\u3002\u4f8b\u5982\u524d\u8ff0GAN+PINN\u67b6\u6784\u4e2d\uff0cPINN\u627f\u62c5\u7684\u6b63\u662f\u9006\u95ee\u9898\u6c42\u89e3\u4efb\u52a1\uff0c\u6839\u636e\u6e29\u5ea6\u3001\u538b\u529b\u7b49\u5de5\u51b5\u6570\u636e<strong>\u56de\u63a8<\/strong>\u53ef\u80fd\u7684\u6545\u969c\u53c2\u6570\u53d8\u5316<a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC12111269\/#:~:text=1,fault%20data%2C%20expanding%20the%20dataset\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov <\/a><a href=\"https:\/\/pmc.ncbi.nlm.nih.gov\/articles\/PMC12111269\/#:~:text=2,temperature%2C%20pressure\" target=\"_blank\" rel=\"noreferrer noopener\">pmc.ncbi.nlm.nih.gov<\/a>\u3002\u9006\u95ee\u9898PINN\u5df2\u6210\u529f\u5e94\u7528\u4e8e\u7ed3\u6784\u635f\u4f24\uff08\u63a8\u65ad\u521a\u5ea6\u8870\u51cf\uff09\u3001\u6d41\u4f53\u6e17\u900f\uff08\u63a8\u65ad\u6e17\u900f\u7387\u573a\uff09\u7b49\u9886\u57df\uff0c\u5c55\u793a\u4e86\u9ad8\u5ea6\u7075\u6d3b\u7684\u9002\u5e94\u6027\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>4.4 PINN\u4e3b\u8981\u53d8\u79cd\u9002\u7528\u6027\u5bf9\u6bd4\uff1a<\/strong>\u7efc\u4e0a\uff0c\u4e0d\u540cPINN\u53d8\u4f53\u9488\u5bf9\u4e0d\u540c\u56f0\u96be\u505a\u51fa\u6539\u8fdb\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u81ea\u9002\u5e94PINN\uff08Adaptive PINN\uff09<\/strong>\uff1a\u5305\u62ec\u81ea\u9002\u5e94\u6743\u91cd\u548c\u81ea\u9002\u5e94\u91c7\u6837\u4e24\u65b9\u5411\u3002\u524d\u8005\u901a\u8fc7\u52a8\u6001\u8c03\u6574\u635f\u5931\u9879\u6743\u91cd\uff0c\u89e3\u51b3\u591a\u635f\u5931\u4e0d\u5e73\u8861\u95ee\u9898\uff0c\u5df2\u5728\u6e4d\u6d41RANS\u6c42\u89e3\u4e2d\u8bc1\u660e\u53ef\u63d0\u5347\u9ad8Re\u6570\u60c5\u51b5\u4e0b\u7684\u7cbe\u5ea6<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=Compressor%20Cascade%20Flow%20%20with,Surrogate%20Modeling%2C%20%20Turbulence%20Modeling\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=compressor%20cascade%20flow%20using%20physics,neural%20networks%20for%20aerodynamics%2C%202024\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=compressor%20cascade%20flow%20using%20physics,neural%20networks%20for%20aerodynamics%2C%202024\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u540e\u8005\u901a\u8fc7\u91cd\u70b9\u91c7\u6837\u6b8b\u5dee\u5927\u7684\u533a\u57df\uff0c\u9002\u7528\u4e8e\u5b58\u5728\u5c40\u90e8\u5267\u70c8\u68af\u5ea6\u6216\u8fb9\u754c\u5c42\u7684\u95ee\u9898\uff0c\u53ef\u52a0\u5feb\u6536\u655b<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=The%20architecture%20and%20training%20strategy,adaptive%20learning%20rate%20schedules%2C%20including\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u603b\u4f53\u6765\u8bf4\uff0c\u81ea\u9002\u5e94PINN\u662f\u4e00\u79cd<strong>\u8bad\u7ec3\u7b56\u7565<\/strong>\u6539\u8fdb\uff0c\u53ef\u4e0e\u5176\u5b83\u53d8\u79cd\u7ed3\u5408\u4f7f\u7528\uff0c\u5728\u673a\u68b0\u3001\u80fd\u6e90\u7b49\u5404\u9886\u57df\u901a\u7528\u3002<\/li>\n\n\n\n<li><strong>Fourier PINN\uff08F-PINN\uff09<\/strong>\uff1a\u9488\u5bf9\u591a\u5c3a\u5ea6\u9ad8\u9891\u95ee\u9898\u3002\u5b83\u5728\u7f51\u7edc\u4e2d\u9884\u8bbe\u7a20\u5bc6\u7684\u5085\u91cc\u53f6\u57fa\u5e95\uff0c\u5c06\u8f93\u5165\u5750\u6807\u6620\u5c04\u5230\u9ad8\u9891\u7279\u5f81\u7a7a\u95f4\uff0c\u4ece\u800c\u663e\u8457\u589e\u5f3a\u7f51\u7edc\u903c\u8fd1\u9ad8\u9891\u632f\u8361\u89e3\u7684\u80fd\u529b<a href=\"https:\/\/openreview.net\/forum?id=KqRnsEMYLx#:~:text=perform%20a%20theoretical%20analysis%20based,learning%20and%20basis%20selection%20algorithm\" target=\"_blank\" rel=\"noreferrer noopener\">openreview.net <\/a><a href=\"https:\/\/openreview.net\/forum?id=KqRnsEMYLx#:~:text=geometries,We%20show%20the%20advantage%20of\" target=\"_blank\" rel=\"noreferrer noopener\">openreview.net<\/a>\u3002\u540c\u65f6\u914d\u5408<strong>\u81ea\u9002\u5e94\u57fa\u9891\u9009\u62e9\u7b97\u6cd5<\/strong>\uff0c\u81ea\u52a8\u6311\u9009\u4e3b\u8981\u9891\u7387\u5206\u91cf<a href=\"https:\/\/openreview.net\/forum?id=KqRnsEMYLx#:~:text=specified%2C%20dense%20Fourier%20bases,a%20set%20of%20systematic%20experiments\" target=\"_blank\" rel=\"noreferrer noopener\">openreview.net<\/a>\u3002Fourier PINN\u5728\u6d89\u53ca\u6ce2\u4f20\u64ad\u3001\u9707\u52a8\u3001\u9ad8\u96f7\u8bfa\u6e4d\u6d41\u7b49\u95ee\u9898\u4e2d\u6548\u679c\u7a81\u51fa\u3002\u4f8b\u5982\u5728\u4e00\u7ef4\u58f0\u5b66\u6ce2\u95ee\u9898\u4e2d\uff0cFourier PINN\u80fd\u6bd4\u6807\u51c6PINN\u66f4\u597d\u5730\u5b66\u4e60\u5230\u5c16\u5cf0\u72b6\u538b\u529b\u5206\u5e03<a href=\"https:\/\/openreview.net\/forum?id=KqRnsEMYLx#:~:text=Abstract%3A%20Interest%20is%20rising%20in,for%20many%20BCs%20and%20domain\" target=\"_blank\" rel=\"noreferrer noopener\">openreview.net<\/a>\u3002\u56e0\u6b64\uff0c\u5b83\u9002\u7528\u4e8e<strong>\u6c42\u89e3\u6ce2\u52a8\u65b9\u7a0b\u3001\u632f\u52a8\u95ee\u9898<\/strong>\u7b49\u5177\u6709\u9ad8\u9891\u7279\u5f81\u7684\u5de5\u4e1a\u573a\u666f\uff08\u5982\u8d85\u58f0\u68c0\u6d4b\u3001\u71c3\u70e7\u632f\u8361\uff09\u3002<\/li>\n\n\n\n<li><strong>XPINN\uff08\u6269\u5c55PINN\uff09<\/strong>\uff1a\u901a\u8fc7<strong>\u7a7a\u95f4\u65f6\u95f4\u5206\u533a<\/strong>\u5b9e\u73b0\u5e76\u884c\u6c42\u89e3\u548c\u9488\u5bf9\u6027\u8bad\u7ec3<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=independent%20computational%20unit,the%20solution%20of%20nonlinear%20PDE\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u975e\u5e38\u9002\u5408<strong>\u5927\u89c4\u6a21\u590d\u6742\u57df<\/strong>\uff08\u5982\u957f\u7ba1\u9053\u3001\u590d\u6742\u7ed3\u6784\uff09\u4ee5\u53ca<strong>\u591a\u7269\u7406\u5f3a\u8026\u5408<\/strong>\uff08\u4e0d\u540c\u533a\u57df\u7269\u7406\u65b9\u7a0b\u4e0d\u540c\uff09\u7684\u95ee\u9898\u3002XPINN\u7684\u5b50\u57df\u72ec\u7acb\u7f51\u7edc\u4f7f\u5176\u5728\u5206\u5e03\u5f0f\u8ba1\u7b97\u73af\u5883\u4e2d\u4f18\u52bf\u660e\u663e\uff0c\u53ef\u5c06\u5927\u578b\u95ee\u9898\u62c6\u89e3\u540e\u5927\u5e45\u964d\u4f4e\u5355\u4e2a\u7f51\u7edc\u8d1f\u62c5<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=on%20arbitrary%20complex%20geometries,along%20with%20its%20flexible%20domain\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002\u4f8b\u5982\uff0c\u5728\u822a\u7a7a\u53d1\u52a8\u673a\u5185\u90e8\u4e0d\u540c\u6a21\u5757\uff08\u538b\u6c14\u673a\u3001\u71c3\u70e7\u5ba4\u3001\u6da1\u8f6e\uff09\u5206\u522b\u8bad\u7ec3PINN\uff0c\u518d\u5728\u754c\u9762\u5904\u5339\u914d\u6d41\u91cf\/\u6e29\u5ea6\uff0c\u80fd\u66f4\u9ad8\u6548\u5730\u5f97\u5230\u6574\u4f53\u89e3\u3002\u8fd9\u5bf9<strong>\u8d85\u5927\u4f53\u7cfb\u7684\u6570\u5b57\u5b6a\u751f<\/strong>\u5177\u6709\u610f\u4e49\u3002\u5f53\u7136\uff0cXPINN\u9700\u8981\u5904\u7406\u5b50\u57df\u63a5\u53e3\u8fde\u7eed\u6027\u7684\u6280\u672f\u7ec6\u8282\uff0c\u4f46\u5df2\u6709\u6587\u732e\u7ed9\u51fa\u4e86\u4e25\u683c\u4fdd\u5b88\u578b(cPINN)\u548c\u5e7f\u4e49XPINN\u6846\u67b6\u4ee5\u4fdd\u8bc1\u89e3\u7684\u7269\u7406\u4e00\u81f4\u6027<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=extended%20PINNs%20%28XPINNs%29%C2%A0%28,interfaces%2C%20cPINNs%20ensure%20the%20conservative\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=characteristics,for%20complex%20scientific%20computing%20problems\" target=\"_blank\" rel=\"noreferrer noopener\"> <\/a><a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=characteristics,for%20complex%20scientific%20computing%20problems\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002<\/li>\n\n\n\n<li><strong>\u5176\u4ed6\u53d8\u4f53\uff1a\u5982\u8d1d\u53f6\u65afPINN<\/strong>\u7528\u4e8e\u4e0d\u786e\u5b9a\u5ea6\u91cf\u5316\uff0c<strong>\u53d8\u5206PINN (VPINN)<\/strong>\u901a\u8fc7\u5f15\u5165\u5f31\u5f62\u5f0f\u79ef\u5206\u63d0\u9ad8\u5bf9\u590d\u6742\u51e0\u4f55\u9002\u5e94\u6027<a href=\"https:\/\/ww2.mathworks.cn\/discovery\/physics-informed-neural-networks.html#:~:text=PINN%20%E4%BA%8E2017%20%E5%B9%B4%E9%A6%96%E6%AC%A1%E6%8E%A8%E5%87%BA%EF%BC%8C%E7%8E%B0%E5%9C%A8%E6%9C%89%E8%AE%B8%E5%A4%9A%E5%8F%98%E4%BD%93%EF%BC%8C%E5%85%B6%E4%B8%AD%E5%8C%85%E6%8B%AC%EF%BC%9A,VPINN%29%EF%BC%8C%E5%AE%83%E4%BB%AC%E5%B0%86PDE%20%E7%9A%84%E5%BC%B1%E5%BD%A2%E5%BC%8F%E7%BA%B3%E5%85%A5%E6%8D%9F%E5%A4%B1%E5%87%BD%E6%95%B0%E4%B8%AD\" target=\"_blank\" rel=\"noreferrer noopener\">ww2.mathworks.cn<\/a>\u7b49\u3002\u8fd9\u4e9b\u53d8\u4f53\u5404\u6709\u9488\u5bf9\u7684\u4fa7\u91cd\u70b9\uff0c\u5728\u6b64\u4e0d\u4e00\u4e00\u5c55\u5f00\u3002<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u88681\uff1a\u4e3b\u8981PINN\u53d8\u4f53\u65b9\u6cd5\u53ca\u5176\u7279\u70b9<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><thead><tr><th><strong>\u53d8\u4f53\u65b9\u6cd5<\/strong><\/th><th><strong>\u6838\u5fc3\u601d\u60f3\u4e0e\u9002\u7528\u573a\u666f<\/strong><\/th><\/tr><\/thead><tbody><tr><td>\u81ea\u9002\u5e94PINN<\/td><td>\u52a8\u6001\u8c03\u6574\u635f\u5931\u6743\u91cd\u6216\u91c7\u6837\u70b9\u5206\u5e03\uff0c\u4f7f\u8bad\u7ec3\u805a\u7126\u96be\u5b66\u90e8\u4f4d<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=The%20architecture%20and%20training%20strategy,adaptive%20learning%20rate%20schedules%2C%20including\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\uff1b\u9002\u7528\u4e8e\u635f\u5931\u68af\u5ea6\u4e0d\u5e73\u8861\u3001\u5c40\u90e8\u68af\u5ea6\u9661\u5ced\u7684\u95ee\u9898\uff0c\u63d0\u9ad8\u6536\u655b\u901f\u5ea6\u548c\u7cbe\u5ea6\u3002\u5e38\u7528\u4e8e\u6e4d\u6d41\u3001\u9ad8\u975e\u7ebf\u6027\u591a\u9879\u635f\u5931\u4efb\u52a1<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=Compressor%20Cascade%20Flow%20%20with,Surrogate%20Modeling%2C%20%20Turbulence%20Modeling\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002<\/td><\/tr><tr><td>Fourier PINN<\/td><td>\u5728\u7f51\u7edc\u4e2d\u52a0\u5165\u5085\u91cc\u53f6\u57fa\u5e95\u4ee5\u6355\u83b7\u9ad8\u9891\/\u591a\u5c3a\u5ea6\u89e3<a href=\"https:\/\/openreview.net\/forum?id=KqRnsEMYLx#:~:text=perform%20a%20theoretical%20analysis%20based,learning%20and%20basis%20selection%20algorithm\" target=\"_blank\" rel=\"noreferrer noopener\">openreview.net<\/a>\uff1b\u9002\u5408\u6ce2\u4f20\u64ad\u3001\u632f\u52a8\u7b49\u9ad8\u9891\u73b0\u8c61\uff08\u58f0\u5b66\u3001\u5f39\u6ce2\u3001\u6da1\u632f\uff09\uff0c\u89e3\u51b3\u6807\u51c6PINN\u5bf9\u9ad8\u9891\u6a21\u5f0f\u5b66\u4e60\u4e0d\u4f73\u7684\u95ee\u9898<a href=\"https:\/\/openreview.net\/forum?id=KqRnsEMYLx#:~:text=equations%20%28PDEs%29,for%20many%20BCs%20and%20domain\" target=\"_blank\" rel=\"noreferrer noopener\">openreview.net<\/a>\u3002<\/td><\/tr><tr><td>XPINN (\u6269\u5c55PINN)<\/td><td>\u5c06\u6c42\u89e3\u57df\u5206\u5757\uff0c\u5b50\u57df\u7f51\u7edc\u5e76\u884c\u8bad\u7ec3\u5e76\u5728\u754c\u9762\u4ea4\u63e1<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=independent%20computational%20unit,the%20solution%20of%20nonlinear%20PDE\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\uff1b\u9002\u7528\u4e8e\u8d85\u5927\u89c4\u6a21\u6216\u591a\u7269\u7406\u8026\u5408\u7cfb\u7edf\uff08\u5206\u533a\u540e\u95ee\u9898\u7b80\u5316\uff09\uff0c\u63d0\u4f9b\u66f4\u597d\u7684\u6269\u5c55\u6027\u548c\u5e76\u884c\u6027\u80fd\uff0c\u5df2\u5728\u590d\u6742\u6c14\u52a8\/\u4f20\u70ed\/\u7ed3\u6784\u8026\u5408\u95ee\u9898\u4e0a\u6210\u529f\u5e94\u7528<a href=\"https:\/\/arxiv.org\/html\/2506.19503v1#:~:text=on%20arbitrary%20complex%20geometries,along%20with%20its%20flexible%20domain\" target=\"_blank\" rel=\"noreferrer noopener\">arxiv.org<\/a>\u3002<\/td><\/tr><tr><td>Bayesian PINN<\/td><td>\u5728PINN\u57fa\u7840\u4e0a\u5f15\u5165\u8d1d\u53f6\u65af\u63a8\u7406\uff0c\u5bf9\u7f51\u7edc\u53c2\u6570\/\u8f93\u51fa\u6784\u5efa\u6982\u7387\u5206\u5e03\uff1b\u9002\u7528\u4e8e\u9700\u8981\u7ed3\u679c\u4e0d\u786e\u5b9a\u5ea6\u8bc4\u4ef7\u7684\u573a\u666f\uff08\u5982\u6838\u5b89\u5168\u88d5\u5ea6\u8bc4\u4f30\uff09\uff0c\u63d0\u9ad8\u6a21\u578b\u51b3\u7b56\u53ef\u9760\u6027\u3002<\/td><\/tr><tr><td>\u53d8\u5206PINN (VPINN)<\/td><td>\u5229\u7528\u53d8\u5206\u5f31\u5f62\u5f0f\uff0c\u5c06\u8bd5\u63a2\u89e3\u4ee5\u9ad8\u6b21\u57fa\u51fd\u6570\u8fd1\u4f3c\uff0c\u7269\u7406\u7ea6\u675f\u4ee5\u79ef\u5206\u5f62\u5f0f\u52a0\u5165\uff1b\u5bf9\u590d\u6742\u51e0\u4f55\u548c\u9ad8\u9636\u8fde\u7eed\u6027\u8981\u6c42\u964d\u4f4e\uff0c\u63d0\u5347\u4e86\u6536\u655b\u6027\u3002\u9002\u7528\u4e8e\u9ad8\u9636PDE\u3001\u590d\u6742\u51e0\u4f55\u7ed3\u6784\u7684\u95ee\u9898<a href=\"https:\/\/www.sciengine.com\/doi\/10.1360\/TB-2024-0683#:~:text=Recent%20progress%20on%20mechanics%20investigations,informed%20neural\" target=\"_blank\" rel=\"noreferrer noopener\">sciengine.com<\/a>\u3002<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><em>\u88681\uff1a\u5217\u51fa\u4e86PINN\u7684\u82e5\u5e72\u91cd\u8981\u53d8\u79cd\u65b9\u6cd5\u53ca\u5176\u7279\u70b9\u3002\u81ea\u9002\u5e94PINN\u7740\u91cd\u89e3\u51b3\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u7684\u68af\u5ea6\u4e0d\u5747\u8861\u4e0e\u91c7\u6837\u4f4e\u6548\u95ee\u9898\uff1bFourier PINN\u65e8\u5728\u589e\u5f3a\u5bf9\u9ad8\u9891\u3001\u591a\u5c3a\u5ea6\u89e3\u7684\u62df\u5408\u80fd\u529b\uff1bXPINN\u901a\u8fc7\u7a7a\u95f4\/\u65f6\u95f4\u5206\u89e3\u6765\u5e94\u5bf9\u5927\u89c4\u6a21\u548c\u591a\u7269\u7406\u8026\u5408\uff1bBayesian PINN\u548cVPINN\u5219\u5206\u522b\u5173\u6ce8\u4e0d\u786e\u5b9a\u5ea6\u91cf\u5316\u548c\u6c42\u89e3\u8fc7\u7a0b\u7684\u6570\u5b66\u5f62\u5f0f\u6539\u8fdb\u7b49\u3002\u5b83\u4eec\u4e3aPINN\u5728\u4e0d\u540c\u5de5\u4e1a\u573a\u666f\u4e0b\u7684\u5e94\u7528\u63d0\u4f9b\u4e86\u6709\u529b\u5de5\u5177\u3002<\/em><\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"1024\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2024\/08\/physics-plus-neural-networks_AI.jpg\" alt=\"\" class=\"wp-image-4288\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2024\/08\/physics-plus-neural-networks_AI.jpg 1024w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/08\/physics-plus-neural-networks_AI-300x300.jpg 300w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/08\/physics-plus-neural-networks_AI-150x150.jpg 150w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/08\/physics-plus-neural-networks_AI-768x768.jpg 768w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/08\/physics-plus-neural-networks_AI-120x120.jpg 120w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-16-9 wp-has-aspect-ratio\"><div class=\"wp-block-embed__wrapper\">\n<iframe loading=\"lazy\" title=\"Physics-Informed Neural Networks (PINNs) - An Introduction - Ben Moseley | Jousef Murad\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/G_hIppUWcsc?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div><figcaption class=\"wp-element-caption\">Physics-Informed Neural Networks (PINNs) &#8211; An Introduction &#8211; <strong><a href=\"https:\/\/benmoseley.blog\/\" target=\"_blank\" rel=\"noreferrer noopener\">Ben Moseley<\/a><\/strong><\/figcaption><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">\u63a8\u8350<a href=\"https:\/\/benmoseley.blog\/\" target=\"_blank\" rel=\"noreferrer noopener\">Ben Moseley<\/a>\u7684\u5173\u4e8ePINNs\u7684\u6587\u7ae0\u548cDemo\u7a0b\u5e8f\uff1a<br><a href=\"https:\/\/benmoseley.blog\/my-research\/so-what-is-a-physics-informed-neural-network\/\" target=\"_blank\" rel=\"noreferrer noopener\">So, what is a physics-informed neural network?<\/a><br><a href=\"https:\/\/github.com\/benmoseley\/harmonic-oscillator-pinn\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/github.com\/benmoseley\/harmonic-oscillator-pinn<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Physics-Informed Neural Networks (PINNs) \u662f\u4e00\u79cd\u5c06\u7269\u7406\u77e5\u8bc6\u878d\u5165\u795e\u7ecf\u7f51\u7edc [&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 center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8,36],"tags":[39,63,48,71],"class_list":["post-4287","post","type-post","status-publish","format-standard","hentry","category-tech","category-36","tag-ai","tag-48","tag-physics"],"rttpg_featured_image_url":null,"rttpg_author":{"display_name":"NullThought","author_link":"https:\/\/nullthought.net\/?author=1"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/nullthought.net\/?cat=8\" rel=\"category\">Tech<\/a> <a href=\"https:\/\/nullthought.net\/?cat=36\" rel=\"category\">\u79d1\u5b66<\/a>","rttpg_excerpt":"Physics-Informed Neural Networks (PINNs) \u662f\u4e00\u79cd\u5c06\u7269\u7406\u77e5\u8bc6\u878d\u5165\u795e\u7ecf\u7f51\u7edc&hellip;","_links":{"self":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/4287","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=4287"}],"version-history":[{"count":6,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/4287\/revisions"}],"predecessor-version":[{"id":6589,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/4287\/revisions\/6589"}],"wp:attachment":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4287"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4287"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4287"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}