{"id":4695,"date":"2024-09-19T10:56:17","date_gmt":"2024-09-19T02:56:17","guid":{"rendered":"https:\/\/nullthought.net\/?p=4695"},"modified":"2024-09-19T10:56:19","modified_gmt":"2024-09-19T02:56:19","slug":"%e7%89%a9%e7%90%86%e4%bf%a1%e6%81%af%e7%a5%9e%e7%bb%8f%e7%ae%97%e5%ad%90%ef%bc%88physics-informed-neural-operator%ef%bc%8cpino%ef%bc%89%e6%9c%89%e6%95%88%e6%8f%90%e5%8d%87%e6%b7%b7%e6%b2%8c%e7%b3%bb","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=4695","title":{"rendered":"\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7b97\u5b50\uff08Physics-Informed Neural Operator\uff0cPINO\uff09\u6709\u6548\u63d0\u5347\u6df7\u6c8c\u7cfb\u7edf\u6a21\u62df\u7684\u6548\u7387\u548c\u7cbe\u5ea6"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2408.05177\" target=\"_blank\" rel=\"noreferrer noopener\">Beyond Closure Models: Learning Chaotic Systems via Physics-Informed Neural Operators<\/a><\/strong>\u8be6\u7ec6\u63a2\u8ba8\u4e86\u5982\u4f55\u4f7f\u7528\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7b97\u5b50\uff08Physics-Informed Neural Operator\uff0cPINO\uff09\u6765\u514b\u670d\u73b0\u6709\u6df7\u6c8c\u7cfb\u7edf\u6a21\u62df\u65b9\u6cd5\u7684\u5c40\u9650\u6027\u3002\u73b0\u6709\u65b9\u6cd5\u4f9d\u8d56\u4e8e\u6240\u8c13\u7684\u201c\u95ed\u5408\u6a21\u578b\u201d\uff08closure models\uff09\uff0c\u4f46\u8fd9\u4e9b\u6a21\u578b\u5b58\u5728\u6839\u672c\u6027\u7684\u4e0d\u8db3\uff0c\u5c24\u5176\u662f\u5728\u957f\u671f\u7edf\u8ba1\u4f30\u8ba1\u4e2d\u7684\u8868\u73b0\u3002<\/p>\n\n\n\n<p>\u5728\u6df7\u6c8c\u7cfb\u7edf\u6a21\u62df\u4e2d\u7684\u91cd\u8981\u5e94\u7528\uff0c\u5c24\u5176\u662f\u5728\u6c14\u5019\u5efa\u6a21\u3001\u6e4d\u6d41\u6a21\u62df\u7b49\u8ba1\u7b97\u91cf\u5e9e\u5927\u7684\u95ee\u9898\u4e0a\uff0cPINO\u65b9\u6cd5\u5c55\u793a\u4e86\u5176\u5de8\u5927\u7684\u6f5c\u529b\u3002\u76f8\u6bd4\u4f20\u7edf\u7684\u95ed\u5408\u6a21\u578b\uff0cPINO\u4e0d\u4ec5\u5728\u6548\u7387\u4e0a\u6709\u663e\u8457\u63d0\u5347\uff0c\u8fd8\u5728\u7cbe\u5ea6\u4e0a\u6709\u5927\u5e45\u6539\u5584\u3002\u672a\u6765\u7684\u5de5\u4f5c\u53ef\u4ee5\u8fdb\u4e00\u6b65\u6269\u5c55PINO\u5728\u5176\u4ed6\u7269\u7406\u7cfb\u7edf\u4e2d\u7684\u5e94\u7528\uff0c\u5e76\u901a\u8fc7\u66f4\u590d\u6742\u7684\u7269\u7406\u4fe1\u606f\u7ea6\u675f\u63d0\u9ad8\u6a21\u578b\u7684\u6027\u80fd\u200b\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aChuwei Wang, Julius Berner, Zongyi Li, Di Zhou, Jiayun Wang, Jane Bae, Anima Anandkumar\uff0c\u5747\u6765\u81ea\u52a0\u5dde\u7406\u5de5\u5b66\u9662\uff08California Institute of Technology\uff0cCaltech\uff09\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">1. \u7814\u7a76\u80cc\u666f\u4e0e\u95ee\u9898\u5b9a\u4e49<\/h5>\n\n\n\n<p>\u5728\u6c14\u5019\u5efa\u6a21\u3001\u98de\u884c\u5668\u8bbe\u8ba1\u548c\u6838\u805a\u53d8\u7b49\u9886\u57df\uff0c\u79d1\u5b66\u5bb6\u4eec\u9762\u4e34\u7684\u4e00\u4e2a\u5171\u540c\u6311\u6218\u662f\u6df7\u6c8c\u7cfb\u7edf\u7684\u957f\u671f\u9884\u6d4b\u3002\u6df7\u6c8c\u7cfb\u7edf\u56e0\u5176\u5bf9\u521d\u59cb\u6761\u4ef6\u6781\u5176\u654f\u611f\u7684\u7279\u6027\uff0c\u4f7f\u5f97\u5c0f\u7684\u8bef\u5dee\u4f1a\u968f\u7740\u65f6\u95f4\u7684\u63a8\u79fb\u8fc5\u901f\u653e\u5927\uff0c\u5bfc\u81f4\u9884\u6d4b\u8f68\u8ff9\u7684\u504f\u79bb\u3002\u56e0\u6b64\uff0c\u4f20\u7edf\u7684\u6570\u503c\u6a21\u62df\u65b9\u6cd5\u5fc5\u987b\u5728\u6781\u7ec6\u7684\u65f6\u7a7a\u7f51\u683c\u4e0a\u8fdb\u884c\u9ad8\u4fdd\u771f\u6a21\u62df\uff0c\u4ee5\u5c06\u79bb\u6563\u5316\u8bef\u5dee\u964d\u5230\u6700\u4f4e\u3002\u8fd9\u79cd\u6a21\u62df\u65b9\u6cd5\u901a\u5e38\u88ab\u79f0\u4e3a\u5168\u89e3\u6790\u6a21\u62df\uff08Fully-Resolved Simulations\uff0cFRS\uff09\uff0c\u4f46\u8fd9\u79cd\u65b9\u6cd5\u7684\u8ba1\u7b97\u6210\u672c\u6781\u9ad8\u3002\u4f8b\u5982\uff0c\u5927\u6c14\u4e2d\u7684\u4e00\u4e2a\u5c0f\u533a\u57df\u7684\u9ad8\u4fdd\u771f\u6a21\u62df\u53ef\u80fd\u9700\u8981\u6570\u6708\u65f6\u95f4\u548c\u6570PB\uff08\u5343\u5146\u5b57\u8282\uff09\u7684\u5185\u5b58\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u51cf\u5c11\u8ba1\u7b97\u6210\u672c\uff0c\u7814\u7a76\u4eba\u5458\u901a\u5e38\u4f7f\u7528\u7c97\u7f51\u683c\u8fdb\u884c\u6a21\u62df\uff0c\u5e76\u901a\u8fc7\u95ed\u5408\u6a21\u578b\uff08closure model\uff09\u6765\u4fee\u6b63\u8bef\u5dee\u3002\u95ed\u5408\u6a21\u578b\u65e8\u5728\u5f25\u8865\u7c97\u7f51\u683c\u4e2d\u672a\u6355\u6349\u5230\u7684\u7cbe\u7ec6\u5c3a\u5ea6\u7684\u4fe1\u606f\u3002\u4f8b\u5982\uff0c\u5728\u6e4d\u6d41\u6a21\u62df\u4e2d\uff0c\u5927\u6da1\u6a21\u62df\uff08LES\uff09\u5c31\u662f\u4e00\u79cd\u5178\u578b\u7684\u7c97\u7f51\u683c\u65b9\u6cd5\u3002\u7136\u800c\uff0c\u95ed\u5408\u6a21\u578b\u7684\u8bbe\u8ba1\u4f9d\u8d56\u4e8e\u7269\u7406\u76f4\u89c9\uff0c\u901a\u5e38\u9700\u8981\u5927\u91cf\u7684\u9886\u57df\u77e5\u8bc6\u548c\u5047\u8bbe\uff0c\u4e14\u8fd9\u4e9b\u6a21\u578b\u5728\u771f\u5b9e\u7684\u590d\u6742\u7cfb\u7edf\u4e2d\u8868\u73b0\u5f80\u5f80\u4e0d\u4f73\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">2. 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class=\"wp-block-heading\">3. PINO\u65b9\u6cd5\u7684\u63d0\u51fa\u4e0e\u4f18\u52bf<\/h5>\n\n\n\n<p>\u4e3a\u4e86\u514b\u670d\u8fd9\u4e9b\u5c40\u9650\u6027\uff0c\u8bba\u6587\u63d0\u51fa\u4e86\u4e00\u79cd\u65b0\u7684\u5b66\u4e60\u6846\u67b6\u2014\u2014\u57fa\u4e8e\u7269\u7406\u4fe1\u606f\u7684\u795e\u7ecf\u7b97\u5b50\uff08PINO\uff09\u3002\u4e0e\u4f20\u7edf\u7684\u95ed\u5408\u6a21\u578b\u4e0d\u540c\uff0cPINO\u65b9\u6cd5\u4e0d\u53d7\u9650\u4e8e\u7c97\u7f51\u683c\uff0c\u800c\u662f\u901a\u8fc7\u5b66\u4e60\u51fd\u6570\u7a7a\u95f4\u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u80fd\u591f\u5728\u4e0d\u540c\u5206\u8fa8\u7387\u4e0b\u8fdb\u884c\u6a21\u62df\u3002\u5177\u4f53\u6765\u8bf4\uff0cPINO\u6a21\u578b\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u5b9e\u73b0\u6df7\u6c8c\u7cfb\u7edf\u7684\u9ad8\u6548\u6a21\u62df\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u8bad\u7ec3\u6b65\u9aa4<\/strong>\uff1a\u9996\u5148\uff0cPINO\u6a21\u578b\u5728\u6765\u81ea\u7c97\u7f51\u683c\u6c42\u89e3\u5668\u7684\u6a21\u62df\u6570\u636e\u4e0a\u8fdb\u884c\u521d\u6b65\u8bad\u7ec3\u3002\u7531\u4e8e\u7c97\u7f51\u683c\u6a21\u62df\u7684\u8ba1\u7b97\u6210\u672c\u76f8\u5bf9\u8f83\u4f4e\uff0c\u56e0\u6b64\u53ef\u4ee5\u83b7\u53d6\u8db3\u591f\u7684\u6570\u636e\u6765\u8fdb\u884c\u521d\u6b65\u8bad\u7ec3\u3002<\/li>\n\n\n\n<li><strong>\u5fae\u8c03\u6b65\u9aa4<\/strong>\uff1a\u5728\u521d\u6b65\u8bad\u7ec3\u540e\uff0cPINO\u6a21\u578b\u901a\u8fc7\u5c11\u91cf\u7684\u5168\u89e3\u6790\u6a21\u62df\u6570\u636e\uff08FRS\uff09\u4ee5\u53ca\u57fa\u4e8e\u7269\u7406\u5b9a\u5f8b\u7684\u635f\u5931\u51fd\u6570\u8fdb\u884c\u5fae\u8c03\u3002\u8fd9\u4e00\u8fc7\u7a0b\u5927\u5927\u51cf\u5c11\u4e86\u5bf9\u9ad8\u4fdd\u771f\u6570\u636e\u7684\u4f9d\u8d56\u200b\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u901a\u8fc7\u8fd9\u79cd\u65b9\u6cd5\uff0cPINO\u6a21\u578b\u80fd\u591f\u9ad8\u6548\u5730\u8fd1\u4f3c\u6df7\u6c8c\u7cfb\u7edf\u7684\u957f\u671f\u7edf\u8ba1\u91cf\uff0c\u5e76\u4e14\u76f8\u8f83\u4e8e\u4f20\u7edf\u7684\u95ed\u5408\u6a21\u578b\u548c\u7c97\u7f51\u683c\u6c42\u89e3\u5668\u5177\u6709\u663e\u8457\u7684\u4f18\u52bf\u3002\u8bba\u6587\u7684\u5b9e\u9a8c\u8868\u660e\uff0cPINO\u6a21\u578b\u5728\u4fdd\u6301\u4f4e\u8bef\u5dee\u7684\u540c\u65f6\uff0c\u901f\u5ea6\u6bd4\u5168\u89e3\u6790\u6a21\u62df\u5feb\u4e86120\u500d\uff0c\u800c\u4f20\u7edf\u7684\u95ed\u5408\u6a21\u578b\u5728\u76f8\u540c\u6570\u636e\u96c6\u4e0a\u7684\u8bef\u5dee\u5374\u9ad8\u8fbe205%\uff0c\u901f\u5ea6\u5219\u6bd4PINO\u616258\u500d\u200b\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4. \u7406\u8bba\u5206\u6790<\/h5>\n\n\n\n<p>\u8bba\u6587\u901a\u8fc7\u4e00\u9879\u91cd\u8981\u7684\u7406\u8bba\u8d21\u732e\u8bc1\u660e\u4e86\uff0c\u4f20\u7edf\u7684\u95ed\u5408\u6a21\u578b\u5728\u7c97\u7f51\u683c\u4e0a\u5b66\u4e60\u7684\u6620\u5c04\u662f\u201c\u75c5\u6001\u201d\u7684\uff0c\u5373\u8fd9\u4e9b\u6a21\u578b\u4e0d\u80fd\u51c6\u786e\u5730\u8fd1\u4f3c\u6df7\u6c8c\u7cfb\u7edf\u7684\u957f\u671f\u7edf\u8ba1\u7279\u6027\u3002\u76f8\u53cd\uff0cPINO\u65b9\u6cd5\u901a\u8fc7\u4f7f\u7528\u795e\u7ecf\u7b97\u5b50\u76f4\u63a5\u5728\u51fd\u6570\u7a7a\u95f4\u4e2d\u5b66\u4e60\u89e3\u51b3\u4e86\u8fd9\u4e00\u95ee\u9898\uff0c\u5e76\u4e14\u8bc1\u660e\u4e86PINO\u6a21\u578b\u53ef\u4ee5\u5728\u8f83\u5c0f\u7684\u8bad\u7ec3\u6570\u636e\u4e0b\uff0c\u903c\u8fd1\u7cfb\u7edf\u7684\u957f\u671f\u7edf\u8ba1\u91cf\u200b\u3002<\/p>\n\n\n\n<p>\u6b64\u5916\uff0c\u8bba\u6587\u63d0\u51fa\u4e86\u4e00\u4e2a\u65b0\u7684\u7406\u8bba\u6846\u67b6\u2014\u2014\u529f\u80fdLiouville\u6d41\uff08Functional Liouville 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\u5b9e\u9a8c\u9a8c\u8bc1<\/h5>\n\n\n\n<p>\u4e3a\u4e86\u9a8c\u8bc1PINO\u65b9\u6cd5\u7684\u6709\u6548\u6027\uff0c\u8bba\u6587\u5728\u6d41\u4f53\u52a8\u529b\u5b66\u9886\u57df\u7684\u4e24\u4e2a\u5178\u578b\u6df7\u6c8c\u7cfb\u7edf\u4e0a\u8fdb\u884c\u4e86\u5b9e\u9a8c\uff1a1D\u7684Kuramoto-Sivashinsky\u65b9\u7a0b\u548c2D\u7684Navier-Stokes\u65b9\u7a0b\u3002\u5b9e\u9a8c\u7ed3\u679c\u8868\u660e\uff0cPINO\u6a21\u578b\u5728\u957f\u65f6\u95f4\u7edf\u8ba1\u9884\u6d4b\u4e2d\u7684\u8868\u73b0\u8fdc\u4f18\u4e8e\u4f20\u7edf\u7684\u95ed\u5408\u6a21\u578b\u548c\u5176\u4ed6\u57fa\u4e8e\u673a\u5668\u5b66\u4e60\u7684\u6a21\u578b\u200b\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Kuramoto-Sivashinsky\u65b9\u7a0b<\/strong>\uff1aPINO\u6a21\u578b\u5728\u8be5\u65b9\u7a0b\u4e0a\u7684\u76f8\u5bf9\u8bef\u5dee\u4e3a12%\uff0c\u76f8\u6bd4\u4e4b\u4e0b\uff0c\u4f20\u7edf\u7684\u95ed\u5408\u6a21\u578b\u548c\u5355\u72b6\u6001\u6a21\u578b\u7684\u8bef\u5dee\u90fd\u8f83\u5927\u3002<\/li>\n\n\n\n<li><strong>Navier-Stokes\u65b9\u7a0b<\/strong>\uff1aPINO\u6a21\u578b\u5728\u8be5\u65b9\u7a0b\u4e0a\u7684\u76f8\u5bf9\u8bef\u5dee\u4e3a19%\uff0c\u5728\u4e0d\u540c\u7684\u7edf\u8ba1\u91cf\uff08\u4f8b\u5982\u80fd\u8c31\u3001\u6da1\u91cf\u5206\u5e03\u7b49\uff09\u4e0a\u4e5f\u4f18\u4e8e\u4f20\u7edf\u65b9\u6cd5\u200b\u3002<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u8bba\u6587Beyond Closure Models: Learning Chaotic Systems via P [&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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