{"id":4266,"date":"2024-08-07T17:34:50","date_gmt":"2024-08-07T09:34:50","guid":{"rendered":"https:\/\/nullthought.net\/?p=4266"},"modified":"2024-08-26T13:16:19","modified_gmt":"2024-08-26T05:16:19","slug":"fourier-neural-operators-fnos%ef%bc%8c%e5%82%85%e9%87%8c%e5%8f%b6%e7%a5%9e%e7%bb%8f%e7%ae%97%e5%ad%90","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=4266","title":{"rendered":"Fourier Neural Operators (FNOs)\uff0c\u5085\u91cc\u53f6\u795e\u7ecf\u7b97\u5b50"},"content":{"rendered":"\n<p>Fourier Neural Operators (FNOs) \u662f\u4e00\u79cd\u7528\u4e8e\u5b66\u4e60\u548c\u6a21\u62df\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDEs\uff09\u7684\u65b0\u578b\u795e\u7ecf\u7f51\u7edc\u67b6\u6784\u3002FNOs \u5229\u7528\u5085\u91cc\u53f6\u53d8\u6362\u6765\u5904\u7406\u9ad8\u7ef4\u51fd\u6570\u7a7a\u95f4\u4e0a\u7684\u64cd\u4f5c\uff0c\u4ece\u800c\u5728\u5904\u7406\u590d\u6742\u7684\u7269\u7406\u548c\u5de5\u7a0b\u95ee\u9898\u65f6\u8868\u73b0\u51fa\u8272\u3002\u4ee5\u4e0b\u662f\u5bf9 Fourier Neural Operators \u7684\u8be6\u7ec6\u89e3\u91ca\uff1a<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. <strong>\u80cc\u666f\u4e0e\u52a8\u673a<\/strong><\/h3>\n\n\n\n<p>\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDEs\uff09\u5728\u7269\u7406\u3001\u5de5\u7a0b\u548c\u79d1\u5b66\u8ba1\u7b97\u4e2d\u5e7f\u6cdb\u5e94\u7528\uff0c\u63cf\u8ff0\u4e86\u8bb8\u591a\u81ea\u7136\u73b0\u8c61\u5982\u6d41\u4f53\u52a8\u529b\u5b66\u3001\u70ed\u4f20\u5bfc\u548c\u7535\u78c1\u573a\u3002\u7136\u800c\uff0c\u6570\u503c\u6c42\u89e3 PDEs \u901a\u5e38\u9700\u8981\u5927\u91cf\u7684\u8ba1\u7b97\u8d44\u6e90\u3002\u4f20\u7edf\u7684\u795e\u7ecf\u7f51\u7edc\u867d\u7136\u5728\u5f88\u591a\u4efb\u52a1\u4e2d\u8868\u73b0\u4f18\u5f02\uff0c\u4f46\u5728\u76f4\u63a5\u5904\u7406\u9ad8\u7ef4\u51fd\u6570\u7a7a\u95f4\u65f6\u6548\u7387\u4e0d\u9ad8\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. <strong>Fourier Neural Operators \u7684\u6838\u5fc3\u601d\u60f3<\/strong><\/h3>\n\n\n\n<p>FNOs \u7684\u6838\u5fc3\u601d\u60f3\u662f\u5229\u7528\u5085\u91cc\u53f6\u53d8\u6362\u5c06\u51fd\u6570\u7a7a\u95f4\u4e2d\u7684\u64cd\u4f5c\u8f6c\u5316\u5230\u9891\u57df\u4e2d\u8fdb\u884c\u5904\u7406\uff0c\u7136\u540e\u518d\u901a\u8fc7\u9006\u5085\u91cc\u53f6\u53d8\u6362\u56de\u5230\u539f\u59cb\u7a7a\u95f4\u3002\u8fd9\u79cd\u65b9\u6cd5\u6709\u4ee5\u4e0b\u51e0\u4e2a\u5173\u952e\u6b65\u9aa4\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u5085\u91cc\u53f6\u53d8\u6362<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5bf9\u8f93\u5165\u6570\u636e\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u5c06\u5176\u4ece\u65f6\u57df\uff08\u7a7a\u95f4\u57df\uff09\u8f6c\u6362\u5230\u9891\u57df\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u9891\u57df\u64cd\u4f5c<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5728\u9891\u57df\u4e2d\uff0c\u4f7f\u7528\u795e\u7ecf\u7f51\u7edc\u5bf9\u5085\u91cc\u53f6\u7cfb\u6570\u8fdb\u884c\u64cd\u4f5c\u3002\u8fd9\u79cd\u65b9\u5f0f\u53ef\u4ee5\u66f4\u6709\u6548\u5730\u6355\u6349\u5168\u5c40\u7279\u5f81\uff0c\u7279\u522b\u662f\u5728\u5904\u7406\u590d\u6742\u7684\u975e\u7ebf\u6027\u95ee\u9898\u65f6\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u9006\u5085\u91cc\u53f6\u53d8\u6362<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5bf9\u7ecf\u8fc7\u64cd\u4f5c\u7684\u5085\u91cc\u53f6\u7cfb\u6570\u8fdb\u884c\u9006\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u6062\u590d\u5230\u65f6\u57df\uff08\u7a7a\u95f4\u57df\uff09\uff0c\u5f97\u5230\u6700\u7ec8\u7684\u8f93\u51fa\u7ed3\u679c\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">3. <strong>\u67b6\u6784\u7ec6\u8282<\/strong><\/h3>\n\n\n\n<p>FNO \u7684\u5177\u4f53\u67b6\u6784\u901a\u5e38\u5305\u62ec\u4ee5\u4e0b\u51e0\u4e2a\u90e8\u5206\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u8f93\u5165\u5c42<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5c06\u8f93\u5165\u6570\u636e\u6620\u5c04\u5230\u4e00\u4e2a\u9ad8\u7ef4\u7a7a\u95f4\uff0c\u901a\u5e38\u4f7f\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u795e\u7ecf\u7f51\u7edc\u5c42\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u5085\u91cc\u53f6\u53d8\u6362\u5c42<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5bf9\u9ad8\u7ef4\u6570\u636e\u8fdb\u884c\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u83b7\u53d6\u5176\u9891\u57df\u8868\u793a\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u9891\u57df\u795e\u7ecf\u7f51\u7edc\u5c42<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5728\u9891\u57df\u4e2d\u5e94\u7528\u795e\u7ecf\u7f51\u7edc\uff0c\u5bf9\u5085\u91cc\u53f6\u7cfb\u6570\u8fdb\u884c\u64cd\u4f5c\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u53ef\u4ee5\u5305\u62ec\u591a\u4e2a\u9891\u57df\u5c42\uff0c\u6bcf\u5c42\u90fd\u8fdb\u884c\u4e00\u5b9a\u7684\u975e\u7ebf\u6027\u53d8\u6362\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u9006\u5085\u91cc\u53f6\u53d8\u6362\u5c42<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5c06\u4fee\u6539\u540e\u7684\u5085\u91cc\u53f6\u7cfb\u6570\u8fdb\u884c\u9006\u5085\u91cc\u53f6\u53d8\u6362\uff0c\u5f97\u5230\u65f6\u57df\uff08\u7a7a\u95f4\u57df\uff09\u4e2d\u7684\u8f93\u51fa\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u8f93\u51fa\u5c42<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5c06\u9891\u57df\u8f6c\u6362\u540e\u7684\u6570\u636e\u6620\u5c04\u56de\u539f\u59cb\u7a7a\u95f4\uff0c\u5f97\u5230\u6700\u7ec8\u7684\u9884\u6d4b\u7ed3\u679c\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">4. <strong>\u4f18\u70b9<\/strong><\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u9ad8\u6548\u6027<\/strong>\uff1a\u901a\u8fc7\u5085\u91cc\u53f6\u53d8\u6362\u5904\u7406\u5168\u5c40\u7279\u5f81\uff0c\u53ef\u4ee5\u5728\u8f83\u5c11\u7684\u8ba1\u7b97\u8d44\u6e90\u4e0b\u89e3\u51b3\u590d\u6742\u7684\u9ad8\u7ef4\u95ee\u9898\u3002<\/li>\n\n\n\n<li><strong>\u51c6\u786e\u6027<\/strong>\uff1a\u80fd\u591f\u6355\u6349\u5230\u8f93\u5165\u6570\u636e\u7684\u5168\u5c40\u7279\u5f81\uff0c\u63d0\u9ad8\u4e86\u5bf9\u590d\u6742\u975e\u7ebf\u6027\u73b0\u8c61\u7684\u5efa\u6a21\u80fd\u529b\u3002<\/li>\n\n\n\n<li><strong>\u6cdb\u5316\u6027<\/strong>\uff1a\u5728\u5904\u7406\u4e0d\u540c\u7c7b\u578b\u7684 PDEs \u65f6\u8868\u73b0\u51fa\u826f\u597d\u7684\u6cdb\u5316\u80fd\u529b\uff0c\u9002\u7528\u4e8e\u591a\u79cd\u7269\u7406\u548c\u5de5\u7a0b\u95ee\u9898\u3002<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">5. <strong>\u5e94\u7528<\/strong><\/h3>\n\n\n\n<p>FNOs \u5728\u8bb8\u591a\u9886\u57df\u4e2d\u90fd\u6709\u5e7f\u6cdb\u7684\u5e94\u7528\uff0c\u5305\u62ec\u4f46\u4e0d\u9650\u4e8e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u6c14\u5019\u548c\u5929\u6c14\u9884\u6d4b<\/li>\n\n\n\n<li>\u6d41\u4f53\u52a8\u529b\u5b66\u6a21\u62df<\/li>\n\n\n\n<li>\u6750\u6599\u79d1\u5b66\u4e2d\u7684\u5fae\u89c2\u7ed3\u6784\u9884\u6d4b<\/li>\n\n\n\n<li>\u5176\u4ed6\u6d89\u53ca PDEs \u6c42\u89e3\u7684\u79d1\u5b66\u8ba1\u7b97\u548c\u5de5\u7a0b\u95ee\u9898<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p><strong><a href=\"https:\/\/neuraloperator.github.io\/neuraloperator\/dev\/index.html\" target=\"_blank\" rel=\"noreferrer noopener\">neuraloperator <\/a><\/strong>is a comprehensive library for learning neural operators in PyTorch. It is the official implementation for <strong>Fourier Neural Operators<\/strong> and Tensorized Neural Operators.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Fourier Neural Operators (FNOs) \u662f\u4e00\u79cd\u7528\u4e8e\u5b66\u4e60\u548c\u6a21\u62df\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDEs\uff09\u7684\u65b0 [&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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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":{"0":"post-4266","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"hentry","6":"category-tech","7":"category-36","8":"tag-ai","10":"tag-48","11":"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":"Fourier Neural Operators (FNOs) \u662f\u4e00\u79cd\u7528\u4e8e\u5b66\u4e60\u548c\u6a21\u62df\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDEs\uff09\u7684\u65b0&hellip;","_links":{"self":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/4266","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=4266"}],"version-history":[{"count":2,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/4266\/revisions"}],"predecessor-version":[{"id":4269,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/4266\/revisions\/4269"}],"wp:attachment":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4266"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4266"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4266"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}