{"id":5286,"date":"2024-12-09T23:23:43","date_gmt":"2024-12-09T15:23:43","guid":{"rendered":"https:\/\/nullthought.net\/?p=5286"},"modified":"2025-02-23T11:08:36","modified_gmt":"2025-02-23T03:08:36","slug":"codomain-attention-neural-operator-coda-no%ef%bc%9a%e5%bc%95%e5%85%a5%e5%80%bc%e5%9f%9f%e6%b3%a8%e6%84%8f%e5%8a%9b%e6%9c%ba%e5%88%b6%e7%9a%84%e7%a5%9e%e7%bb%8f%e7%ae%97%e5%ad%90","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=5286","title":{"rendered":"Codomain Attention Neural Operator (CoDA-NO)\uff1a\u5f15\u5165\u503c\u57df\u6ce8\u610f\u529b\u673a\u5236\u7684\u795e\u7ecf\u7b97\u5b50"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2403.12553\" target=\"_blank\" rel=\"noreferrer noopener\">Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs<\/a><\/strong>\u63d0\u51fa\u65b0\u578b\u67b6\u6784Codomain Attention Neural Operator (CoDA-NO)\u3002CoDA-NO\u901a\u8fc7\u5f15\u5165\u503c\u57df\u6ce8\u610f\u529b\u673a\u5236\u548c\u81ea\u76d1\u7763\u9884\u8bad\u7ec3\u7b56\u7565\uff0c\u4e3a\u591a\u7269\u7406\u573aPDE\u6c42\u89e3\u63d0\u4f9b\u4e86\u4e00\u79cd\u5168\u65b0\u7684\u89e3\u51b3\u65b9\u6848\u3002\u8be5\u67b6\u6784\u80fd\u591f\u6709\u6548\u5904\u7406\u5177\u6709\u590d\u6742\u4ea4\u4e92\u4f5c\u7528\u7684\u591a\u7269\u7406\u573a\u95ee\u9898\uff0c\u5e76\u901a\u8fc7\u5c11\u6837\u672c\u5fae\u8c03\u5b9e\u73b0\u5feb\u901f\u9002\u5e94\u3002CoDA-NO\u7684\u8868\u73b0\u4f18\u4e8e\u4f20\u7edf\u7684\u6570\u503c\u65b9\u6cd5\u548c\u73b0\u6709\u7684\u795e\u7ecf\u7f51\u7edc\u65b9\u6cd5\uff0c\u5c24\u5176\u5728\u6570\u636e\u7a00\u7f3a\u548c\u591a\u7269\u7406\u573a\u95ee\u9898\u4e2d\u5c55\u73b0\u51fa\u8f83\u5f3a\u7684\u80fd\u529b\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3a\uff0c\u6765\u81eaMd Ashiqur Rahman, Robert Joseph George, Mogab Elleithy, Daniel Leibovici, Zongyi Li, Boris Bonev, Colin White, Julius Berner, Raymond A. Yeh, Jean Kossaifi, Kamyar Azizzadenesheli, Anima Anandkumar\uff0c\u6765\u81eaPurdue University, Caltech, NVIDIA\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"389\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-8-1024x389.png\" alt=\"\" class=\"wp-image-5287\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-8-1024x389.png 1024w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-8-300x114.png 300w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-8-768x292.png 768w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-8.png 1036w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><strong><a href=\"https:\/\/arxiv.org\/abs\/2403.12553\" target=\"_blank\" rel=\"noreferrer noopener\">Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs<\/a><\/strong><\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>\u4e00\u3001\u80cc\u666f\u4e0e\u95ee\u9898<\/strong><\/h4>\n\n\n\n<p>\u5728\u8bb8\u591a\u79d1\u5b66\u548c\u5de5\u7a0b\u95ee\u9898\u4e2d\uff0c\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u7528\u6765\u63cf\u8ff0\u7269\u7406\u73b0\u8c61\uff0c\u5982\u6d41\u4f53\u52a8\u529b\u5b66\u3001\u6ce2\u4f20\u64ad\u3001\u6750\u6599\u53d8\u5f62\u7b49\u3002\u5c24\u5176\u5728\u6d89\u53ca\u591a\u4e2a\u7269\u7406\u8fc7\u7a0b\u76f8\u4e92\u4f5c\u7528\u7684\u591a\u7269\u7406\u573a\uff08multiphysics\uff09\u95ee\u9898\u4e2d\uff0c\u591a\u4e2aPDE\u7cfb\u7edf\u9700\u8981\u8026\u5408\u6c42\u89e3\u3002\u4f20\u7edf\u7684\u6570\u503c\u65b9\u6cd5\uff0c\u5982\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\u7b49\uff0c\u867d\u7136\u80fd\u591f\u89e3\u51b3\u8fd9\u4e9b\u95ee\u9898\uff0c\u4f46\u5b83\u4eec\u901a\u5e38\u9762\u4e34\u8ba1\u7b97\u91cf\u5927\u3001\u5185\u5b58\u9700\u6c42\u9ad8\u3001\u96be\u4ee5\u5e94\u5bf9\u590d\u6742\u51e0\u4f55\u5f62\u72b6\u548c\u9ad8\u5206\u8fa8\u7387\u6570\u636e\u7684\u95ee\u9898\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u8026\u5408PDE\u95ee\u9898\u7684\u590d\u6742\u6027<\/strong>\uff1a\u591a\u7269\u7406\u573a\u95ee\u9898\u6d89\u53ca\u591a\u4e2a\u8026\u5408\u7684\u7269\u7406\u65b9\u7a0b\uff0c\u8fd9\u4e9b\u65b9\u7a0b\u4e4b\u95f4\u7684\u590d\u6742\u4ea4\u4e92\u4f7f\u5f97\u6c42\u89e3\u8fc7\u7a0b\u66f4\u4e3a\u56f0\u96be\uff0c\u5c24\u5176\u662f\u5f53\u9700\u8981\u8003\u8651\u590d\u6742\u51e0\u4f55\u5f62\u72b6\u548c\u8fb9\u754c\u6761\u4ef6\u65f6\u3002<\/li>\n\n\n\n<li><strong>\u6570\u636e\u7a00\u7f3a\u95ee\u9898<\/strong>\uff1a\u4f20\u7edf\u6570\u503c\u65b9\u6cd5\u9700\u8981\u5927\u91cf\u7684\u9ad8\u5206\u8fa8\u7387\u6570\u636e\u8fdb\u884c\u8bad\u7ec3\uff0c\u4f46\u83b7\u53d6\u8fd9\u4e9b\u6570\u636e\u4e0d\u4ec5\u6210\u672c\u9ad8\uff0c\u8fd8\u9700\u8981\u5927\u91cf\u8ba1\u7b97\u8d44\u6e90\u3002\u5728\u4e00\u4e9b\u591a\u7269\u7406\u573a\u95ee\u9898\u4e2d\uff0c\u5c24\u5176\u662f\u6a21\u62df\u4eff\u771f\u975e\u5e38\u6602\u8d35\uff0c\u83b7\u53d6\u7684\u8bad\u7ec3\u6570\u636e\u5e38\u5e38\u5341\u5206\u6709\u9650\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u73b0\u6709\u7684\u795e\u7ecf\u7f51\u7edc\u65b9\u6cd5\uff0c\u5982<a href=\"https:\/\/nullthought.net\/?s=%E7%A5%9E%E7%BB%8F%E7%AE%97%E5%AD%90\" target=\"_blank\" rel=\"noreferrer noopener\"><strong>\u795e\u7ecf\u7b97\u5b50<\/strong>\uff08Neural Operators\uff09<\/a>\uff0c\u867d\u7136\u80fd\u591f\u5728\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u53d6\u4ee3\u4f20\u7edf\u6570\u503c\u65b9\u6cd5\uff0c\u4f46\u5728\u5904\u7406\u591a\u7269\u7406\u573a\u8026\u5408\u95ee\u9898\u65f6\u5b58\u5728\u663e\u8457\u4e0d\u8db3\uff0c\u4e3b\u8981\u8868\u73b0\u4e3a\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u65e0\u6cd5\u5904\u7406\u4e0d\u540c\u7269\u7406\u53d8\u91cf\u548c\u4e0d\u540c\u7f51\u683c\u5206\u8fa8\u7387\u7684\u591a\u7269\u7406\u573a\u8026\u5408\u95ee\u9898\u3002<\/li>\n\n\n\n<li>\u65e0\u6cd5\u6709\u6548\u5229\u7528\u73b0\u6709\u7684\u6709\u9650\u6570\u636e\u8fdb\u884c\u9884\u8bad\u7ec3\u548c\u8fc1\u79fb\u5b66\u4e60\u3002<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>\u4e8c\u3001CoDA-NO\u67b6\u6784\uff1aCodomain Attention Neural Operator<\/strong><\/h4>\n\n\n\n<p>\u4e3a\u4e86\u5e94\u5bf9\u4e0a\u8ff0\u6311\u6218\uff0c\u8bba\u6587\u63d0\u51fa\u4e86\u4e00\u79cd\u65b0\u578b\u7684\u67b6\u6784\u2014\u2014<strong>CoDA-NO<\/strong>\uff08Codomain Attention Neural Operator\uff09\u3002\u8be5\u65b9\u6cd5\u901a\u8fc7\u5f15\u5165<strong>\u503c\u57df\u6ce8\u610f\u529b\u673a\u5236\uff08codomain attention\uff09<\/strong>\uff0c\u80fd\u591f\u6709\u6548\u5904\u7406\u591a\u7269\u7406\u573aPDE\u95ee\u9898\uff0c\u89e3\u51b3\u4e86\u4f20\u7edf\u795e\u7ecf\u7b97\u5b50\u5728\u591a\u7269\u7406\u573a\u95ee\u9898\u4e2d\u9047\u5230\u7684\u4e00\u4e9b\u5173\u952e\u96be\u9898\u3002<\/p>\n\n\n\n<p><strong>CoDA-NO\u7684\u521b\u65b0\u70b9<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u503c\u57df\u6ce8\u610f\u529b\u673a\u5236<\/strong>\uff1aCoDA-NO\u901a\u8fc7\u5728<strong>\u503c\u57df<\/strong>\u7a7a\u95f4\u5185\u5bf9\u7269\u7406\u53d8\u91cf\u8fdb\u884c\u6807\u8bb0\u548c\u81ea\u6211\u76d1\u7763\u5b66\u4e60\uff0c\u907f\u514d\u4e86\u4f20\u7edf\u65b9\u6cd5\u7684\u5c40\u9650\u3002\u901a\u8fc7\u6269\u5c55\u4f4d\u7f6e\u7f16\u7801\u3001\u6ce8\u610f\u529b\u673a\u5236\u548c\u6b63\u5219\u5316\u65b9\u6cd5\uff0cCoDA-NO\u80fd\u591f\u5904\u7406\u8026\u5408\u7684PDE\u95ee\u9898\uff0c\u5e76\u80fd\u540c\u65f6\u5904\u7406\u591a\u4e2a\u7269\u7406\u7cfb\u7edf\u3002<\/li>\n\n\n\n<li><strong>\u81ea\u76d1\u7763\u9884\u8bad\u7ec3\u548c\u5c11\u6837\u672c\u5fae\u8c03<\/strong>\uff1aCoDA-NO\u80fd\u591f\u901a\u8fc7\u81ea\u76d1\u7763\u5b66\u4e60\u5bf9\u591a\u4e2aPDE\u7cfb\u7edf\u8fdb\u884c\u9884\u8bad\u7ec3\uff0c\u5b66\u4e60\u4e0d\u540cPDE\u7cfb\u7edf\u7684\u8868\u793a\uff0c\u7136\u540e\u4f7f\u7528\u5c11\u91cf\u6570\u636e\u8fdb\u884c\u5fae\u8c03\uff0c\u4ece\u800c\u89e3\u51b3\u6570\u636e\u4e0d\u8db3\u7684\u95ee\u9898\u3002<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>\u4e09\u3001\u67b6\u6784\u8bbe\u8ba1\u4e0e\u65b9\u6cd5<\/strong><\/h4>\n\n\n\n<p>CoDA-NO\u67b6\u6784\u57fa\u4e8eTransformer\uff0c\u5e76\u5bf9\u5176\u8fdb\u884c\u4e86\u6269\u5c55\uff0c\u4f7f\u5176\u80fd\u591f\u5728<strong>\u51fd\u6570\u7a7a\u95f4<\/strong>\u5185\u6267\u884c\u64cd\u4f5c\uff0c\u4ece\u800c\u5904\u7406\u591a\u7269\u7406\u573aPDE\u95ee\u9898\u4e2d\u7684\u51fd\u6570\u6620\u5c04\u95ee\u9898\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u8f93\u5165\u51fd\u6570\u7684\u8868\u793a<\/strong>\uff1a \u8f93\u5165\u7684\u7269\u7406\u53d8\u91cf\u901a\u8fc7<strong>\u4f4d\u7f6e\u7f16\u7801<\/strong>\uff08variable-specific positional encoder, VSPE\uff09\u548c<strong>\u56fe\u795e\u7ecf\u7b97\u5b50\uff08graph neural operator, GNO\uff09<\/strong>\u8fdb\u884c\u5904\u7406\u3002\u6bcf\u4e2a\u7269\u7406\u53d8\u91cf\u88ab\u89c6\u4e3a\u4e00\u4e2a\u51fd\u6570\uff0c\u901a\u8fc7\u5b66\u4e60\u5230\u7684\u5085\u91cc\u53f6\u7cfb\u6570\u8868\u793a\u76f8\u5e94\u7684\u7269\u7406\u7279\u5f81\u3002<\/li>\n\n\n\n<li><strong>\u81ea\u6ce8\u610f\u529b\u673a\u5236\u6269\u5c55<\/strong>\uff1a \u5728\u4f20\u7edf\u7684Transformer\u67b6\u6784\u4e2d\uff0c\u6ce8\u610f\u529b\u673a\u5236\u901a\u8fc7\u67e5\u8be2\uff08query\uff09\u3001\u952e\uff08key\uff09\u548c\u503c\uff08value\uff09\u6765\u8ba1\u7b97\u8f93\u51fa\uff0c\u800cCoDA-NO\u5c06\u8fd9\u4e00\u673a\u5236\u6269\u5c55\u5230\u51fd\u6570\u7a7a\u95f4\u3002\n<ul class=\"wp-block-list\">\n<li>\u5728CoDA-NO\u4e2d\uff0c\u6bcf\u4e2a\u7269\u7406\u53d8\u91cf\u751f\u6210\u5bf9\u5e94\u7684<strong>\u952e\u3001\u67e5\u8be2\u548c\u503c\u51fd\u6570<\/strong>\uff0c\u901a\u8fc7\u5085\u91cc\u53f6\u795e\u7ecf\u7b97\u5b50\uff08FNO\uff09\u751f\u6210\u5b83\u4eec\u7684\u8868\u793a\u3002<\/li>\n\n\n\n<li>\u901a\u8fc7\u8ba1\u7b97\u67e5\u8be2\u4e0e\u952e\u4e4b\u95f4\u7684\u70b9\u79ef\uff0c\u5f97\u5230\u7684\u6ce8\u610f\u529b\u6743\u91cd\u5e94\u7528\u5230\u503c\u51fd\u6570\u4e0a\uff0c\u4ece\u800c\u5f97\u5230\u8f93\u51fa\u51fd\u6570\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u6b63\u5219\u5316\u65b9\u6cd5<\/strong>\uff1a CoDA-NO\u91c7\u7528\u4e86\u4e00\u79cd\u5728\u51fd\u6570\u7a7a\u95f4\u4e2d\u7684<strong>\u6807\u51c6\u5316<\/strong>\u65b9\u6cd5\uff0c\u786e\u4fdd\u6a21\u578b\u5728\u4e0d\u540c\u7269\u7406\u7cfb\u7edf\u4e0b\u7a33\u5b9a\u5de5\u4f5c\u3002\u6bcf\u4e2a\u51fd\u6570\u7684\u6807\u51c6\u5316\u662f\u57fa\u4e8e\u5176\u5747\u503c\u548c\u6807\u51c6\u5dee\u6765\u8fdb\u884c\u7684\u3002<\/li>\n<\/ol>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>\u56db\u3001\u5b9e\u9a8c\u4e0e\u7ed3\u679c<\/strong><\/h4>\n\n\n\n<p><strong>\u591a\u7269\u7406\u573a\u95ee\u9898\u7684\u5e94\u7528<\/strong>\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u6d41\u4f53-\u7ed3\u6784\u76f8\u4e92\u4f5c\u7528\u95ee\u9898<\/strong>\uff1a \u5728\u6d41\u4f53-\u7ed3\u6784\u8026\u5408\u95ee\u9898\u4e2d\uff0c\u9996\u5148\u5bf9Navier-Stokes\u65b9\u7a0b\u8fdb\u884c\u9884\u8bad\u7ec3\u3002\u901a\u8fc7\u5c11\u6837\u672c\u5fae\u8c03\uff0c\u6a21\u578b\u80fd\u591f\u9002\u5e94\u5f39\u6027\u6ce2\u65b9\u7a0b\u7684\u4f4d\u79fb\u573a\u3002\u5b9e\u9a8c\u8868\u660e\uff0cCoDA-NO\u76f8\u8f83\u4e8e\u73b0\u6709\u65b9\u6cd5\u63d0\u9ad8\u4e8636.8%\u7684\u51c6\u786e\u5ea6\u3002<\/li>\n\n\n\n<li><strong>Rayleigh-B\u00e9nard\u5bf9\u6d41\u95ee\u9898<\/strong>\uff1a CoDA-NO\u901a\u8fc7\u5bf9Navier-Stokes\u65b9\u7a0b\u7684\u9884\u8bad\u7ec3\uff0c\u6210\u529f\u5730\u9002\u5e94\u4e86Rayleigh-B\u00e9nard\u5bf9\u6d41\u95ee\u9898\u3002\u5b9e\u9a8c\u7ed3\u679c\u663e\u793a\uff0c\u5728\u6d41\u4f53\u6e29\u5ea6\u4e0e\u901f\u5ea6\u8026\u5408\u7684\u60c5\u5883\u4e0b\uff0cCoDA-NO\u80fd\u663e\u8457\u63d0\u9ad8\u9884\u6d4b\u7cbe\u5ea6\uff0c\u51cf\u5c1150%\u7684\u8bef\u5dee\u3002<\/li>\n<\/ol>\n\n\n\n<p><strong>\u4e0e\u73b0\u6709\u65b9\u6cd5\u7684\u6bd4\u8f83<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>CoDA-NO\u5728\u5c11\u6837\u672c\u5b66\u4e60\u4e2d\u7684\u8868\u73b0\u4f18\u4e8e\u5176\u4ed6\u65b9\u6cd5\u3002\u901a\u8fc7\u9884\u8bad\u7ec3\uff0cCoDA-NO\u80fd\u591f\u8fc5\u901f\u9002\u5e94\u65b0\u7684\u7269\u7406\u573a\u7cfb\u7edf\uff0c\u800c\u4e0d\u9700\u8981\u4ece\u5934\u5f00\u59cb\u8bad\u7ec3\u3002<\/li>\n\n\n\n<li>\u5728\u5b9e\u9a8c\u4e2d\uff0cCoDA-NO\u4e0e\u73b0\u6709\u7684<strong><a href=\"https:\/\/nullthought.net\/?p=4266\" target=\"_blank\" rel=\"noreferrer noopener\">FNO<\/a><\/strong>\u3001DeepONet\u7b49\u65b9\u6cd5\u76f8\u6bd4\uff0c\u5728\u5c11\u91cf\u6570\u636e\u4e0b\u7684\u6027\u80fd\u5927\u5e45\u63d0\u9ad8\uff0c\u5e76\u4e14\u5728\u591a\u4e2a\u7269\u7406\u7cfb\u7edf\u4e0a\u8868\u73b0\u51fa\u8272\u3002<\/li>\n<\/ul>\n\n\n\n<p><strong>\u53c2\u6570\u6548\u7387\u4e0e\u8ba1\u7b97\u6548\u7387<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>CoDA-NO\u5728\u591a\u4e2a\u5b9e\u9a8c\u4e2d\u8868\u73b0\u51fa\u9ad8\u6548\u7684\u53c2\u6570\u5229\u7528\u7387\uff0c\u6bd4\u4f20\u7edf\u7684FNO\u67b6\u6784\u51cf\u5c11\u4e86\u6700\u591a43%\u7684\u6d4b\u8bd5\u8bef\u5dee\uff0c\u4e14\u4ec5\u4f7f\u75282%\u7684\u53c2\u6570\u3002<\/li>\n\n\n\n<li>CoDA-NO\u80fd\u591f\u5904\u7406\u4e0d\u540c\u5206\u8fa8\u7387\u7684\u7f51\u683c\uff0c\u9002\u5e94\u591a\u7269\u7406\u573a\u7cfb\u7edf\u7684\u9700\u6c42\uff0c\u907f\u514d\u4e86\u4f20\u7edf\u65b9\u6cd5\u5728\u5904\u7406\u4e0d\u89c4\u5219\u7f51\u683c\u65f6\u7684\u6027\u80fd\u4e0b\u964d\u3002<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\"><strong>\u4e94\u3001\u672a\u6765\u5de5\u4f5c<\/strong><\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u7ed3\u5408\u7269\u7406\u542f\u53d1\u7684\u5b66\u4e60\u65b9\u6cd5<\/strong>\uff1a\u8fdb\u4e00\u6b65\u96c6\u6210\u7269\u7406\u77e5\u8bc6\u548c\u6a21\u578b\u7ea6\u675f\uff0c\u4ee5\u63d0\u9ad8\u6a21\u578b\u5728\u66f4\u590d\u6742\u7269\u7406\u7cfb\u7edf\u4e2d\u7684\u8868\u73b0\u3002<\/li>\n\n\n\n<li><strong>\u6269\u5c55\u5e94\u7528\u9886\u57df<\/strong>\uff1a\u672a\u6765\u53ef\u4ee5\u5c06CoDA-NO\u5e94\u7528\u4e8e\u66f4\u5e7f\u6cdb\u7684\u9886\u57df\uff0c\u5982\u6c14\u5019\u6a21\u62df\u3001\u6750\u6599\u8bbe\u8ba1\u548c\u751f\u7269\u533b\u5b66\u5de5\u7a0b\u7b49\uff0c\u4ee5\u9a8c\u8bc1\u5176\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u6027\u80fd\u3002<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>CoDA-NO on GitHub: <a href=\"https:\/\/github.com\/neuraloperator\/CoDA-NO\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/github.com\/neuraloperator\/CoDA-NO<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8bba\u6587Pretraining Codomain Attention Neural Operators for S [&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":[39,84,95,71],"class_list":["post-5286","post","type-post","status-publish","format-standard","hentry","category-tech","category-36","tag-ai","tag-nvidia","tag-transformer","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":"\u8bba\u6587Pretraining Codomain Attention Neural Operators for S&hellip;","_links":{"self":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5286","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=5286"}],"version-history":[{"count":3,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5286\/revisions"}],"predecessor-version":[{"id":5290,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5286\/revisions\/5290"}],"wp:attachment":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5286"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5286"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}