{"id":5969,"date":"2025-06-12T10:40:08","date_gmt":"2025-06-12T02:40:08","guid":{"rendered":"https:\/\/nullthought.net\/?p=5969"},"modified":"2025-06-12T10:59:12","modified_gmt":"2025-06-12T02:59:12","slug":"%e5%88%9b%e6%96%b0%e7%9a%84%e7%a5%9e%e7%bb%8f%e7%ae%97%e5%ad%90neural-operator%e6%9e%b6%e6%9e%84%ef%bc%9agaot%ef%bc%88geometry-aware-operator-transformer%ef%bc%89","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=5969","title":{"rendered":"\u521b\u65b0\u7684\u795e\u7ecf\u7b97\u5b50(neural operator)\u67b6\u6784\uff1aGAOT\uff08Geometry Aware Operator Transformer\uff09"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/camlab-ethz.github.io\/GAOT\/static\/pdfs\/gaot.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Geometry Aware Operator Transformer As An Efficient And Accurate Neural Surrogate For PDEs On Arbitrary Domains<\/a><\/strong>\u63d0\u51fa\u4e86\u4e00\u79cd\u65b0\u7684<a href=\"https:\/\/nullthought.net\/?p=4270\" target=\"_blank\" rel=\"noreferrer noopener\">\u795e\u7ecf\u7b97\u5b50(neural operator)<\/a>\u67b6\u6784\uff1aGAOT\uff08Geometry Aware Operator Transformer\uff09\u3002GAOT\u5c06\u65b0\u9896\u7684\u591a\u5c3a\u5ea6\u6ce8\u610f\u529b\u56fe\u795e\u7ecf\u7b97\u5b50\u7f16\u7801\u5668\u4e0e\u89e3\u7801\u5668\u3001\u51e0\u4f55\u5d4c\u5165\u6a21\u5757\u4ee5\u53ca\u89c6\u89c9Transformer\u5904\u7406\u5668\u76f8\u7ed3\u5408\uff0c\u80fd\u591f\u5c06\u6709\u5173\u57df\u548c\u8f93\u5165\u7684\u4fe1\u606f\u51c6\u786e\u5730\u6620\u5c04\u4e3aPDE\u89e3\u7684\u7a33\u5065\u8fd1\u4f3c\u3002\u5728GAOT\u7684\u5b9e\u73b0\u4e2d\u8fd8\u5f15\u5165\u4e86\u591a\u9879\u521b\u65b0\uff0c\u4ee5\u786e\u4fdd\u5176\u8ba1\u7b97\u6548\u7387\u4e0e\u53ef\u6269\u5c55\u6027\u3002\u5728\u591a\u4e2a\u6765\u6e90\u591a\u6837\u7684PDE\u5b66\u4e60\u4efb\u52a1\u4e0a\u5bf9GAOT\u8fdb\u884c\u4e86\u5e7f\u6cdb\u6d4b\u8bd5\uff0c\u7ed3\u679c\u663e\u793a\u5176\u5728\u7cbe\u5ea6\u548c\u6548\u7387\u65b9\u9762\u76f8\u6bd4\u591a\u4e2a\u57fa\u51c6\u6a21\u578b\u5747\u6709\u663e\u8457\u63d0\u5347\uff0c\u5e76\u5728\u4e00\u4e2a\u5927\u89c4\u6a21\u4e09\u7ef4\u5de5\u4e1a\u8ba1\u7b97\u6d41\u4f53\u529b\u5b66\uff08CFD\uff09\u6570\u636e\u96c6\u4e0a\u8fbe\u5230\u4e86\u5f53\u524d\u6700\u4f18\u6027\u80fd\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aShizheng Wen, Arsh Kumbhat, Levi Lingsch, Sepehr Mousavi, Yizhou Zhao, Praveen Chandrashekar, Siddhartha Mishra\uff0c\u6765\u81eaETH Zurich Switzerland, CMU USA\u548cTIFR, India\u3002<\/p>\n\n\n\n<p>\u4e00\u3001\u7814\u7a76\u80cc\u666f\u4e0e\u95ee\u9898\u5b9a\u4e49<\/p>\n\n\n\n<p>\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u662f\u6a21\u62df\u81ea\u7136\u73b0\u8c61\u4e0e\u5de5\u7a0b\u95ee\u9898\u7684\u57fa\u7840\u5de5\u5177\u3002\u7136\u800c\uff0c\u5728\u5177\u6709\u590d\u6742\u51e0\u4f55\u8fb9\u754c\u7684\u5b9e\u9645\u5de5\u4e1a\u573a\u666f\u4e2d\uff0c\u4f20\u7edf\u7684\u6570\u503c\u89e3\u6cd5\uff08\u5982\u6709\u9650\u5143\/\u5dee\u5206\u6cd5\uff09\u5728\u5904\u7406\u591a\u6b21\u67e5\u8be2\u95ee\u9898\uff08\u5982\u4e0d\u786e\u5b9a\u6027\u91cf\u5316\u3001\u63a7\u5236\u3001\u53cd\u95ee\u9898\uff09\u65f6\u8ba1\u7b97\u6210\u672c\u6781\u9ad8\u3002\u4e3a\u4e86\u63d0\u5347\u8ba1\u7b97\u6548\u7387\uff0c\u8fd1\u5e74\u6765\u6d8c\u73b0\u51fa\u5927\u91cf\u201c\u795e\u7ecf\u7b97\u5b50\u201d\u65b9\u6cd5\uff0c\u8bd5\u56fe\u7528\u6df1\u5ea6\u5b66\u4e60\u6784\u9020PDE\u89e3\u7b97\u5668\u7684\u8fd1\u4f3c\u66ff\u4ee3\u3002<\/p>\n\n\n\n<p>\u5df2\u6709\u7684\u795e\u7ecf\u7b97\u5b50\u5982<a href=\"https:\/\/nullthought.net\/?p=4266\" target=\"_blank\" rel=\"noreferrer noopener\">FNO<\/a>\u3001DeepONet\u7b49\u591a\u57fa\u4e8e\u89c4\u5219\u7f51\u683c\u6784\u5efa\uff0c\u96be\u4ee5\u5904\u7406\u4efb\u610f\u51e0\u4f55\u57df\u4e0a\u7684\u70b9\u4e91\u6570\u636e\u3002\u56e0\u6b64\uff0c\u672c\u7814\u7a76\u81f4\u529b\u4e8e\u8bbe\u8ba1\u4e00\u79cd\u80fd\u540c\u65f6\u5177\u5907\u9ad8\u7cbe\u5ea6\u4e0e\u9ad8\u6548\u7387\u7684\u795e\u7ecf\u7b97\u5b50\uff0c\u5728\u4efb\u610f\u51e0\u4f55\u57df\u4e0a\u8fd1\u4f3cPDE\u89e3\u7b97\u5668\u3002<\/p>\n\n\n\n<p>\u4e8c\u3001\u6838\u5fc3\u8d21\u732e\u4e0e\u6a21\u578b\u6982\u8ff0<\/p>\n\n\n\n<p>GAOT\uff08Geometry Aware Operator Transformer\uff09\u6a21\u578b\u7684\u7279\u70b9\u5982\u4e0b\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u9488\u5bf9\u4efb\u610f\u51e0\u4f55\u7ed3\u6784\u8bbe\u8ba1\uff0c\u8f93\u5165\u4e3a\u65e0\u7ed3\u6784\u70b9\u4e91\uff0c\u8f93\u51fa\u4e3a\u4efb\u610f\u67e5\u8be2\u70b9\u7684PDE\u89e3\uff1b<\/li>\n\n\n\n<li>\u5728\u4fdd\u6301Transformer\u5904\u7406\u6548\u7387\u7684\u57fa\u7840\u4e0a\uff0c\u5f15\u5165\u591a\u5c3a\u5ea6\u56fe\u795e\u7ecf\u6a21\u5757MAGNO\u589e\u5f3a\u7a7a\u95f4\u611f\u77e5\u80fd\u529b\uff1b<\/li>\n\n\n\n<li>\u878d\u5408\u5c40\u90e8\u51e0\u4f55\u7edf\u8ba1\u4fe1\u606f\u5f62\u6210\u201c\u51e0\u4f55\u5d4c\u5165\u201d\uff0c\u63d0\u5347\u6a21\u578b\u5bf9\u590d\u6742\u57df\u5f62\u72b6\u7684\u7406\u89e3\uff1b<\/li>\n\n\n\n<li>\u652f\u6301\u65f6\u95f4\u76f8\u5173PDE\u4e0e\u5927\u89c4\u6a21\u5de5\u4e1a\u7ea7\u4e09\u7ef4\u4efb\u52a1\uff08\u5982DrivAerNet++\uff09\uff1b<\/li>\n\n\n\n<li>\u572824\u4e2a\u6570\u636e\u96c6\u4e0a\u53d6\u5f97SOTA\u8868\u73b0\uff0c\u5e76\u663e\u8457\u63d0\u5347\u8ba1\u7b97\u6548\u7387\u4e0e\u6cdb\u5316\u80fd\u529b\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u4e09\u3001\u65b9\u6cd5\u8bbe\u8ba1\u4e0e\u6280\u672f\u7ec6\u8282<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u95ee\u9898\u5f62\u5f0f\u5316<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u65f6\u95f4\u65e0\u5173PDE\u5f62\u5f0f\uff1aD(c, u) = f\uff0c\u76ee\u6807\u662f\u5b66\u4e60\u4ece\u8f93\u5165a=(c,f,u<sub>b<\/sub>,\u03c7D)\u5230\u89e3u\u7684\u7b97\u5b50S\uff1b<\/li>\n\n\n\n<li>\u65f6\u95f4\u76f8\u5173PDE\u5f62\u5f0f\uff1au<sub>t<\/sub>+D(c,u)=0\uff0c\u76ee\u6807\u662f\u5b66\u4e60\u89e3\u7b97\u5b50\u5e8f\u5217 S<sub>t<\/sub>(a)\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u603b\u4f53\u67b6\u6784<\/strong>\uff1a\u5178\u578b\u7684\u7f16\u7801\u5668-\u5904\u7406\u5668-\u89e3\u7801\u5668\u7ed3\u6784\uff08encode-process-decode\uff09\uff1a\n<ul class=\"wp-block-list\">\n<li>\u7f16\u7801\u5668\uff1a\u8f93\u5165\u539f\u59cb\u70b9\u4e91\uff0c\u5c06\u5176\u6620\u5c04\u5230\u6f5c\u7a7a\u95f4\u7684\u7ed3\u6784\u5316\u70b9\u4e91\uff1b<\/li>\n\n\n\n<li>\u5904\u7406\u5668\uff1aTransformer\uff08ViT\uff09\u5bf9\u6f5c\u7a7a\u95f4Token\u8fdb\u884c\u5904\u7406\uff1b<\/li>\n\n\n\n<li>\u89e3\u7801\u5668\uff1a\u5c06\u6f5c\u7a7a\u95f4\u7279\u5f81\u6620\u5c04\u56de\u539f\u59cb\u70b9\u4e91\u4efb\u610f\u67e5\u8be2\u70b9\uff0c\u8f93\u51faPDE\u89e3\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u7f16\u7801\u5668\uff1a\u591a\u5c3a\u5ea6\u6ce8\u610f\u529b\u56fe\u795e\u7ecf\u7b97\u5b50\uff08MAGNO\uff09<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u76f8\u6bd4\u4f20\u7edfGNO\u4ec5\u805a\u5408\u5355\u4e00\u5c3a\u5ea6\u90bb\u57df\u4fe1\u606f\uff0cMAGNO\u5f15\u5165\u591a\u4e2a\u5c3a\u5ea6\uff08\u5982\u534a\u5f84\u4e3ar, 2r, 4r\u7b49\uff09\uff0c\u6355\u6349\u5c40\u90e8\u4e0e\u5168\u5c40\u7279\u5f81\uff1b<\/li>\n\n\n\n<li>\u6bcf\u4e2a\u5c3a\u5ea6\u901a\u8fc7\u6ce8\u610f\u529b\u673a\u5236\u8fdb\u884c\u52a0\u6743\u79ef\u5206\uff08Attentional Quadrature\uff09\uff0c\u66f4\u7cbe\u7ec6\u63a7\u5236\u5c40\u90e8\u7279\u5f81\u91cd\u8981\u6027\uff1b<\/li>\n\n\n\n<li>\u6240\u6709\u5c3a\u5ea6\u7279\u5f81\u901a\u8fc7softmax\u52a0\u6743\u878d\u5408\u5f97\u5230\u6700\u7ec8Token\u8868\u793a\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u51e0\u4f55\u5d4c\u5165\uff08Geometry Embedding\uff09<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u4f7f\u7528\u5c40\u90e8\u7edf\u8ba1\u91cf\u6784\u5efa\u51e0\u4f55\u4fe1\u606f\u5411\u91cf\uff08\u90bb\u70b9\u6570\u3001\u5e73\u5747\u8ddd\u79bb\u3001\u8ddd\u79bb\u65b9\u5dee\u3001\u4e3b\u65b9\u5411\u7b49\uff09\uff0c<\/li>\n\n\n\n<li>\u7ecf\u8fc7\u5f52\u4e00\u5316\u540e\u4f5c\u4e3a\u51e0\u4f55\u63cf\u8ff0\u8f93\u5165MLP\uff0c\u5f97\u5230\u51e0\u4f55\u611f\u77e5\u5411\u91cf\uff0c\u5e76\u4e0eGNO\u8f93\u51fa\u62fc\u63a5\uff0c\u63d0\u4f9b\u663e\u5f0f\u51e0\u4f55\u5148\u9a8c\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u5904\u7406\u5668\uff1a\u57fa\u4e8eViT\u7684Transformer<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u82e5\u6f5c\u7a7a\u95f4\u4e3a\u89c4\u5219\u7f51\u683c\uff0c\u5219\u53ef\u4f7f\u7528\u56fe\u50cfPatch\u5212\u5206\u65b9\u5f0f\u6784\u5efaToken\uff1b<\/li>\n\n\n\n<li>\u91c7\u7528\u591a\u5934\u6ce8\u610f\u529b\u3001RMS\u5f52\u4e00\u5316\u548c\u65cb\u8f6c\u4f4d\u7f6e\u7f16\u7801\uff08RoPE\uff09\u63d0\u9ad8\u6548\u7387\u548c\u7a33\u5b9a\u6027\uff1b<\/li>\n\n\n\n<li>\u652f\u6301\u8df3\u8dc3\u8fde\u63a5\uff0c\u4ee5\u589e\u5f3a\u7279\u5f81\u4f20\u9012\u80fd\u529b\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u89e3\u7801\u5668\uff1a\u5bf9\u79f0MAGNO\u7ed3\u6784<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u7ed9\u5b9a\u4efb\u610f\u67e5\u8be2\u70b9x\uff0c\u4ece\u6f5c\u7a7a\u95f4Token\u4e2d\u6784\u5efa\u591a\u4e2a\u5c3a\u5ea6\u90bb\u57df\uff0c\u8ba1\u7b97AGNO\u8f93\u51fa\uff1b<\/li>\n\n\n\n<li>\u878d\u5408\u51e0\u4f55\u5d4c\u5165\u540e\u9884\u6d4bx\u5904\u7684\u89e3\u503cu(x)\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u65f6\u95f4\u76f8\u5173PDE\u6269\u5c55<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u5c06\u5f53\u524d\u65f6\u523bt\u4e0e\u524d\u8fdb\u6b65\u957f\u03c4\u52a0\u5165\u8f93\u5165\uff1b<\/li>\n\n\n\n<li>\u901a\u8fc7\u53ef\u914d\u7f6e\u7b56\u7565\uff08\u5982\u76f4\u63a5\u9884\u6d4b\u3001\u6b8b\u5dee\u9884\u6d4b\u3001\u65f6\u95f4\u5bfc\u6570\uff09\u5b9e\u73b0\u4e0d\u540c\u7684\u65f6\u95f4\u63a8\u8fdb\u65b9\u6848\uff1b<\/li>\n\n\n\n<li>\u652f\u6301all2all\u7b56\u7565\uff0c\u4ece\u8f68\u8ff9\u6570\u636e\u4e2d\u5b66\u4e60\u4efb\u610f\u65f6\u523b\u89e3\u7b97\u5b50\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u9ad8\u6548\u5b9e\u73b0\u7b56\u7565<\/strong>\n<ul class=\"wp-block-list\">\n<li>\u9884\u5904\u7406\u56fe\u7ed3\u6784\uff0c\u51cf\u5c11\u8bad\u7ec3\u65f6\u56fe\u6784\u5efa\u5f00\u9500\uff1b<\/li>\n\n\n\n<li>\u7f16\u7801\u5668\u4e0e\u89e3\u7801\u5668\u4f7f\u7528\u5e8f\u5217\u5904\u7406\u8282\u7701\u663e\u5b58\uff0c\u5904\u7406\u5668\u4ecd\u6279\u91cf\u5e76\u884c\uff1b<\/li>\n\n\n\n<li>\u8fb9\u7f18\u4e22\u5f03\uff08Edge Dropping\uff09\u7f13\u89e3\u5185\u5b58\u74f6\u9888\uff0c\u652f\u6301\u66f4\u5927\u56fe\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p>\u56db\u3001\u5b9e\u9a8c\u8bc4\u4f30\u4e0e\u6027\u80fd\u8868\u73b0<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u6570\u636e\u96c6<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u65f6\u95f4\u65e0\u5173\u4efb\u52a1\uff1aPoisson\u65b9\u7a0b\u3001\u5f39\u6027\u65b9\u7a0b\u3001\u7a7a\u6c14\u52a8\u529b\u5b66\u6d41\u52a8\u7b49\uff1b<\/li>\n\n\n\n<li>\u65f6\u95f4\u76f8\u5173\u4efb\u52a1\uff1aEuler\u65b9\u7a0b\u3001Navier-Stokes\u65b9\u7a0b\u3001\u6ce2\u52a8\u65b9\u7a0b\uff1b<\/li>\n\n\n\n<li>\u7279\u522b\u5305\u62ec\u4e09\u7ef4\u5de5\u4e1a\u7ea7DrivAerNet++\u6570\u636e\u96c6\uff088K\u8f66\u8f86\u5f62\u72b6 \u00d7 \u6bcf\u5f62\u72b650\u4e07\u70b9\uff09\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u7cbe\u5ea6\u4e0e\u9c81\u68d2\u6027<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>GAOT\u572824\u4e2a\u6570\u636e\u96c6\u4e2d\u670914\u4e2a\u4e3a\u6700\u4f18\u6216\u6b21\u4f18\uff1b<\/li>\n\n\n\n<li>\u5728\u65f6\u95f4\u65e0\u5173\u4efb\u52a1\u4e0a\u6bd4\u7b2c\u4e8c\u4f18\u6a21\u578bRIGNO-18\u63d0\u5347\u7ea650%\u7cbe\u5ea6\uff1b<\/li>\n\n\n\n<li>\u9c81\u68d2\u6027\u6307\u6807\u663e\u793aGAOT\u5728\u4e0d\u540c\u6570\u636e\u96c6\u8868\u73b0\u7a33\u5b9a\uff0c\u4e0d\u6613\u9000\u5316\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u8ba1\u7b97\u6548\u7387<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>GAOT\u8bad\u7ec3\u541e\u5410\u7387\u4e3a97.5 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class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>GAOT on GitHub: <a href=\"https:\/\/github.com\/camlab-ethz\/GAOT\" target=\"_blank\" rel=\"noreferrer noopener\">https:\/\/github.com\/camlab-ethz\/GAOT<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8bba\u6587Geometry Aware Operator Transformer As An Efficient A [&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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