{"id":42790,"date":"2023-05-26T18:41:11","date_gmt":"2023-05-26T10:41:11","guid":{"rendered":"https:\/\/swarma.org\/?p=42790"},"modified":"2023-05-26T18:41:11","modified_gmt":"2023-05-26T10:41:11","slug":"%e5%82%85%e7%ab%8b%e5%8f%b6%e7%a5%9e%e7%bb%8f%e7%ae%97%e5%ad%90%ef%bc%9a%e5%82%85%e7%ab%8b%e5%8f%b6%e5%8f%98%e6%8d%a2%e5%ba%94%e7%94%a8%e4%ba%8e%e6%b7%b1%e5%ba%a6%e5%ad%a6%e4%b9%a0%ef%bc%8c%e9%a1%bd","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=42790","title":{"rendered":"\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\uff1a\u5085\u7acb\u53f6\u53d8\u6362\u5e94\u7528\u4e8e\u6df1\u5ea6\u5b66\u4e60\uff0c\u987d\u5f3a\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b"},"content":{"rendered":"
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\u4ece\u7269\u7406\u5b66\u4e2d\u7684\u70ed\u6269\u6563\u65b9\u7a0b\u3001\u63cf\u8ff0\u6d41\u4f53\u8fd0\u52a8\u7684\u7eb3\u7ef4\u8036-\u65af\u6258\u514b\u65af\u65b9\u7a0b\u3001\u523b\u753b\u91cf\u5b50\u6ce2\u51fd\u6570\u7684\u859b\u5b9a\u8c14\u65b9\u7a0b\uff0c\u5230\u751f\u7269\u5b66\u4e2d\u63cf\u8ff0\u6591\u56fe\u5f62\u6210\u7684\u53cd\u5e94\u6269\u6563\u65b9\u7a0b\uff0c\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u7684\u5e94\u7528\u975e\u5e38\u5e7f\u6cdb\u3002\u4f46\u6c42\u89e3 PDE \u5374\u662f\u975e\u5e38\u56f0\u96be\u7684\u95ee\u9898\u3002\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\u4e3a\u6b64\u63d0\u4f9b\u4e86\u4fbf\u5229\u3002<\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/p>\n


<\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/p>\n

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\u5728\u96c6\u667a\u4ff1\u4e50\u90e8\u300cAI+Science<\/a>\u300d\u8bfb\u4e66\u4f1a\u672c\u5468\u65e5\uff085\u670828\u65e5\uff09\u7684\u5206\u4eab\u4e2d\uff0c\u674e\u5b97\u5b9c\u535a\u58eb\u5c06\u5206\u4eab\u81ea\u5df1\u5728\u795e\u7ecf\u7b97\u5b50\u5b66\u4e60\u65b9\u9762\u7684\u6700\u65b0\u5de5\u4f5c\uff0c\u540c\u65f6\u65af\u5766\u798f\u5927\u5b66\u7684\u8d75\u9752\u9752\u535a\u58eb\u5c06\u4ecb\u7ecd\u57fa\u4e8e\u56fe\u795e\u7ecf\u7f51\u7edc\uff08Graph Neural Network\uff09\u8fdb\u884c\u7269\u7406\u4eff\u771f\u7684\u7cfb\u5217\u5de5\u4f5c\uff0c\u6b22\u8fce\u611f\u5174\u8da3\u7684\u670b\u53cb\u53c2\u4e0e\uff5e<\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/span><\/strong><\/span><\/strong><\/strong><\/strong><\/p>\n<\/section>\n

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\u4ec7\u7ade\u7eac\uff08NCC lab<\/span>\uff09<\/span> <\/span>| \u4f5c\u8005<\/strong><\/span><\/span><\/p>\n

\u5218\u6cc9\u5f71<\/span> <\/span>| \u6821\u5bf9<\/strong><\/span><\/span><\/strong><\/span><\/p>\n

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\u795e\u7ecf\u7f51\u7edc<\/span><\/strong> <\/span><\/span><\/section>\n
\u795e\u7ecf\u7f51\u7edc<\/span> (<\/span>neural network<\/span>) <\/span><\/span>\u662f\u4e00\u79cd\u7531\u5927\u91cf\u8282\u70b9\u6784\u6210\u7684\u8fd0\u7b97\u6a21\u578b\u3002\u6bcf\u4e2a\u8282\u70b9\u4ee3\u8868\u4e00\u79cd\u7279\u5b9a\u7684\u8f93\u51fa\u51fd\u6570\uff0c\u79f0\u4e3a\u6fc0\u52b1\u51fd\u6570<\/span> (<\/span>activation function<\/span>)<\/span><\/span>\u3002\u5404\u4e2a\u8282\u70b9<\/span> (\u6216\u79f0\u795e\u7ecf\u5143) <\/span>\u4e4b\u95f4\u76f8\u4e92\u8fde\u63a5\uff0c\u5f62\u6210\u795e\u7ecf\u7f51\u7edc\u7684\u5c42<\/span> (<\/span>layer<\/span>)<\/span><\/span>\u3002\u795e\u7ecf\u7f51\u7edc\u7684\u8f93\u5165\u548c\u8f93\u51fa\u4e00\u822c\u90fd\u662f\u6709\u9650\u7ef4\u7684\u5411\u91cf\u3002<\/span><\/section>\n

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\u795e\u7ecf\u7b97\u5b50<\/span><\/strong> <\/span><\/span><\/section>\n
\u795e\u7ecf\u7b97\u5b50<\/span> (<\/span>neural operator<\/span>) <\/span><\/span>\u662f\u5bf9\u795e\u7ecf\u7f51\u7edc\u7684\u4e00\u79cd\u6269\u5c55\uff0c\u5176\u8f93\u5165\u548c\u8f93\u51fa\u90fd\u662f\u4e00\u4e2a\u51fd\u6570\u3002 <\/span><\/section>\n

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\u504f\u5fae\u5206\u65b9\u7a0b<\/span><\/strong> <\/span><\/span><\/section>\n
\u504f\u5fae\u5206\u65b9\u7a0b<\/span> (<\/span>PDE<\/span>) <\/span><\/span>\u662f\u5305\u542b\u4e86\u672a\u77e5\u51fd\u6570\u7684\u504f\u5bfc\u6570\u7684\u65b9\u7a0b\u3002\u504f\u5fae\u5206\u65b9\u7a0b\u5728\u6570\u5b66\uff0c\u7269\u7406\u53ca\u5de5\u7a0b\u6280\u672f\u4e0a\u5e94\u7528\u5e7f\u6cdb\uff0c\u859b\u5b9a\u8c14\u65b9\u7a0b\uff0c\u4f2f\u683c\u65af\u65b9\u7a0b\uff0c\u7231\u56e0\u65af\u5766\u573a\u65b9\u7a0b\u7b49\u90fd\u5c5e\u4e8e\u504f\u5fae\u5206\u65b9\u7a0b\u3002<\/span><\/section>\n

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\u5085\u7acb\u53f6\u7a7a\u95f4<\/span> (<\/span>Fourier space<\/span>)<\/span><\/span>\uff0c\u5373\u9891\u7387\u7a7a\u95f4\u3002\u73b0\u4ee3\u4eba\u542c\u7684Fm\u7535\u53f0\u5c31\u662f\u7528\u7535\u53f0\u6ce2\u6bb5\u7684\u9891\u7387\u6765\u547d\u540d\uff0c\u5982\u4e2d\u592e\u4eba\u2ea0\u5e7f\u64ad\u7535\u53f0\u9891\u7387\u5c31\u662f<\/span>89.3<\/span>\u5146\u8d6b\u3002<\/span><\/span><\/section>\n

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PDE<\/span>\u4f5c\u4e3a\u4e00\u79cd\u5efa\u6a21\u5de5\u5177\uff0c\u4ece<\/span>19<\/span>\u4e16\u7eaa\u5f00\u59cb\u5c31\u88ab\u5e7f\u6cdb\u5e94\u7528\u4e8e\u7269\u7406\uff0c\u6570\u5b66\u4e0a\u3002\u70ed\u65b9\u7a0b\u3001\u859b\u5b9a\u8c14\u65b9\u7a0b\u3001\u7eb3\u7ef4<\/span>–<\/span>\u65af\u6258\u514b\u65af\u65b9\u7a0b\u7b49\u90fd\u662f\u7269\u7406\u4e0a\u975e\u5e38\u6709\u540d\u7684\u4f8b\u5b50\u3002\u8fd1\u6765<\/span>PDE<\/span>\u4e5f\u88ab\u9010\u6e10\u5e94\u7528\u4e8e\u751f\u7269\u5b66\uff0c\u53cd\u5e94\u6269\u6563\u65b9\u7a0b<\/span><\/span> (<\/span>reaction-diffusion equation<\/span>) <\/span><\/span>\u5c31\u88ab\u5e94\u7528\u4e8e\u6a21\u5f0f\u5f62\u6210<\/span> (<\/span>pattern formation<\/span>) <\/span><\/span>\u7684\u7814\u7a76\uff0c\u6591\u2ee2\u3001\u8001\u2ec1\u8eab\u4e0a\u6591\u7eb9\u7684\u5f62\u6210\u5c31\u53ef\u4ee5\u7528\u53cd\u5e94\u6269\u6563\u65b9\u7a0b\u6765\u89e3\u91ca\u3002<\/span><\/section>\n

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\u4f46<\/span>PDE<\/span>\u7684\u6c42\u89e3\u4e5f\u662f\u4e00\u4e2a\u5341\u5206\u8270\u96be\u7684\u95ee\u9898\uff0c\u4f20\u7edf\u7684<\/span>PDE<\/span>\u6c42\u89e3\u65b9\u6cd5\u5305\u62ec\u6709\u9650\u5dee\u5206\u6cd5<\/span><\/span> (<\/span>finite difference<\/span>)<\/span><\/span>\u3001\u6709\u9650\u5143\u6cd5<\/span> (<\/span>finite element<\/span>) <\/span><\/span>\u7b49 \uff0c\u8fd9\u4e9b\u65b9\u6cd5\u901f\u5ea6\u8f83\u6162\uff0c\u6709\u65f6\u6548\u7387\u4f4e\u4e0b\u3002\u4f8b\u5982\uff0c\u5f53\u8bbe\u8ba1\u673a\u7ffc\u7b49\u6750\u6599\u65f6\uff0c\u6211\u4eec\u9700\u8981\u5bf9\u524d\u5411\u6a21\u578b\u8fdb\u884c\u6570\u5343\u6b21\u8bc4\u4f30\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u4e9f\u9700\u4e00\u79cd\u66f4\u6709\u6548\u7387\u7684\u65b9\u6cd5\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u673a\u5668\u5b66\u4e60\u65b9\u6cd5\u4e3a<\/span>PDE<\/span>\u548c\u8bb8\u591a\u6570\u5b66\u95ee\u9898\u7684\u6c42\u89e3\u63d0\u4f9b\u4e86\u4fbf\u5229\u7684\u9014\u5f84\uff0c\u6210\u4e3a\u4e86\u9769\u65b0\u8bb8\u591a\u79d1\u5b66\u4e0e\u5de5\u7a0b\u5b66\u79d1\u7684\u5173\u952e\u6240\u5728\u3002\u4f46\u662f\u7ecf\u5178\u7684\u795e\u7ecf\u7f51\u7edc\u662f\u6709\u9650\u7ef4\u6b27\u6c0f\u7a7a\u95f4\u4e4b\u95f4\u7684\u6620\u5c04<\/strong>\uff0c\u5982\u679c\u6211\u4eec\u60f3\u7528\u795e\u7ecf\u7f51\u7edc\u6765\u5bf9<\/span>PDE<\/span>\u8fdb\u884c\u6c42\u89e3\uff0c\u5219<\/span><\/span>\u9700\u8981\u5bf9<\/span>PDE<\/span>\u7684\u51fd\u6570\u5b9a\u4e49\u57df\u8fdb\u884c\u79bb\u6563\u5316\uff0c\u5e76\u4e14\u6bcf\u6b21\u53ea\u80fd\u5b66\u4e60\u5230\u7279\u5b9a\u79bb\u6563\u533a\u95f4\u4e0b\u7684<\/span>PDE<\/span>\u7684\u89e3<\/span><\/span>\u3002\u8fd9\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u5f80\u5f80\u662f\u4e0d\u591f\u7684\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e00\u79cd\u80fd\u591f<\/span>\u5bf9\u4e0d\u540c\u79bb\u6563\u533a\u95f4\u90fd\u9002\u7528\u7684\u795e\u7ecf\u7f51\u7edc (<\/span>mesh-invariant neural network<\/span>)<\/span><\/strong>\u3002\u5728\u7814\u7a76\u4eba\u5458\u53d1\u660e\u795e\u7ecf\u7b97\u5b50\u4e4b\u524d\uff0c\u4e3b\u6d41\u7684\u57fa\u4e8e\u795e\u7ecf\u7f51\u7edc\u6c42\u89e3<\/span>PDE<\/span>\u7684\u65b9\u6cd5\u5305\u62ec\u4e86\u6709\u9650\u7ef4\u7b97\u5b50<\/span><\/span> (<\/span>finite- dimensional operator<\/span>) <\/span><\/span>\u548c\u795e\u7ecf<\/span>FEM<\/span><\/span> <\/span>(<\/span>neural-FEM<\/span>)<\/span><\/span>\u3002<\/span><\/section>\n

<\/section>\n
\u6709\u9650\u7ef4\u7b97\u5b50<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u7531<\/span>Guo Xiaoxiao<\/span>\u7b49\u4eba\u63d0\u51fa\u7684\u6709\u9650\u7ef4\u7b97\u5b50 \u901a\u8fc7\u628a<\/span>PDE<\/span>\u7684\u89e3\u7b97\u5b50\u53c2\u6570\u5316\u4e3a\u6709\u9650\u7ef4\u6b27\u6c0f\u7a7a\u95f4\u4e4b\u95f4\u7684\u6df1\u5ea6\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff0c\u5b9e\u73b0\u4e86\u5bf9<\/span>PDE<\/span>\u7684\u6c42\u89e3\u3002\u8fd9\u4e9b\u65b9\u6cd5\u5e76\u4e0d\u662f<\/span>mesh-invariant<\/span>\u7684\uff0c\u5373\u5bf9\u4e0d\u540c\u7684\u79bb\u6563\u533a\u95f4\u6211\u4eec\u5f80\u5f80\u9700\u8981\u5bf9\u6709\u9650\u7ef4\u7b97\u5b50\u8fdb\u884c\u4fee\u6539\u548c\u8c03\u6574\u3002\u9664\u6b64\u4e4b\u5916\uff0c\u8fd9\u79cd\u65b9\u6cd5\u4e5f\u53d7\u5230\u8bad\u7ec3\u6570\u636e\u7684\u51e0\u4f55\u5f62\u72b6\u4e0e\u79bb\u6563\u533a\u95f4\u7684\u5c40\u9650\uff1a\u5bf9\u4e8e<\/span>PDE<\/span>\u7684\u5b9a\u4e49\u57df\u4e0a\u5b58\u5728\u4f46\u6570\u503c\u7f51\u683c\u4e0d\u80fd\u8986\u76d6\u5230\u7684\u70b9\uff0c\u6211\u4eec\u65e0\u6cd5\u7528\u6709\u9650\u7ef4\u7b97\u5b50\u6c42\u5f97\u5bf9\u5e94\u7684\u89e3\u3002<\/span><\/span><\/section>\n

<\/span><\/section>\n
\u795e\u7ecf<\/span>FEM<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u4e0e\u6709\u9650\u7ef4\u7b97\u5b50\u4e0d\u540c\uff0c\u795e\u7ecf<\/span>FEM<\/span>\u76f4\u63a5\u628a<\/span>PDE<\/span>\u7684\u89e3\u53c2\u6570\u5316\u4e3a\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u3002\u56e0\u6b64\uff0c\u5f53<\/span>PDE<\/span>\u7684\u7cfb\u6570\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u6211\u4eec\u90fd\u9700\u8981\u8bad\u7ec3\u4e00\u4e2a\u65b0\u7684\u795e\u7ecf<\/span>FEM<\/span>\u6765\u5bf9<\/span>PDE<\/span>\u6c42\u89e3\u3002\u8fd9\u79cd\u65b9\u6cd5\u4e0e\u7ecf\u5178\u7684\u6709\u9650\u5143\u65b9\u6cd5\u975e\u5e38\u76f8\u4f3c\uff0c\u4e0d\u540c\u7684\u5730\u65b9\u5c31\u5728\u4e8e\u795e\u7ecf<\/span>FEM<\/span>\u628a\u6709\u9650\u5143\u65b9\u6cd5\u4e2d\u7684\u57fa\u51fd\u6570\u6362\u6210\u4e86\u795e\u7ecf\u7f51\u7edc\u3002\u540c\u65f6\uff0c\u795e\u7ecf<\/span>FEM<\/span>\u4e0e\u7ecf\u5178\u6570\u503c\u65b9\u6cd5\u7684\u5c40\u9650\u6027\u76f8\u4f3c\uff1a\u9700\u8981\u9488\u5bf9\u6bcf\u4e00\u4e2a\u65b0\u7684\u5b9e\u4f8b\u89e3\u51b3\u4f18\u5316\u95ee\u9898\u5e76\u4e14\u5fc5\u987b\u77e5\u9053<\/span>PDE<\/span>\u7684\u5177\u4f53\u8868\u8fbe\u5f0f\u3002<\/span><\/span><\/section>\n

<\/span><\/section>\n
\u795e\u7ecf\u7f51\u7edc<\/span><\/strong> <\/span><\/span><\/section>\n

<\/span><\/section>\n
\u7531\u4e8e\u795e\u7ecf\u7b97\u5b50\u662f\u5bf9\u795e\u7ecf\u7f51\u7edc\u7684\u4e00\u4e2a\u63a8\u5e7f\uff0c\u6211\u4eec\u5728\u8fd9\u91cc\u5148\u4ecb\u7ecd\u795e\u7ecf\u7f51\u7edc\u7684\u57fa\u672c\u67b6\u6784\u3002\u987e\u540d\u601d\u4e49\uff0c\u795e\u7ecf\u7f51\u7edc\u6a21\u4eff\u7684\u662f\u4eba\u7c7b\u5927\u8111\u795e\u7ecf\u5143\u7684\u7ed3\u6784\uff0c\u4f46\u540c\u65f6\u53c8\u4e0e\u751f\u7269\u795e\u7ecf\u5143\u6709\u7740\u663e\u8457\u7684\u5dee\u522b\u3002\u5728\u6570\u5b66\u4e0a\uff0c\u6211\u4eec\u53ef\u628a\u795e\u7ecf\u7f51\u7edc\u5b9a\u4e49\u4e3a\u4e00\u4e2a\u8fde\u7eed\u7684\u590d\u5408\u51fd\u6570<\/span><\/section>\n

<\/span><\/section>\n
F <\/span>= <\/span>\u03c3 <\/span>\u2218 <\/span>W<\/span>K<\/span>\u22121 <\/span>\u2218 . . . \u2218 <\/span>\u03c3 <\/span>\u2218 <\/span>W<\/span>1 <\/span>\u2218 <\/span>\u03c3 <\/span>\u2218 <\/span>W<\/span>0<\/span><\/span><\/section>\n

<\/span><\/section>\n
\u5176\u4e2d<\/span>K<\/span>\u4e3a\u795e\u7ecf\u7f51\u7edc\u7684\u6df1\u5ea6\uff0c<\/span>{<\/span>W<\/span>0<\/span>, . . . , <\/span>W<\/span>K<\/span>\u22121<\/span>}<\/span>\u4e3a\u53ef\u8bad\u7ec3\u7684\u6743\u91cd\uff0c\u6fc0\u6d3b\u51fd\u6570<\/span>\u03c3<\/span>\u5e94\u7528\u5230\u5411\u91cf\u7684\u6bcf\u4e00\u4e2a\u5143\u7d20\u4e2d\u3002\u53ef\u2ec5\uff0c\u795e\u7ecf\u7f51\u7edc\u53ef\u88ab\u770b\u4f5c\u6709\u9650\u7ef4\u6b27\u6c0f\u7a7a\u95f4\u4e4b\u95f4\u7684\u4e00\u4e2a\u6620\u5c04\u3002\u4f46<\/span>PDE<\/span>\u5904\u7406\u7684\u662f\u65e0\u7a77\u7ef4\u7684\u51fd\u6570\uff0c\u56e0\u6b64\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u65b0\u578b\u7684\u80fd\u5904\u7406\u51fd\u6570\u7684\u795e\u7ecf\u7f51\u7edc\u3002<\/span><\/span><\/section>\n

<\/span><\/section>\n
\u683c\u6797\u51fd\u6570\u6cd5 <\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u5bf9\u4e8e\u4e00\u4e2a<\/span>PDE <\/span>L<\/span>a<\/span>u <\/span>= <\/span>f <\/span>(\u5f88\u591a<\/span>PDE<\/span>\u53ef\u88ab\u8868\u8fbe\u6210\u8fd9\u4e2a\u5f62\u5f0f\uff0c\u6bd4\u5982\u8fbe\u2ec4\u6d41\u65b9\u7a0b <\/span>\u2212<\/span>div<\/span>(<\/span>a<\/span>(<\/span>x<\/span>)\u2207<\/span>u<\/span>(<\/span>x<\/span>)) = <\/span>f<\/span>(<\/span>x<\/span>)<\/span>)\uff0c\u6211\u4eec\u53ef\u4ee5\u7528\u683c\u6797\u51fd\u6570\u5199\u51fa\u5b83\u7684\u89e3\u3002\u683c\u6797\u51fd\u6570\u662f\u4e00\u79cd\u7528\u6765\u89e3\u6709\u521d\u59cb\u6761\u4ef6\u6216\u8fb9\u754c\u6761\u4ef6\u7684\u975e\u2eec\u6b21\u5fae\u5206\u65b9\u7a0b\u7684\u51fd\u6570\u3002\u5bf9\u4e00\u4e2a\u5fae\u5206\u7b97\u5b50<\/span>L<\/span>a<\/span>\uff0c\u5b83\u7684\u683c\u6797\u51fd\u6570\u6ee1\u8db3<\/span><\/span><\/section>\n

<\/section>\n
<\/section>\n

<\/span>PDE <\/span>L<\/span>a<\/span>u <\/span>= <\/span>f<\/span> \u7684\u89e3 \u53ef\u4ee5\u7528\u8fd9\u4e2a\u683c\u6797\u51fd\u6570\u8868\u8fbe\u51fa\u6765\uff1a<\/span><\/span><\/section>\n

<\/span><\/section>\n
<\/section>\n

<\/section>\n
\u4e0a\u9762\u8fd9\u4e2a\u7ed3\u8bba\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u76f4\u63a5\u4ee3\u5165\u53bb\u9a8c\u8bc1:<\/span><\/section>\n

<\/section>\n
<\/section>\n

<\/section>\n
\u7531\u683c\u6797\u51fd\u6570\u6cd5\u6211\u4eec\u77e5\u9053\uff0c\u4e00\u4e2a<\/span>PDE<\/span>\u7684\u89e3\u53ef\u4ee5\u8868\u793a\u6210\u4ee5\u4e0a\u79ef\u5206\u7684\u5f62\u5f0f\u3002\u901a\u8fc7\u8fd9\u4e2a\u542f\u793a\uff0c\u4f5c\u8005\u4eec\u5728\u795e\u7ecf\u7b97\u5b50\u4e2d\u5f15\u5165\u4e86\u4e00\u4e2a\u79ef\u5206\u7b97\u5b50\uff0c\u4f7f\u5f97\u795e\u7ecf\u7b97\u5b50\u80fd\u591f\u5904\u7406\u5173\u4e8e\u51fd\u6570\u7684\u6570\u636e\u3002<\/span><\/span><\/section>\n

<\/span><\/section>\n
\u5085\u91cc\u53f6\u53d8\u6362<\/span><\/strong><\/span><\/section>\n

<\/span><\/section>\n
\u6570\u5b66\u5bb6\u4eec\u5728\u6c42\u89e3<\/span>PDE<\/span>\u7684\u7ecf\u9a8c\u4e2d\u53d1\u73b0\uff0c\u5bf9\u4e00\u4e2a<\/span>PDE<\/span>\u505a\u5085\u7acb\u53f6\u53d8\u6362\u5f80\u5f80\u80fd\u591f\u4f7f\u95ee\u9898\u53d8\u5f97\u7b80\u5355\u3002\u8fd9\u4e00\u4e2a\u79d8\u8bc0\u85cf\u5728\u8fd9\u4e2a\u5085\u7acb\u53f6\u53d8\u6362\u516c\u5f0f\u4e2d\uff0c\u5373\u7269\u7406\u7a7a\u95f4<\/span><\/span> (\u5373\u4ee5<\/span>x<\/span>\u4e3a\u53d8\u91cf\u7684\u7a7a\u95f4)<\/span><\/span>\u4e0a\u7684\u5fae\u5206<\/strong>\u7b49\u6548\u4e8e\u5085\u7acb\u53f6\u7a7a\u95f4\u4e2d\u7684\u4e58\u6cd5<\/strong>\uff1a<\/span><\/section>\n

<\/section>\n
              <\/span><\/section>\n

<\/section>\n
\u901a\u8fc7\u5085\u7acb\u53f6\u53d8\u6362\uff0c<\/span>PDE<\/span>\u4e0a\u7684\u4e00\u90e8\u5206\u504f\u5fae\u5206\u88ab\u6d88\u9664\uff0c\u800c<\/span>PDE<\/span>\u4e5f\u88ab\u8f6c\u5316\u6210\u4e00\u4e2a<\/span>ODE<\/span><\/strong>\uff0c\u95ee\u9898\u7531\u6b64\u5f97\u5230\u7b80\u5316\u3002\u7c7b\u4f3c\u5730\uff0c\u5085\u7acb\u53f6\u53d8\u6362\u4e5f\u88ab\u5c1d\u8bd5\u7528\u5728\u6df1\u5ea6\u5b66\u4e60\u4e2d\uff0c\u795e\u7ecf\u7b97\u5b50\u7684\u5de5\u4f5c\u5c31\u501f\u52a9\u4e86\u5085\u7acb\u53f6\u53d8\u6362\u7684\u5e2e\u52a9\u3002<\/span><\/span><\/section>\n

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\u5085\u91cc\u53f6\u53d8\u6362\u5728\u6df1\u5ea6\u5b66\u4e60\u7684\u53d1\u5c55\u4e2d\u53d1\u6325\u4e86\u91cd\u8981\u4f5c\u7528\u3002\u7406\u8bba\u4e0a\uff0c\u5b83\u4eec\u51fa\u73b0\u5728\u901a\u7528\u903c\u8fd1\u5b9a\u7406\u7684\u8bc1\u660e\u4e2d <\/span>[Hornik et al., 1989]<\/span>\uff0c\u540c\u65f6\u4e5f\u88ab\u7528\u4e8e\u52a0\u901f\u5377\u79ef\u795e\u7ecf\u7f51\u7edc <\/span>[Mathieu et al., 2013]<\/span>\u3002<\/span>Bengio et al.<\/span>\uff0c<\/span>2007<\/span>\uff0c <\/span>Mingo et al.<\/span>\uff0c<\/span>2004<\/span>\uff0c<\/span>Sitzmann et al.<\/span>\uff0c<\/span>2020<\/span>\u7b49\u8fd8\u63d0\u51fa\u5e76\u7814\u7a76\u4e86\u6d89\u53ca\u5085\u7acb\u53f6\u53d8\u6362\u6216\u4f7f\u7528\u6b63\u5f26\u6fc0\u6d3b\u51fd\u6570\u7684\u795e\u7ecf\u7f51\u7edc\u67b6\u6784\u3002\u6700\u8fd1\uff0c\u4e00\u4e9b\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u8c31\u65b9\u6cd5\u5df2\u6269\u5c55\u5230\u795e\u7ecf\u7f51\u7edc <\/span>[Fan et al., 2019a, Fan et al., 2019b]<\/span>\u3002\u672c\u6587\u4f5c\u8005<\/span>Li Zongyi<\/span>\u5728\u8fd9\u4e9b\u524d\u4eba\u5de5\u4f5c\u7684\u57fa\u7840\u4e0a\uff0c\u63d0\u51fa\u4e86\u4e00\u79cd\u76f4\u63a5\u5728\u5085\u7acb\u53f6\u7a7a\u95f4\u4e2d\u5b9a\u4e49\u7684\u795e\u7ecf\u7b97\u5b50\u67b6\u6784\u3002<\/span><\/span><\/section>\n

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\u6765\u8868\u793a\uff0c\u5176\u4e2d<\/span>{<\/span>W<\/span>0<\/span>, . . . , <\/span>W<\/span>T<\/span>\u22121<\/span>}<\/span>\u4e3a\u8bad\u7ec3\u7684\u6743\u91cd\uff0c<\/span>v<\/span>j<\/span>\u662f\u4e00\u7cfb\u5217\u51fd\u6570\uff0c\u6bcf\u4e00\u4e2a\u51fd\u6570\u53d6\u503c\u4e8e<\/span>R<\/span>dv<\/span>\u3002\u5176\u4e2d\u7531<\/span>a<\/span>\u5230 <\/span>v<\/span>0<\/span>\u7684\u8f6c\u6362\u53ef\u4ee5\u7531\u5c40\u90e8\u53d8\u6362<\/span>P <\/span>: R<\/span>d<\/span>a <\/span>\u2192 R<\/span>d<\/span>v<\/span>\u6765\u5b8c\u6210\uff0c\u5373<\/span>v<\/span>0 <\/span>= <\/span>P<\/span>(<\/span>a<\/span>)<\/span>\u3002\u8fd9\u4e00\u5c40\u90e8\u53d8\u6362<\/span>P<\/span>\u4e00\u822c\u901a\u8fc7\u6d45\u5c42\u5168\u8fde\u63a5\u7f51\u7edc\u8fdb\u884c\u53c2\u6570\u5316\u3002\u7c7b\u4f3c\u5730\uff0c\u8f93\u51fa<\/span>u <\/span>= <\/span>Q<\/span>(<\/span>v<\/span>T <\/span>)<\/span>\u662f<\/span>v<\/span>T<\/span>\u901a\u8fc7\u5c40\u90e8\u53d8\u6362<\/span>Q <\/span>: R<\/span>d<\/span>v <\/span>\u2192 R<\/span>d<\/span>u<\/span>\u540e\u7684\u6295\u5f71\u3002\u5728\u6bcf\u6b21\u8fed\u4ee3\u4e2d\u66f4\u65b0<\/span>v<\/span>t<\/span>+1<\/span>\u7531\u524d\u4e00\u65f6\u523b<\/span>v<\/span>t<\/span>\u901a\u8fc7\u975e\u5c40\u90e8\u79ef\u5206\u7b97\u5b50<\/span>K<\/span>\u548c\u5c40\u90e8\u975e\u7ebf\u6027\u6fc0\u6d3b\u51fd\u6570<\/span>\u03c3<\/span>\u7684\u8fde\u7eed\u53d8\u6362\u5f97\u5230\u3002<\/span>k<\/span>\u03c6<\/span>\u662f\u4ece\u6570\u636e\u4e2d\u5b66\u5f97\u7684\u6838\u51fd\u6570\u3002\u901a\u8fc7\u5f15\u5165\u79ef\u5206\u7b97\u5b50\uff0c\u6211\u4eec\u6210\u529f\u5730\u5b8c\u6210\u4e86\u795e\u7ecf\u7f51\u7edc\u5411\u65e0\u9650\u7ef4\u7a7a\u95f4\u7684\u63a8\u5e7f\u3002<\/span><\/span><\/section>\n

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\u5728\u5377\u79ef\u795e\u7ecf\u7f51\u7edc<\/span> (<\/span>CNN<\/span>) <\/span><\/span>\u4e2d\uff0c\u6bcf\u4e2a\u5377\u79ef\u5c42\u5b66\u7684\u662f\u5c40\u90e8\u7684\u4fe1\u606f\uff0c\u6bd4\u5982\u5bf9\u4e00\u5f20\u56fe\u7247\u6765\u8bf4\uff0c\u5377\u79ef\u5c42\u62bd\u53d6\u7684\u662f<\/span>3 \u00d7 3<\/span>\u6216<\/span>5 \u00d7 5<\/span>\u7684\u4e00\u5757\u3002\u4f46\u5bf9\u4e8e\u8fde\u7eed\u51fd\u6570\u6765\u8bb2\uff0c\u6211\u4eec\u66f4\u5173\u6ce8\u5b83\u5168\u5c40\u7684\u4fe1\u606f\uff0c\u5085\u7acb\u53f6\u53d8\u6362\u5728\u5904\u7406\u5168\u5c40\u4fe1\u606f\u65f6\u66f4\u6709\u4f18\u52bf\u3002<\/span><\/span><\/section>\n

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\u5bf9\u6b64\uff0c\u7814\u7a76\u4eba\u5458\u63d0\u51fa\u4e86\u7528\u5085\u7acb\u53f6\u53d8\u6362\u5b9a\u4e49\u795e\u7ecf\u7b97\u5b50\u4e2d\u7684\u79ef\u5206\u7b97\u5b50\u3002\u4ee4<\/span>F<\/span>\u8868\u793a\u5bf9\u51fd\u6570\u7684\u5085\u7acb\u53f6\u53d8\u6362\uff0c <\/span>F<\/span>\u22121<\/span>\u8868\u793a\u5085\u7acb\u53f6\u53cd\u53d8\u6362\uff0c\u5219\u5bf9\u51fd\u6570<\/span>f <\/span>: <\/span>D <\/span>\u2192 R<\/span>dv<\/span>\u7684\u5085\u7acb\u53f6\u53d8\u6362\u548c\u53cd\u53d8\u6362\u53ef\u4ee5\u5b9a\u4e49\u4e3a<\/span><\/span><\/section>\n

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\u5176\u4e2d<\/span>i<\/span>\u4e3a\u865a\u6570\u5355\u4f4d\uff0c<\/span>j <\/span>= 1,…,<\/span>d<\/span>v<\/span>\u4e3a\u51fd\u6570\u7684\u6307\u6807\u3002\u82e5\u6211\u4eec\u4ee4\u79ef\u5206\u7b97\u5b50(<\/span>2<\/span>)\u4e2d\u7684\u6838
<\/span>k<\/span>\u03c6<\/span>(<\/span>x<\/span>, <\/span>y<\/span>, <\/span>a<\/span>(<\/span>x<\/span>), <\/span>a<\/span>(<\/span>y<\/span>)) = <\/span>k<\/span>\u03c6<\/span>(<\/span>x <\/span>\u2212 <\/span>y<\/span>)<\/span>\uff0c\u5219\u79ef\u5206\u7b97\u5b50(<\/span>2<\/span>)\u53d8\u6210\u4e00\u4e2a\u5377\u79ef<\/span>(<\/span>k<\/span>\u03c6 <\/span>* <\/span>v<\/span>t<\/span>)(<\/span>x<\/span>)<\/span>\u3002\u8fd0\u7528\u5377\u79ef\u5b9a\u7406<\/span>F(<\/span>k<\/span>\u03c6 <\/span>* <\/span>v<\/span>t<\/span>) = F(<\/span>k<\/span>\u03c6<\/span>)F(<\/span>v<\/span>t<\/span>) <\/span>\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u79ef\u5206\u7b97\u5b50\u7684\u65b0\u5b9a\u4e49<\/span><\/span><\/section>\n

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\u8fd0\u7528\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\u5b66\u4e60<\/span>PDE<\/span>\u7684\u89e3 <\/span><\/strong><\/span><\/section>\n

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\u4e3e\u4e00\u4e2a\u786e\u5207\u7684\u4f8b\u5b50\uff0c\u4e3a\u4e86\u5b66\u4e60\u4e0b\u9762\u8fd9\u4e2a<\/span>PDE<\/span>\u7684\u89e3\uff0c<\/span><\/span><\/section>\n

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\u795e\u7ecf\u7b97\u5b50\u6240\u9700\u8981\u7684\u8bad\u7ec3\u6570\u636e\u662f<\/span>{<\/span>a<\/span>j<\/span>, <\/span>u<\/span>j<\/span>}<\/span>, <\/span>\u5176\u4e2d<\/span>a<\/span>j<\/span>\u662f\u65b9\u7a0b\u4e2d\u7684\u7cfb\u6570<\/span><\/span>(<\/span>coefficient function<\/span>)<\/span><\/span>\uff0c<\/span>u<\/span>j<\/span>\u5219\u662f\u6240\u5bf9\u5e94\u7684\u65b9\u7a0b\u7684\u89e3\u3002\u795e\u7ecf\u7b97\u5b50\u5c06\u4f1a\u5b66\u4e60<\/span>a<\/span>j<\/span>\u548c<\/span>u<\/span>j<\/span>\u4e4b\u95f4\u7684\u5bf9\u5e94\u5173\u7cfb\uff0c\u5f53\u8f93\u5165\u4e00\u4e2a\u65b0\u7684\u7cfb\u6570<\/span>a<\/span>j<\/span>\u65f6\u795e\u7ecf\u7b97\u5b50\u5c06\u4f1a\u9884\u6d4b\u76f8\u5bf9\u5e94\u7684<\/span>PDE <\/span>\u89e3<\/span>u<\/span>j<\/span>\u3002\u8fd9\u79cd\u601d\u8def\u53ef\u4ee5\u7c7b\u6bd4\u56fe\u50cf\u5230\u56fe\u50cf\u7684\u56de\u5f52\u5b66\u4e60<\/em>\uff0c\u5982\u56fe 1 \u6240\u793a<\/span><\/span><\/section>\n


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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

\u6587\u7ae0\u4ecb\u7ecd\u4e86\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\uff0c\u8fd9\u662f\u4e00\u79cd\u80fd\u591f\u5b66\u4e60\u65e0\u9650\u7ef4\u51fd\u6570\u7a7a\u95f4\u4e4b\u95f4\u6620\u5c04\u7684\u6df1\u5ea6\u5b66\u4e60\u67b6\u6784\u3002<\/span><\/p>\n

\u2022 \u7b97\u6cd5\u4f18\u8d8a\u6027\uff1a<\/span>\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\u662f<\/span>mesh-invariant<\/span>\u7684\uff0c\u53ef\u4ee5\u5b66\u4e60\u4e0d\u540c\u79bb\u6563\u533a\u95f4\u6761\u4ef6\u4e0b<\/span>PDE<\/span>\u7684\u89e3\u800c\u4e0d\u9700\u8981\u9488\u5bf9\u6bcf\u4e2a\u79bb\u6563\u533a\u95f4\u8fdb\u884c\u8c03\u6574\u3002<\/span><\/span><\/p>\n

\u2022 <\/span>\u6027\u80fd\u4f18\u8d8a\u6027\uff1a\u5728\u6587\u7ae0\u5c55\u793a\u7684\u5b9e\u9a8c\u4e2d\uff0c\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\u7684\u5b66\u4e60\u6548\u679c\u4f18\u4e8e\u6240\u6709\u73b0\u6709\u7684<\/span>PDE <\/span>\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\u3002\u76f8\u6bd4\u5176\u5b83\u65b9\u6cd5\uff0c\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\u7684\u9519\u8bef\u7387\u5728\u4f2f\u683c\u65af\u65b9\u7a0b\u4e0a\u964d\u4f4e\u4e86 <\/span>30%<\/span>\uff0c\u5728\u8fbe\u2ec4\u6d41\u65b9\u7a0b\u4e0a\u964d\u4f4e\u4e86 <\/span>60%<\/span>\uff0c\u5728\u7eb3\u7ef4<\/span>–<\/span>\u65af\u6258\u514b\u65af\u65b9\u7a0b <\/span><\/span>(\u96f7\u8bfa\u6570\u4e3a <\/span>10000 <\/span>\u7684\u6e4d\u6d41\u72b6\u6001) <\/span><\/span>\u4e0a\u964d\u4f4e\u4e86 <\/span>30%<\/span>\u3002\u5728\u5b66\u4e60\u6574\u4e2a\u65f6\u95f4\u5e8f\u5217\u7684\u6620\u5c04\u65f6\uff0c\u8be5\u65b9\u6cd5\u5728\u96f7\u8bfa\u6570\u4e3a<\/span>1000 <\/span>\u65f6\u7684\u8bef\u5dee\u5c0f\u4e8e <\/span>1%<\/span>\uff0c\u5728\u96f7\u8bfa\u6570\u4e3a<\/span>10000 <\/span>\u65f6\u7684\u8bef\u5dee\u4e3a <\/span>8%<\/span>\u3002<\/span><\/span><\/p>\n

\u2022 <\/span>\u8ba1\u7b97\u9ad8\u6548\u6027\uff1a\u5728 <\/span>256 \u00d7 256 <\/span>\u7f51\u683c\u4e0a\uff0c\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\u7684\u8ba1\u7b97\u65f6\u95f4\u4ec5\u4e3a <\/span>0.005 <\/span>\u79d2\uff0c\u800c\u7528\u4e8e\u6c42\u89e3\u7eb3\u7ef4<\/span>–<\/span>\u65af\u6258\u514b\u65af\u7684\u4f2a\u8c31\u65b9\u6cd5\u7684\u8ba1\u7b97\u65f6\u95f4\u4e3a <\/span>2.2 <\/span>\u79d2\u3002<\/span><\/span><\/section>\n

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\u5176\u4e2d <\/span>u<\/span>0 <\/span>\u2208 <\/span>L<\/span>2<\/span>((0,1); R) <\/span>\u662f\u521d\u59cb\u6761\u4ef6\uff0c<\/span>\u03bd <\/span>\u662f\u7c98\u5ea6\u7cfb\u6570\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bad\u7ec3\u4e00\u4e2a\u795e\u7ecf\u7b97\u5b50\uff0c\u4f7f\u5b83\u5c3d\u53ef\u80fd\u5730\u903c\u8fd1\u89e3\u7b97\u5b50<\/span>G <\/span>: <\/span>L<\/span>2<\/span>((0,1); R) \u2192 <\/span>H<\/span>r<\/span>((0,1); R)<\/span>\uff0c\u89e3\u7b97\u5b50<\/span>G <\/span>: <\/span>u<\/span>0 <\/span>\u2192 <\/span>u<\/span>(\u22c5,1)<\/span>\u4ece\u521d\u59cb\u6761\u4ef6<\/span>u<\/span>0<\/span>\u6620\u5c04\u5230\u65f6\u95f4<\/span>t <\/span>= 1<\/span>\u65f6\u7684\u89e3<\/span>u<\/span>(\u22c5,1)<\/span>\u3002\u521d\u59cb\u6761\u4ef6 <\/span>u<\/span>0<\/span>(<\/span>x<\/span>) <\/span>\u7b26\u5408\u4e00\u4e2a\u9ad8\u65af\u5206\u5e03\uff0c\u5373 <\/span>u<\/span>0 <\/span>\u223c <\/span>N<\/span>(0,625(\u2212\u0394 + 25<\/span>I <\/span>)<\/span>\u22122<\/span>)<\/span>\uff0c\u7c98\u5ea6<\/span>\u03bd<\/span>\u8bbe\u7f6e\u4e3a<\/span>0.1\u3002<\/span><\/span><\/section>\n

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\u5b9e\u9a8c\u7684\u7ed3\u679c\u5982\u56fe <\/span>2 <\/span>\u6240\u793a\u3002\u4e0e\u4efb\u4f55\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\u76f8\u6bd4\uff0c\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50<\/span><\/span> (<\/span>FNO<\/span>) <\/span><\/span>\u83b7\u5f97\u4e86\u6700\u4f4e\u7684\u76f8\u5bf9\u8bef\u5dee\u3002\u57fa\u4e8e\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7684\u65b9\u6cd5<\/span> (<\/span>FCN<\/span>) <\/span><\/span>\u7684\u8bef\u5dee\u968f\u7740\u8bad\u7ec3\u6570\u636e\u7684\u5206\u8fa8\u7387\u7684\u589e\u52a0\u800c\u589e\u2ed3\uff0c\u9664\u6b64\u4e4b\u5916\u5176\u5b83 \u65b9\u6cd5\u7684\u8bef\u5dee\u57fa\u672c\u72ec\u7acb\u4e8e\u8bad\u7ec3\u6570\u636e\u7684\u5206\u8fa8\u7387\u3002\u4e0e\u5728\u7269\u7406\u7a7a\u95f4\u4e2d\u4f7f\u7528 <\/span>Nystro\u0308m <\/span>\u91c7\u6837\u7684 <\/span>GNO <\/span>\u548c <\/span>MGNO <\/span>\u7b49\u5176\u4ed6\u795e\u7ecf\u7b97\u5b50\u65b9\u6cd5\u76f8\u6bd4\uff0c\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\u66f4\u51c6\u786e\u4e14\u8ba1\u7b97\u6548\u7387\u66f4\u9ad8\u3002<\/span><\/span><\/section>\n

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\u56fe2. \u5404\u79cd\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\u5728\u4f2f\u683c\u65af\u65b9\u7a0b\u4e0a\u7684\u5b66\u4e60\u6548\u679c<\/span><\/section>\n

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\u5176\u4e2d
<\/span>FCN<\/span>\uff1a\u4e00\u4e2a\u57fa\u4e8e\u5168\u5377\u79ef\u7f51\u7edc\u7684\u795e\u7ecf\u7f51\u7edc\u67b6\u6784 <\/span><\/span><\/section>\n
PCANN\uff1a<\/span>\u4e00\u79cd\u4f7f\u7528 <\/span>PCA <\/span>\u4f5c\u4e3a\u8f93\u5165\u548c\u8f93\u51fa\u6570\u636e\u7684\u81ea\u52a8\u7f16\u7801\u5668 <\/span><\/span><\/section>\n
GNO<\/span>\uff1a\u4e00\u79cd\u56fe\u795e\u7ecf\u7b97\u5b50<\/span>
LNO<\/span>\uff1a\u57fa\u4e8e\u6838<\/span>k <\/span>(<\/span>x <\/span>, <\/span>y<\/span>)<\/span>\u7684\u4f4e\u79e9\u5206\u89e3\u7684\u795e\u7ecf\u7b97\u5b50\u65b9\u6cd5
<\/span>MGNO<\/span>\uff1a\u4e00\u79cd\u591a\u6781\u56fe\u795e\u7ecf\u7b97\u5b50<\/span>
FNO<\/span>\uff1a\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50<\/span><\/span><\/section>\n

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\u8fbe\u2ec4\u6d41\u65b9\u7a0b <\/span><\/strong><\/span><\/section>\n

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\u8fbe\u2ec4\u6d41\u65b9\u7a0b\u662f\u4e00\u4e2a\u4e8c\u9636\u7ebf\u6027\u692d\u5706\u504f\u5fae\u5206\u65b9\u7a0b<\/span><\/section>\n

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\u5176\u4e2d <\/span>a <\/span>\u2208 <\/span>L<\/span>\u221e<\/span>((0,1)<\/span>2<\/span>; R<\/span>+<\/span>)<\/span>\u662f\u6269\u6563\u7cfb\u6570\uff0c<\/span>f <\/span>\u2208 <\/span>L<\/span>2<\/span>((0,1)<\/span>2<\/span>; R) <\/span>\u662f\u5f3a\u5236\u51fd\u6570\u3002\u8be5 <\/span>PDE <\/span>\u6709\u8bb8\u591a\u5e94\u7528\uff0c\u5305\u62ec\u6a21\u62df\u5730\u4e0b\u6d41\u4f53\u7684\u538b\u529b\u3001\u7ebf\u5f39\u6027\u6750\u6599\u7684\u53d8\u5f62\u4ee5\u53ca\u5bfc\u7535\u6750\u6599\u7684\u7535\u52bf\u3002\u4f5c\u8005\u901a\u8fc7\u795e\u7ecf\u7b97\u5b50\u5b66\u4e60\u4e86\u8fbe\u2ec4\u6d41\u65b9\u7a0b\u7684\u89e3\uff0c\u5e76\u4e0e\u5176\u5b83\u7684\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\u8fdb\u884c\u4e86\u6bd4\u8f83\u3002\u5b83\u4eec\u7684\u8ba1\u7b97\u7ed3\u679c\u4e0e<\/span>PDE<\/span>\u7684\u771f\u5b9e\u89e3\u4e4b\u95f4\u7684\u8bef\u5dee\u53ef\u53c2\u2ec5\u56fe\uff083\uff09<\/span><\/span><\/section>\n

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 \u56fe3. \u5404\u79cd\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\u5728\u8fbe\u897f\u6d41\u65b9\u7a0b\u4e0a\u7684\u5b66\u4e60\u6548\u679c<\/span><\/section>\n

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\u5176\u4e2d<\/span>RBM<\/span>\u6307\u7684\u662f\u4e00\u79cd\u7ecf\u5178\u7684\u7b80\u5316\u57fa\u65b9\u6cd5<\/span><\/span> (\u4f7f\u7528 <\/span>POD <\/span>\u57fa)<\/span><\/span>\uff0c\u5176\u4f59\u7684\u7f29\u5199\u53ef\u53c2\u8003\u4e0a\u6587\u3002<\/span><\/section>\n

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\u7eb3\u7ef4<\/span>–<\/span>\u65af\u6258\u514b\u65af\u65b9\u7a0b<\/span><\/strong><\/span><\/section>\n

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\u7eb3\u7ef4<\/span>–<\/span>\u65af\u6258\u514b\u65af\u65b9\u7a0b<\/span><\/strong>\u662f\u6d41\u4f53\u529b\u5b66\u4e0a\u7684\u8457\u540d\u65b9\u7a0b\uff0c\u53ef\u4ee5\u7528\u4e8e\u63cf\u8ff0\u5404\u79cd\u6d41\u4f53\u7684\u8fd0\u52a8\uff0c\u5982\u9ad8\u7a7a\u6c14\u6d41\u3001\u6d0b\u6d41\u3001\u7ba1\u9053\u4e2d\u7684\u6c34\u6d41\u3001\u673a\u7ffc\u9644\u8fd1\u7a7a\u6c14\u7684\u6d41\u52a8\u3001\u661f\u7cfb\u4e2d\u5929\u4f53\u7684\u8fd0\u52a8\u7b49\u3002\u4e0b\u9762\u8fd9\u4e2a\u7eb3\u7ef4<\/span>–<\/span>\u65af\u6258\u514b\u65af\u65b9\u7a0b\u53ef\u4ee5\u7528\u4e8e\u63cf\u8ff0\u4e8c\u7ef4\u7684\u7c98\u7a20\u4e0d\u53ef\u538b\u7f29\u6d41\u4f53<\/span><\/span><\/section>\n

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\u5176\u4e2d <\/span>u <\/span>\u2208 <\/span>C<\/span>([0,<\/span>T <\/span>]; <\/span>H<\/span>r<\/span>((0,1)<\/span>2<\/span>; R<\/span>2<\/span>)) <\/span>\u4e3a\u901f\u5ea6\u573a\uff0c<\/span>w <\/span>= \u2207 \u00d7 <\/span>u<\/span>\u662f\u6da1\u5ea6\uff0c<\/span>w<\/span>0 <\/span>\u2208 <\/span>L<\/span>2<\/span>((0,1)<\/span>2<\/span>; R) <\/span>\u662f\u521d\u59cb\u6da1\u91cf\uff0c <\/span>\u03bd <\/span>\u2208 R<\/span>+ <\/span>\u662f\u7c98\u6027\u7cfb\u6570\uff0c<\/span>f <\/span>\u2208 <\/span>L<\/span>2<\/span>((0,1)<\/span>2<\/span>; R)<\/span>\u662f\u5f3a\u8feb\u51fd\u6570\u3002\u6211\u4eec\u7684\u76ee\u6807\u662f\u8bad\u7ec3\u4e00\u4e2a\u795e\u7ecf\u7b97\u5b50\u6765\u5b66\u4e60\u6620\u5c04<\/span>G <\/span>: <\/span>C<\/span>([0,10]; <\/span>H<\/span>r<\/span>((0,1)<\/span>2<\/span>; R)) \u2192 <\/span>C<\/span>([10,<\/span>T <\/span>]; <\/span>H<\/span>r<\/span>((0,1)<\/span>2<\/span>; R))<\/span>\uff0c\u5373\u4e00\u4e2a\u4ece<\/span>[0,10]<\/span>\u7684\u6da1\u5ea6 \u6620\u5c04\u5230 <\/span>[10,<\/span>T<\/span>]<\/span>\u7684\u6da1\u5ea6\u7684\u7b97\u5b50\u3002\u8bad\u7ec3\u7684\u6548\u679c\u5982\u4e0b\u56fe\u6240\u793a<\/span><\/span><\/section>\n
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\u56fe4. \u5404\u79cd\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\u5728\u7eb3\u7ef4-\u65af\u6258\u514b\u65af\u65b9\u7a0b\u4e0a\u7684\u5b66\u4e60\u6548\u679c<\/span><\/section>\n

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\u5176\u4e2d
<\/span>ResNet<\/span>\uff1a\u5177\u6709\u6b8b\u5dee\u8fde\u63a5\u7684 <\/span>18 <\/span>\u5c42\u4e8c\u7ef4\u5377\u79ef\u7f51\u7edc\u3002 <\/span><\/span><\/section>\n
U-Net<\/span>\uff1a\u4e00\u79cd\u7528\u4e8e\u5904\u7406\u56fe\u50cf\u5230\u56fe\u50cf\u56de\u5f52\u4efb\u52a1\u7684\u6d41\u884c\u65b9\u6cd5\u3002 <\/span><\/span><\/section>\n
TF-Net<\/span>\uff1a\u4e00\u79cd\u7528\u4e8e\u5b66\u4e60\u6e4d\u6d41\u7684\u7f51\u7edc\u3002
<\/span>FNO-2d<\/span>\uff1a\u5177\u6709 <\/span>RNN <\/span>\u7ed3\u6784\u7684\u4e8c\u7ef4\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\u3002 <\/span><\/span><\/section>\n
FNO-3d<\/span>\uff1a\u76f4\u63a5\u5728\u65f6\u7a7a\u4e2d\u8fdb\u884c\u5377\u79ef\u7684\u4e09\u7ef4\u5085\u7acb\u53f6\u795e\u7ecf\u7b97\u5b50\u3002<\/span><\/span><\/section>\n

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\u53c2\u8003\u6587\u732e<\/span><\/strong><\/span><\/section>\n
Bengio, Y., LeCun, Y., et al. (2007). Scaling learning algorithms towards ai. Large-scale kernel machines, 34(5):1\u201341.<\/span><\/section>\n
Fan, Y., Bohorquez, C. O., and Ying, L. (2019a). Bcr-net: A neural network based on the nonstandard wavelet form. Journal of Computational Physics, 384:1\u201315.<\/span><\/section>\n
Fan, Y., Lin, L., Ying, L., and Zepeda-Nu \u0301nez, L. (2019b). A multiscale neural networkbased on hierarchical matrices. Multiscale Modeling & Simulation, 17(4):1189\u20131213.<\/span><\/section>\n
Hornik, K., Stinchcombe, M., White, H., et al. (1989). Multilayer feedforward networks are universal approximators. Neural networks, 2(5):359\u2013366.<\/span><\/section>\n
Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A. (2020a). Fourier neural operator for parametric partial differential equations.<\/span><\/section>\n
Mingo, L., Aslanyan, L., Castellanos, J., Diaz, M., and Riazanov, V. (2004). Fourier neural networks: An approach with sinusoidal activation functions.<\/span><\/section>\n
Mathieu, M., Henaff, M., and LeCun, Y. (2013). Fast training of convolutional net- works through ffts.<\/span><\/section>\n

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AI+Science \u662f\u8fd1\u5e74\u5174\u8d77\u7684\u5c06\u4eba\u5de5\u667a\u80fd\u548c\u79d1\u5b66\u76f8\u7ed3\u5408\u7684\u4e00\u79cd\u8d8b\u52bf\u3002\u4e00\u65b9\u9762\u662f AI for Science\uff0c\u673a\u5668\u5b66\u4e60\u548c\u5176\u4ed6 AI \u6280\u672f\u53ef\u4ee5\u7528\u6765\u89e3\u51b3\u79d1\u5b66\u7814\u7a76\u4e2d\u7684\u95ee\u9898\uff0c\u4ece\u9884\u6d4b\u5929\u6c14\u548c\u86cb\u767d\u8d28\u7ed3\u6784\uff0c\u5230\u6a21\u62df\u661f\u7cfb\u78b0\u649e\u3001\u8bbe\u8ba1\u4f18\u5316\u6838\u805a\u53d8\u53cd\u5e94\u5806\uff0c\u751a\u81f3\u50cf\u79d1\u5b66\u5bb6\u4e00\u6837\u8fdb\u884c\u79d1\u5b66\u53d1\u73b0\uff0c\u88ab\u79f0\u4e3a\u79d1\u5b66\u53d1\u73b0\u7684\u201c\u7b2c\u4e94\u8303\u5f0f\u201d\u3002\u53e6\u4e00\u65b9\u9762\u662f Science for AI\uff0c\u79d1\u5b66\u5c24\u5176\u662f\u7269\u7406\u5b66\u4e2d\u7684\u89c4\u5f8b\u548c\u601d\u60f3\u542f\u53d1\u673a\u5668\u5b66\u4e60\u7406\u8bba\uff0c\u4e3a\u4eba\u5de5\u667a\u80fd\u7684\u53d1\u5c55\u63d0\u4f9b\u5168\u65b0\u7684\u89c6\u89d2\u548c\u65b9\u6cd5\u3002<\/span><\/section>\n

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\u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u65af\u5766\u798f\u5927\u5b66\u8ba1\u7b97\u673a\u79d1\u5b66\u7cfb\u535a\u58eb\u540e\u7814\u7a76\u5458\u5434\u6cf0\u9716\uff08Jure Leskovec \u6559\u6388\u6307\u5bfc\uff09\u3001\u54c8\u4f5b\u91cf\u5b50\u8ba1\u5212\u7814\u7a76\u5458\u6248\u9e3f\u4e1a\u3001\u9ebb\u7701\u7406\u5de5\u5b66\u9662\u7269\u7406\u7cfb\u535a\u58eb\u751f\u5218\u5b50\u9e23\uff08Max Tegmark \u6559\u6388\u6307\u5bfc\uff09\uff0c\u5171\u540c\u53d1\u8d77\u4ee5\u201cAI+Science<\/a>\u201d\u4e3a\u4e3b\u9898\u7684\u8bfb\u4e66\u4f1a\uff0c\u63a2\u8ba8\u8be5\u9886\u57df\u7684\u91cd\u8981\u95ee\u9898\uff0c\u5171\u5b66\u5171\u7814\u76f8\u5173\u6587\u732e\u3002\u8bfb\u4e66\u4f1a\u4ece2023\u5e743\u670826\u65e5\u5f00\u59cb\uff0c\u6bcf\u5468\u65e5\u65e9\u4e0a 9:00-11:00 \u7ebf\u4e0a\u4e3e\u884c\uff0c\u6301\u7eed\u65f6\u95f4\u9884\u8ba110\u5468\u3002\u6b22\u8fce\u5bf9\u63a2\u7d22\u8fd9\u4e2a\u6fc0\u52a8\u4eba\u5fc3\u7684\u524d\u6cbf\u9886\u57df\u6709\u5174\u8da3\u7684\u670b\u53cb\u62a5\u540d\u53c2\u4e0e\u3002<\/span><\/section>\n

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\u62a5\u540d\u94fe\u63a5\uff1ahttps:\/\/pattern.swarma.org\/study_group\/24?from=wechat<\/span><\/section>\n

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\u4e3a\u4e86\u63a2\u8ba8\u56fe\u795e\u7ecf\u7f51\u7edc\u5728\u7b97\u6cd5\u795e\u7ecf\u5316\u6c42\u89e3\u7684\u53d1\u5c55\u4e0e\u73b0\u5b9e\u5e94\u7528\uff0c\u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u56fd\u9632\u79d1\u6280\u5927\u5b66\u7cfb\u7edf\u5de5\u7a0b\u5b66\u9662\u526f\u6559\u6388\u8303\u957f\u4fca\u3001\u4e2d\u56fd\u4eba\u6c11\u5927\u5b66\u9ad8\u74f4\u4eba\u5de5\u667a\u80fd\u5b66\u9662\u52a9\u7406\u6559\u6388\u9ec4\u6587\u70b3\uff0c\u5171\u540c\u53d1\u8d77\u300c\u56fe\u795e\u7ecf\u7f51\u7edc\u4e0e\u7ec4\u5408\u4f18\u5316\u300d\u8bfb\u4e66\u4f1a\u3002\u8bfb\u4e66\u4f1a\u5c06\u805a\u7126\u4e8e\u56fe\u795e\u7ecf\u7f51\u7edc\u4e0e\u7b97\u6cd5\u795e\u7ecf\u5316\u6c42\u89e3\u7684\u76f8\u5173\u9886\u57df\uff0c\u5305\u62ec\u795e\u7ecf\u7b97\u6cd5\u63a8\u7406\u3001\u7ec4\u5408\u4f18\u5316\u95ee\u9898\u6c42\u89e3\u3001\u51e0\u4f55\u56fe\u795e\u7ecf\u7f51\u7edc\uff0c\u4ee5\u53ca\u7b97\u6cd5\u795e\u7ecf\u5316\u6c42\u89e3\u5728 AI for Science \u4e2d\u7684\u5e94\u7528\u7b49\u65b9\u9762\uff0c\u5e0c\u671b\u4e3a\u53c2\u4e0e\u8005\u63d0\u4f9b\u4e00\u4e2a\u5b66\u672f\u4ea4\u6d41\u5e73\u53f0\uff0c\u6fc0\u53d1\u53c2\u4e0e\u8005\u7684\u5b66\u672f\u5174\u8da3\uff0c\u8fdb\u4e00\u6b65\u63a8\u52a8\u76f8\u5173\u9886\u57df\u7684\u7814\u7a76\u548c\u5e94\u7528\u53d1\u5c55\u3002\u8bfb\u4e66\u4f1a\u4ece2023\u5e746\u670814\u65e5\u5f00\u59cb\uff0c\u6bcf\u5468\u4e09\u665a 19:00-21:00 \u4e3e\u884c\uff0c\u6301\u7eed\u65f6\u95f4\u9884\u8ba18\u5468\u3002\u6b22\u8fce\u611f\u5174\u8da3\u7684\u670b\u53cb\u62a5\u540d\u53c2\u4e0e\uff01<\/span><\/section>\n

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\u8be6\u60c5\u8bf7\u89c1\uff1a<\/span><\/section>\n
\u52a0\u901f\u7ecf\u5178\u7b97\u6cd5\u6548\u7387\uff0c\u7a81\u7834\u73b0\u5b9e\u6280\u672f\u74f6\u9888\uff1a\u56fe\u795e\u7ecf\u7f51\u7edc\u4e0e\u7ec4\u5408\u4f18\u5316\u8bfb\u4e66\u4f1a\u542f\u52a8<\/a><\/span>
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\u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/p>\n

1. <\/strong>2023\u5e74\uff0c\u8bf7\u4e0d\u8981\u5ffd\u7565\uff1a\u56fe\u795e\u7ecf\u7f51\u7edc\uff08GNN\uff09\u8fd9\u4e00\u5339\u9ed1\u9a6c\uff01<\/a><\/strong><\/span><\/section>\n
2.<\/strong> Barab\u00e1si \u6700\u65b0\u7814\u7a76\uff1a\u5229\u7528\u56fe\u795e\u7ecf\u7f51\u7edc\u52a0\u901f\u7f51\u7edc\u5e03\u5c40<\/a><\/strong><\/span><\/strong><\/section>\n
3. <\/strong>Nat. Commun. \u901f\u9012\uff1a\u56fe\u795e\u7ecf\u7f51\u7edc\u9884\u6d4b\u5355\u4e2a\u86cb\u767d\u8d28\u7ed3\u6784\u4e2d\u9690\u85cf\u53e3\u888b\u7684\u4f4d\u7f6e<\/a><\/strong><\/span><\/section>\n
4. \u300a\u5f20\u6c5f\u00b7\u590d\u6742\u79d1\u5b66\u524d\u6cbf27\u8bb2\u300b\u5b8c\u6574\u4e0a\u7ebf\uff01<\/strong><\/a><\/strong><\/strong><\/strong><\/section>\n
<\/strong><\/a><\/strong><\/strong><\/strong><\/section>\n
5. \u6210\u4e3a\u96c6\u667aVIP\uff0c\u89e3\u9501\u5168\u7ad9\u8bfe\u7a0b\uff0f\u8bfb\u4e66\u4f1a<\/span><\/strong><\/a><\/strong><\/strong><\/strong><\/strong><\/section>\n
6.<\/strong><\/strong><\/strong> <\/strong><\/strong><\/strong>\u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/strong><\/a><\/strong><\/strong><\/strong><\/span><\/section>\n

<\/section>\n


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\u70b9\u51fb\u201c\u9605\u8bfb\u539f\u6587\u201d\uff0c\u62a5\u540d\u8bfb\u4e66\u4f1a<\/span><\/strong><\/p>\n

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