{"id":54014,"date":"2024-12-05T21:23:10","date_gmt":"2024-12-05T13:23:10","guid":{"rendered":"https:\/\/swarma.org\/?p=54014"},"modified":"2024-12-05T21:23:10","modified_gmt":"2024-12-05T13:23:10","slug":"nat-rev-phys-%e8%a7%82%e7%82%b9%ef%bc%9a%e8%b6%85%e8%b6%8a%e4%bc%a0%e7%bb%9f%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9c%ef%bc%8c%e7%a5%9e%e7%bb%8f%e7%ae%97%e5%ad%90%e5%8a%a0%e9%80%9f%e7%a7%91%e5%ad%a6","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=54014","title":{"rendered":"Nat.Rev.Phys.\u89c2\u70b9\uff1a\u8d85\u8d8a\u4f20\u7edf\u795e\u7ecf\u7f51\u7edc\uff0c\u795e\u7ecf\u7b97\u5b50\u52a0\u901f\u79d1\u5b66\u4eff\u771f\u4e0e\u8bbe\u8ba1"},"content":{"rendered":"
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\u5173\u952e\u8bcd\uff1aAI for Science\uff0c\u4eba\u5de5\u667a\u80fd\uff0c\u590d\u6742\u7cfb\u7edf\uff0c\u795e\u7ecf\u7f51\u7edc\uff0c\u795e\u7ecf\u7b97\u5b50<\/span><\/strong><\/span><\/strong><\/strong><\/span><\/strong><\/strong><\/strong><\/section>\n
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Kamyar Azizzadenesheli, Nikola Kovachki, Zongyi Li, Miguel Liu-Schiaffini, Jean Kossaifi & Anima Anandkumar<\/span> <\/span>| \u4f5c\u8005<\/span><\/p>\n

\u6768\u9e64<\/span> <\/span>| \u8bd1\u8005<\/span><\/span><\/p>\n

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1. \u795e\u7ecf\u7b97\u5b50\u5728\u79d1\u5b66\u4eff\u771f\u4e0a\u7684\u4f18\u52bf<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u81ea\u7136\u79d1\u5b66\u7684\u7814\u7a76\u76ee\u7684\u5728\u4e8e\u63ed\u793a\u7269\u8d28\u4e16\u754c\u53d1\u751f\u7684\u73b0\u8c61\u4ee5\u53ca\u7269\u8d28\u73b0\u8c61\u53d1\u751f\u8fc7\u7a0b\u7684\u5b9e\u8d28\u3002\u800c\u4e00\u4e2a\u66f4\u7b80\u660e\u7684\u7406\u89e3\u662f\u2014\u2014\u6211\u4eec\u5728\u6316\u6398\u81ea\u7136\u4e2d\u4e00\u5207\u4ee5\u65f6\u95f4\u4e3a\u81ea\u53d8\u91cf\u7684\u51fd\u6570\u3002\u5728\u7269\u7406\u3001\u5316\u5b66\u548c\u5de5\u7a0b\u9886\u57df\uff0c\u6211\u4eec\u53d1\u73b0\u5927\u90e8\u5206\u65f6\u95f4\u51fd\u6570<\/span>\uff08\u6216\u65f6\u95f4\u4e3a\u9690\u53d8\u91cf\uff09<\/span>\u662f\u7531\u67d0\u4e2a\u7b2c\u4e00\u6027\u539f\u7406\u5bfc\u51fa\u7684\u504f\u5fae\u5206\u65b9\u7a0b<\/span> (Partial Differential Equations, PDE) <\/span>\u3002\u8fd9\u4e9b\u539f\u7406\u5305\u62ec\u70ed\u529b\u5b66\u5b9a\u5f8b\u3001\u5316\u5b66\u76f8\u4e92\u4f5c\u7528\u548c\u529b\u5b66\u5b9a\u5f8b\u7b49\u3002\u5982\u679c\u5df2\u77e5\u8fd9\u6837\u7684\u539f\u7406\u6216\u5176\u8fd1\u4f3c\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u901a\u8fc7\u5927\u91cf\u7684\u6570\u503c\u4eff\u771f\u6765\u6c42\u51fa\u7b26\u5408\u7cbe\u5ea6\u8981\u6c42\u7684\u89e3\u4ee5\u89e3\u51b3\u5b9e\u9645\u95ee\u9898\u3002\u4f8b\u5982\uff0c\u7eb3\u7ef4-\u65af\u6258\u514b\u65af\u65b9\u7a0b <\/span>(Navier-Stokes PDE) <\/span>\u53ef\u7528\u4e8e\u6a21\u62df\u6c7d\u8f66\u548c\u98de\u673a\u7684\u7a7a\u6c14\u52a8\u529b\u5b66\u7279\u6027\u3002\u7136\u800c\u5bf9\u4e8e\u89e3\u6790\u5730\u7403\u6c14\u5019\u53d8\u5316\u7684\u8fd9\u7c7b\u5de8\u590d\u6742\u7cfb\u7edf\uff0c\u901a\u8fc7\u5b9e\u9a8c\u83b7\u53d6\u5927\u91cf\u6570\u636e\u8fdb\u884c\u6570\u503c\u4eff\u771f\u662f\u5b8c\u5168\u4e0d\u73b0\u5b9e\u7684\u3002\u901a\u8fc7\u4f7f\u7528\u5927\u6a21\u578b\u7b97\u6cd5\uff0c\u6211\u4eec\u53ef\u4ee5\u5efa\u7acb\u5feb\u901f\u3001\u9ad8\u4fdd\u771f\u7684\u4eff\u771f\u5b9e\u9a8c\u3002\u4ee5\u5f80\u7684AI\u6a21\u578b\uff0c\u5982\u7a00\u758f\u8868\u793a\u3001\u9012\u5f52\u795e\u7ecf\u7f51\u7edc\u3001\u50a8\u5907\u6c60\u8ba1\u7b97\u7b49\uff0c\u5df2\u7ecf\u5728\u5efa\u6a21\u52a8\u6001\u7cfb\u7edf\u4e2d\u663e\u793a\u51fa\u4e86\u5e0c\u671b\u3002AI \u7684\u6700\u65b0\u8fdb\u5c55\u2014\u2014\u795e\u7ecf\u7b97\u5b50\uff0c\u5728\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u4e0a\u5c55\u73b0\u51fa\u5f3a\u5927\u7684\u80fd\u529b\uff0c\u53ef\u4ee5\u6a21\u62df\u590d\u6742\u73b0\u8c61\u5e76\u6355\u6349\u66f4\u7ec6\u5fae\u7684\u5c3a\u5ea6\uff0c\u4ece\u800c\u5f97\u5230\u9ad8\u4fdd\u771f\u5ea6\u7684\u89e3<\/strong>\u3002<\/span><\/section>\n

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\u4e00\u822c\u7684\u795e\u7ecf\u7f51\u7edc\u5728\u6b64\u7c7b\u4eff\u771f\u5b9e\u9a8c\u4e2d\u8868\u73b0\u5982\u4f55\u5462\uff1f\u5927\u5bb6\u719f\u77e5\u7684 ChatGPT \u548c\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u7b49\u6a21\u578b\u5728\u7ed8\u56fe\u548c\u4f5c\u6587\u4e0a\u8868\u73b0\u51fa\u4e86\u4ee4\u4eba\u5370\u8c61\u6df1\u523b\u7684\u6027\u80fd\uff0c\u4f46\u4f5c\u8005\u6307\u51fa\uff0c\u8fd9\u4e00\u7c7b\u6a21\u578b\u5177\u6709\u6839\u672c\u7684\u5c40\u9650\u6027\uff0c\u963b\u788d\u4e86\u5b83\u4eec\u6210\u529f\u7528\u4e8e\u79d1\u5b66\u8ba1\u7b97\u3002\u53d7\u9650\u4e8e\u8bad\u7ec3\u6570\u636e\u96c6\u7684\u6027\u8d28\uff0c\u7531\u4e8e\u5b9e\u9a8c\u91c7\u96c6\u7684\u901a\u5e38\u662f\u79bb\u6563\u6570\u636e<\/span>\uff08\u56fe1a\u3001\u56fe1b\uff09<\/span>\uff0c\u56e0\u800c\u4f20\u7edf\u795e\u7ecf\u7f51\u7edc\u65e0\u6cd5\u9884\u6d4b\u6a21\u62df\u8bb8\u591a\u53d1\u751f\u5728\u8fde\u7eed\u57df<\/span> (continuous domains) <\/span>\u4e0a\u7684\u79d1\u5b66\u73b0\u8c61<\/span>\uff08\u4f8b\u5982\u5927\u90e8\u5206\u52a8\u529b\u5b66\u7cfb\u7edf\u3001\u6ce2\u4f20\u5bfc\u548c\u6750\u6599\u5f62\u53d8\uff09<\/span>\u3002\u56e0\u4e3a\u53ea\u652f\u6301\u56fa\u5b9a\u5206\u8fa8\u7387\u7684\u8f93\u5165\u548c\u8f93\u51fa\uff0c\u6240\u4ee5\u4f20\u7edf\u795e\u7ecf\u7f51\u7edc\u6781\u6613\u88ab\u79bb\u6563\u7684\u8bad\u7ec3\u96c6\u9650\u5236\u5e76\u8499\u853d[1]\u3002\u800c\u795e\u7ecf\u7b97\u5b50\u5219\u5177\u5907\u4ee5\u65e0\u9650\u5206\u8fa8\u7387\u9884\u6d4b\u8f93\u51fa\u7684\u80fd\u529b<\/span>\uff08\u56fe1c\uff09<\/span>\u3002\u5b83\u4eec\u901a\u8fc7\u8fd1\u4f3c\u5e95\u5c42\u7684\u7b97\u5b50\u6765\u5b9e\u73b0\u8fd9\u4e00\u70b9\uff0c\u76f4\u63a5\u7ed9\u51fa\u8f93\u5165\u548c\u8f93\u51fa\u51fd\u6570\u7a7a\u95f4\u4e4b\u95f4\u7684\u6620\u5c04\uff0c\u800c\u8fd9\u4e24\u4e2a\u7a7a\u95f4\u90fd\u53ef\u4ee5\u662f\u65e0\u9650\u7ef4\u7684\u3002\u7b80\u5355\u6765\u8bf4\uff0c\u795e\u7ecf\u7b97\u5b50\u662f\u4e00\u7c7b\u6cdb\u5316\u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u5177\u6709\u6bd4\u4f20\u7edf\u795e\u7ecf\u7f51\u7edc\u66f4\u7cbe\u51c6\u7684\u9884\u6d4b\u80fd\u529b\u3002<\/strong>\u5373\u4f7f\u4ec5\u4f7f\u7528\u79bb\u6563\u6570\u636e\u8fdb\u884c\u8bad\u7ec3\uff0c\u795e\u7ecf\u7b97\u5b50\u4e5f\u80fd\u8fd1\u4f3c\u8fde\u7eed\u6620\u5c04\uff0c\u56e0\u6b64\u80fd\u591f\u66f4\u51c6\u786e\u5730\u6355\u6349\u66f4\u7cbe\u7ec6\u5c3a\u5ea6\u7684\u4fe1\u606f\uff0c\u7ed3\u679c\u4e0d\u53d7\u8bad\u7ec3\u6570\u636e\u7684\u79bb\u6563\u7a0b\u5ea6\u9650\u5236\uff0c\u4f7f\u5f97\u5b83\u4eec\u5728\u89e3\u51b3\u5b9e\u9a8c\u6a21\u62df\u4e0a\u5177\u5907\u4ee5\u5f80\u6a21\u578b\u65e0\u53ef\u6bd4\u62df\u7684\u4f18\u52bf\u3002<\/span><\/section>\n

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\u56fe1 a, \u795e\u7ecf\u7f51\u7edc\u5b66\u4e60\u56fa\u5b9a\u7f51\u683c\u3001\u79bb\u6563\u7f51\u683c\u4e0a\u7684\u8f93\u5165\u548c\u8f93\u51fa\u70b9\u4e4b\u95f4\u7684\u6620\u5c04\u3002b, \u795e\u7ecf\u7b97\u5b50\u5728\u8fde\u7eed\u57df\u4e4b\u95f4\u7684\u51fd\u6570\u4e0a\u8fdb\u884c\u6620\u5c04\uff0c\u5373\u4f7f\u8bad\u7ec3\u6570\u636e\u5728\u56fa\u5b9a\u7f51\u683c\u4e0a\u4e5f\u662f\u5982\u6b64\u3002c, \u56e0\u4e3a\u795e\u7ecf\u7b97\u5b50\u5728\u51fd\u6570\u4e4b\u95f4\u8fdb\u884c\u6620\u5c04\uff0c\u5b83\u63a5\u53d7\u8bad\u7ec3\u7f51\u683c\u4e4b\u5916\u7684\u8f93\u5165\uff0c\u5e76\u80fd\u8fdb\u884c\u8d85\u5206\u8fa8\u7387\u5904\u7406\u3002\uff5c\u56fe\u7247\u6765\u6e90\uff1a\u539f\u6587\u56fe2<\/span><\/p>\n


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2. \u795e\u7ecf\u7b97\u5b50\u7684\u57fa\u672c\u67b6\u6784\u548c\u7c7b\u578b<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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2.1 \u57fa\u672c\u67b6\u6784<\/strong><\/span><\/section>\n

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\u795e\u7ecf\u7b97\u5b50\u4e0e\u6807\u51c6\u795e\u7ecf\u7f51\u7edc\u5177\u6709\u7c7b\u4f3c\u7684\u7ed3\u6784\uff0c\u53ef\u5206\u4e3a\u7ebf\u6027\u90e8\u5206\u548c\u975e\u7ebf\u6027\u90e8\u5206\u3002\u524d\u6587\u63d0\u5230\uff0c\u6807\u51c6\u795e\u7ecf\u7f51\u7edc\u53ea\u80fd\u8f93\u5165\u548c\u8f93\u51fa\u5177\u6709\u56fa\u5b9a\u5206\u8fa8\u7387\u7684\u6570\u636e\uff0c\u8fd9\u662f\u56e0\u4e3a\u5b83\u4eec\u901a\u5e38\u7531\u4e00\u7cfb\u5217\u7ebf\u6027\u51fd\u6570\u6a21\u5757\u7ec4\u6210\uff0c\u5982\u5145\u5206\u5168\u8fde\u63a5\u6216\u5377\u79ef\u7684\u5c42\uff0c\u540e\u9762\u8ddf\u7740\u975e\u7ebf\u6027\u6fc0\u6d3b\u51fd\u6570\uff0c\u4f8b\u5982\u4fee\u6b63\u7ebf\u6027\u5355\u5143<\/span> (ReLU) <\/span>\u3002\u6807\u51c6\u795e\u7ecf\u7f51\u7edc\u50cf\u4e00\u5f20\u94fa\u5f00\u7684\u86db\u7f51\uff0c\u6bcf\u4e00\u4e2a\u8282\u70b9\u662f\u7b80\u5355\u7684\u53c2\u6570\u3002\u795e\u7ecf\u7b97\u5b50\u7684\u57fa\u672c\u67b6\u6784\u7c7b\u4f3c\uff0c\u5373\u7ebf\u6027\u6a21\u5757\u540e\u9762\u8ddf\u7740\u975e\u7ebf\u6027\u53d8\u6362\uff0c\u4e0d\u540c\u7684\u662f\u795e\u7ecf\u7b97\u5b50\u4e2d\u7684\u7ebf\u6027\u90e8\u5206\u8ba1\u7b97\u7684\u662f\u79ef\u5206\u7b97\u5b50\u800c\u4e0d\u662f\u56fa\u5b9a\u7ef4\u5ea6\u7684\u7ebf\u6027\u51fd\u6570<\/strong>\uff0c\u4f8b\u5982\u683c\u6797\u51fd\u6570\u79ef\u5206\u5b9e\u73b0\u7684\u7ebf\u6027 PDE \u7684\u89e3\u7b97\u5b50\uff1b\u8fd9\u5c31\u50cf\u86db\u7f51\u6bcf\u4e2a\u8282\u70b9\u88ab\u671d\u9732\u6d78\u6da6\uff0c\u5728\u9633\u5149\u4e0b\u6298\u5c04\u51fa\u4e1d\u7ebf\u7684\u65e0\u9650\u7ec6\u8282\u3002<\/span><\/section>\n

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<\/section>\n

\u56fe2 \u7528\u86db\u7f51\u88ab\u671d\u9732\u6d78\u6da6\u6765\u8868\u73b0\u795e\u7ecf\u7b97\u5b50\u5728\u5904\u7406\u65e0\u9650\u5206\u8fa8\u7387\u6570\u636e\u4efb\u52a1\u7684\u6f5c\u529b\uff5c\u56fe\u7247\u6765\u6e90\uff1aStable Diffusion3 \u751f\u6210<\/span><\/p>\n


<\/span><\/section>\n
\u4ee5\u4e0b\u7ed9\u51fa\u7ebf\u6027\u79ef\u5206\u7b97\u5b50\u7684\u8868\u8fbe\u5f0f\uff1a<\/span><\/section>\n

<\/section>\n

<\/p>\n


<\/section>\n
\u5176\u4e2d a(\u2009\u22c5\u2009) <\/span>\u662f\u7b97\u5b50\u5757\u7684\u8f93\u5165\u51fd\u6570\uff0c\u03ba(x,y) <\/span>\u5206\u522b\u8868\u793a\u8f93\u51fa\u57df\u548c\u8f93\u5165\u57df\u4e2d\u4efb\u610f\u4e24\u70b9 x \u548c y \u4e4b\u95f4\u7684\u53ef\u5b66\u4e60\u6838\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u5728\u7ebf\u6027\u79ef\u5206\u7b97\u5b50\u4e2d\uff0c\u65b9\u7a0b\u8f93\u51fa\u4e0d\u4f1a\u5c40\u9650\u4e8e\u8bad\u7ec3\u6570\u636e\u7684\u79bb\u6563\u5ea6\uff0c\u53ef\u4ee5\u662f\u8fde\u7eed\u57df\u4e2d\u7684\u4efb\u4f55\u70b9<\/span>\uff08\u56fe3a\uff09<\/span>\u3002\u800c\u5bf9\u4e8e\u8f93\u5165\u6765\u8bf4\uff0c\u79bb\u6563\u5ea6\u8d8a\u5c0f\uff0c\u8fd1\u4f3c\u5c06\u8d8a\u51c6\u786e\u3002\u7f51\u683c\u662f\u6570\u503c\u8ba1\u7b97\u4e2d\u5c06\u6c42\u89e3\u57df\u79bb\u6563\u5316\u7684\u4e00\u79cd\u8bbe\u8ba1\uff0c\u800c\u795e\u7ecf\u7b97\u5b50\u5177\u6709\u79f0\u4e3a\u79bb\u6563\u5316\u6536\u655b<\/strong>\uff08discretization convergence\uff09\u7684\u6027\u8d28\uff0c\u5373\uff0c\u5728\u8f93\u5165\u7f51\u683c\u5927\u5c0f\u8d8b\u4e8e\u96f6\u65f6\uff0c\u795e\u7ecf\u7b97\u5b50\u53ef\u4ee5\u6536\u655b\u5230\u552f\u4e00\u7684\u7b97\u5b50\uff08\u56fe3b\uff09\u3002<\/strong>\u8fd9\u4e00\u7279\u6027\u6765\u81ea\u4e8e\u795e\u7ecf\u7b97\u5b50\u4e2d\u4f7f\u7528\u7684\u79ef\u5206\u8fd1\u4f3c\u65b9\u6cd5\uff0c\u5982\u9ece\u66fc\u548c<\/span>(Riemannian sum)<\/span>\u3001\u4f3d\u8fbd\u91d1\u8c31<\/span> (Galerkin spectral)<\/span> \u6216\u5085\u91cc\u53f6\u8c31<\/span> (Fourier spectral)<\/span>\u3002\u8fd9\u662f\u8861\u91cf\u4efb\u4f55\u79d1\u5b66\u8ba1\u7b97\u7684\u53ef\u9760\u6027\u6240\u5fc5\u9700\u7684\u5173\u952e\u7279\u6027\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u5728\u7ebf\u6027\u79ef\u5206\u7b97\u5b50\u540e\uff0c\u8ddf\u7740\u9010\u70b9\u975e\u7ebf\u6027\u6fc0\u6d3b\u6a21\u5757\uff0c\u4f8b\u5982\u9ad8\u65af\u8bef\u5dee\u7ebf\u6027\u5355\u5143<\/span> (Gaussian Error Linear Units\uff0cGeLU)<\/span>\u3002\u9700\u8981\u6ce8\u610f\uff0c\u795e\u7ecf\u7b97\u5b50\u4e0d\u4f7f\u7528\u4fee\u6b63\u7ebf\u6027\u5355\u5143 <\/span>(ReLU) <\/span>\uff0c\u56e0\u4e3a\u5b83\u4eec\u4e0d\u5149\u6ed1\u3002<\/span><\/section>\n

<\/span><\/section>\n
2.2 \u5177\u4f53\u7c7b\u578b<\/strong><\/span><\/section>\n

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\u57fa\u4e8e\u5bf9\u65b9\u7a0b (1) \u4e2d\u79ef\u5206\u64cd\u4f5c\u7684\u53c2\u6570\u5316\u65b9\u6cd5\uff0c\u5df2\u6709\u8bb8\u591a\u4e0d\u540c\u67b6\u6784\u7684\u795e\u7ecf\u7b97\u5b50\u3002\u4f8b\u5982\uff1a<\/span><\/section>\n

<\/strong><\/section>\n
1. DeepONet\u6a21\u578b<\/span><\/strong>
<\/section>\n

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DeepONet \u6a21\u578b\u57fa\u4e8e\u73b0\u6709\u6570\u503c\u65b9\u6cd5\uff0c\u5e76\u5c06\u65b9\u7a0b (1) \u4e2d\u7684\u6838 \u03ba(x,y) \u9650\u5236\u5728 x \u548c y\u53ef\u5206\u79bb\u7684\u60c5\u5f62<\/span>\uff08\u5373\u03ba(x<\/em>,y<\/em>)= \u03ba1<\/sub>(x<\/em>) \u03ba2<\/sub>(y<\/em>)\uff09<\/span>\u3002\u5c3d\u7ba1\u6700\u521d\u7684DeepONet \u6a21\u578b\u88ab\u9650\u5236\u5728\u56fa\u5b9a\u7684\u8f93\u5165\u7f51\u683c\u4e0a\uff0c\u4f46\u540e\u6765\u7684 DeepONet \u6269\u5c55\u5374\u53bb\u9664\u4e86\u8fd9\u4e00\u9650\u5236\uff0c\u4fdd\u7559\u4e86\u79bb\u6563\u5316\u6536\u655b\u6027\uff0c\u56e0\u6b64\u662f\u795e\u7ecf\u7b97\u5b50\u7684\u4e00\u4e2a\u7279\u4f8b\u3002<\/span><\/section>\n

<\/strong><\/section>\n
2. \u56fe\u795e\u7ecf\u7b97<\/span><\/strong>\u5b50\uff08GNO\uff09<\/span><\/strong>
<\/section>\n

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\u56fe\u795e\u7ecf\u7b97\u5b50<\/span> (Graph Neural Operators, GNO) <\/span>\u662f\u56fe\u795e\u7ecf\u7f51\u7edc\u7684\u7b97\u5b50\u6269\u5c55\uff0c\u652f\u6301\u5728\u56fa\u5b9a\u534a\u5f84\u7403\u4f53\u4e0a\u7684\u6838\u79ef\u5206\u3002\u56fe\u795e\u7ecf\u7b97\u5b50\u5584\u4e8e\u5904\u7406\u56fe\u5206\u7c7b\u3001\u94fe\u63a5\u9884\u6d4b\u7b49\u4efb\u52a1\uff0c\u4f46\u5bf9\u4e8e\u5168\u5c40\u6027\u8d28\u7684\u628a\u63a7\u53d7\u9650\u3002\u5f53\u771f\u5b9e\u7b97\u5b50\u662f\u975e\u5c40\u90e8\u65f6\uff0c\u5982\u679c\u534a\u5f84\u8f83\u5c0f\uff0cGNO\u4f1a\u56e0\u611f\u53d7\u91ce\u6709\u9650\u800c\u8868\u8fbe\u80fd\u529b\u53d7\u9650\uff0c\u65e0\u6cd5\u6355\u6349\u5168\u5c40\u6548\u5e94\u3002\u800c\u5982\u679c\u589e\u52a0\u611f\u53d7\u91ce\u5e76\u589e\u5927\u90bb\u57df\u534a\u5f84\uff0c\u4f7f\u7528GNO\u4f1a\u53d8\u5f97\u8ba1\u7b97\u6210\u672c\u9ad8\u6602\u3002\u9488\u5bf9\u8fd9\u4e2a\u95ee\u9898\uff0c\u5176\u4ed6\u5177\u6709\u5168\u5c40\u6027\u7684\u795e\u7ecf\u7b97\u5b50\u88ab\u63d0\u51fa\u3002<\/span><\/section>\n

<\/section>\n
3. \u5085\u91cc\u53f6\u795e\u7ecf\u7b97\u5b50\uff08FNO\uff09<\/span><\/strong>
<\/section>\n

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\u5085\u91cc\u53f6\u795e\u7ecf\u7b97\u5b50<\/span> (Fourier Neural Operators, FNO) <\/span>\u5c31\u662f\u53e6\u4e00\u79cd\u83b7\u53d6\u5168\u5c40\u7b97\u5b50\u7684\u65b9\u6cd5\u3002FNO \u901a\u8fc7\u5feb\u901f\u5085\u91cc\u53f6\u53d8\u6362<\/span> (Fast Fourier Transform, FFT) <\/span>\u5b9e\u73b0\u4e00\u7cfb\u5217\u8ba1\u7b97\u5168\u5c40\u5377\u79ef\u7b97\u5b50\u7684\u5c42\uff0c\u7136\u540e\u5728\u9891\u57df\u4e2d\u6df7\u5408\u6743\u91cd\u5e76\u8fdb\u884c\u9006\u5085\u91cc\u53f6\u53d8\u6362\u3002\u5373\u901a\u8fc7\u5728\u5085\u91cc\u53f6\u57df<\/span>\uff08\u9891\u57df\uff09<\/span>\u4e2d\u4f7f\u7528\u5b66\u4e60\u5f97\u5230\u7684\u7cfb\u6570\u8fdb\u884c\u9010\u5143\u7d20\u76f8\u4e58<\/span>\uff08\u5bf9\u5e94\u4e8e\u65f6\u57df\u4e2d\u7684\u5377\u79ef\uff09<\/span>\uff0c\u7136\u540e\u901a\u8fc7\u9006\u5085\u91cc\u53f6\u53d8\u6362\u8fd4\u56de\u5230\u539f\u59cb\u57df<\/span>\uff08\u65f6\u57df\uff09<\/span>\uff0c\u5b9e\u73b0\u65b9\u7a0b (1) \u4e2d\u7684\u6838\u79ef\u5206\u8ba1\u7b97\u3002\u901a\u8fc7\u7ec4\u5408\u5168\u5c40\u5377\u79ef\u7b97\u5b50\u548c\u975e\u7ebf\u6027\u6fc0\u6d3b\uff0cFNO \u53ef\u4ee5\u903c\u8fd1\u9ad8\u5ea6\u975e\u7ebf\u6027\u548c\u975e\u5c40\u90e8\u89e3\u7b97\u5b50\u3002FNO \u53ca\u5176\u53d8\u4f53\u80fd\u591f\u6a21\u62df\u8bb8\u591a\u504f\u5fae\u5206\u65b9\u7a0b\uff0c\u4f8b\u5982\u7eb3\u7ef4-\u65af\u6258\u514b\u65af\u65b9\u7a0b\u548c\u5730\u9707\u6ce2\uff0c\u8fdb\u884c\u9ad8\u5206\u8fa8\u7387\u5929\u6c14\u9884\u62a5\uff0c\u5e76\u4ee5\u524d\u6240\u672a\u6709\u7684\u6210\u672c\u4e0e\u7cbe\u5ea6\u6743\u8861\u6765\u9884\u6d4b\u4e8c\u6c27\u5316\u78b3\u8fc1\u79fb[1]\u3002FNO \u65b9\u6cd5\u53d7\u5230\u4f2a\u8c31\u6c42\u89e3\u5668\u7684\u542f\u53d1\uff0c\u8fd9\u4e9b\u6c42\u89e3\u5668\u4f7f\u7528\u5085\u91cc\u53f6\u57fa\uff0c\u5e76\u8fed\u4ee3\u6267\u884c\u64cd\u4f5c\u3002\u5c3d\u7ba1\u53d7\u5230\u8fd9\u79cd\u65b9\u6cd5\u7684\u542f\u53d1\uff0cFNO \u7684\u975e\u7ebf\u6027\u7ec4\u4ef6\u4f7f\u5b83\u4eec\u5177\u5907\u66f4\u5f3a\u5927\u7684\u8868\u8fbe\u80fd\u529b\u3002\u5728\u5904\u7406\u89c4\u5219\u7f51\u683c\u65f6\uff0cFNO\u4f9d\u9760FFT\u5c55\u73b0\u4e86\u9ad8\u6548\u7684\u8ba1\u7b97\u4f18\u52bf\u3002\u4f46\u5728\u5904\u7406\u4e0d\u89c4\u5219\u7f51\u683c\u65f6\uff0cFNO\u7684\u5c40\u9650\u6027\u53d8\u5f97\u660e\u663e\u3002\u7531\u4e8eFFT\u4ec5\u9002\u7528\u4e8e\u5747\u5300\u7f51\u683c\uff0c\u56e0\u6b64\u5728\u5904\u7406\u975e\u5747\u5300\u6216\u4e0d\u89c4\u5219\u7f51\u683c\u65f6\uff0cFNO\u9700\u8981\u8fdb\u884c\u63d2\u503c\u6216\u5176\u4ed6\u53d8\u5f62\u64cd\u4f5c\uff0c\u8fd9\u4f1a\u5bfc\u81f4\u8ba1\u7b97\u590d\u6742\u5ea6\u589e\u52a0\uff0c\u5e76\u4e14\u53ef\u80fd\u5f15\u5165\u8f83\u5927\u7684\u63d2\u503c\u8bef\u5dee\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e00\u95ee\u9898\uff0c\u4e00\u4e9b\u7814\u7a76\u63d0\u51fa\u4e86\u5c06\u4e0d\u89c4\u5219\u7f51\u683c\u8f6c\u6362\u4e3a\u89c4\u5219\u7f51\u683c\u7684\u65b9\u6cd5\uff0c\u4f8b\u5982\u901a\u8fc7\u53d8\u5f62<\/span>\uff08deformation\uff09<\/span>[2]\u6216\u56fe\u6838\u51fd\u6570 <\/span>(graph kernels) <\/span>\u6765\u5b9e\u73b0\u3002\u6b64\u5916\uff0c\u4e5f\u6709\u7814\u7a76\u5c1d\u8bd5\u901a\u8fc7\u51e0\u4f55\u611f\u77e5\u5085\u91cc\u53f6\u53d8\u6362<\/span>\uff08\u5982Geo-FNO\uff09<\/span>\u6765\u5904\u7406\u4e0d\u89c4\u5219\u51e0\u4f55\u5f62\u72b6\uff0c\u8fd9\u79cd\u65b9\u6cd5\u901a\u8fc7\u53d8\u5f62\u5c06\u7269\u7406\u7a7a\u95f4\u6620\u5c04\u5230\u89c4\u5219\u7684\u8ba1\u7b97\u7a7a\u95f4\u4e0a\uff0c\u4ece\u800c\u53ef\u4ee5\u5728\u89c4\u5219\u7f51\u683c\u4e0a\u5e94\u7528FFT\u3002\u7136\u800c\uff0c\u8fd9\u4e9b\u65b9\u6cd5\u901a\u5e38\u9700\u8981\u5927\u91cf\u7684\u6570\u636e\u8bad\u7ec3\uff0c\u5e76\u4e14\u5728\u9762\u5bf9\u9ad8\u5ea6\u590d\u6742\u7684\u51e0\u4f55\u5f62\u72b6\u65f6\uff0c\u4ecd\u5b58\u5728\u4e00\u5b9a\u7684\u5c40\u9650\u6027\u3002<\/span><\/section>\n

<\/span><\/section>\n
\u603b\u4e4b\uff0cFNO\u5728\u89c4\u5219\u7f51\u683c\u4e0a\u7684\u9ad8\u6548\u6027\u662f\u5176\u4e00\u5927\u4f18\u52bf\uff0c\u4f46\u5728\u5904\u7406\u4e0d\u89c4\u5219\u7f51\u683c\u65f6\uff0c\u5176\u6269\u5c55\u6027\u548c\u8ba1\u7b97\u6548\u7387\u53d7\u5230\u9650\u5236\uff0c\u8fd9\u9700\u8981\u8fdb\u4e00\u6b65\u7684\u7814\u7a76\u548c\u6539\u8fdb\u4ee5\u63d0\u9ad8\u5176\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u7684\u9002\u7528\u6027\u3002<\/span><\/section>\n

<\/section>\n
4. \u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7b97\u5b50\uff08PINO\uff09<\/span><\/strong><\/section>\n

<\/section>\n
\u5728\u795e\u7ecf\u7f51\u7edc\u7684\u65e9\u671f\u5e94\u7528\u9636\u6bb5\uff0c\u7528\u795e\u7ecf\u7f51\u7edc\u8868\u793a\u8fde\u7eed\u51fd\u6570\u5c31\u662f\u4e00\u91cd\u8981\u7684\u547d\u9898\u3002\u4f8b\u5982\u5728\u56fe\u5f62\u548c\u573a\u666f\u5904\u7406\u4e2d\uff0c\u8fde\u7eed\u51fd\u6570\u7684\u4f7f\u7528\u53ef\u4ee5\u4f7f\u8ba1\u7b97\u673a\u83b7\u5f97\u8fde\u7eed\u89c6\u89c9\uff0c\u800c\u4e0d\u662f\u4e00\u4e2a\u4e2a\u50cf\u7d20\u70b9\u62fc\u63a5\u7684\u9a6c\u8d5b\u514b\u3002\u8fd9\u88ab\u79f0\u4e3a\u9690\u5f0f\u795e\u7ecf\u8868\u793a\u3002\u800c\u901a\u8fc7\u795e\u7ecf\u7f51\u7edc\u8868\u793a\u5355\u4e00\u504f\u5fae\u5206\u65b9\u7a0b <\/span>(PDE) <\/span>\u7684\u89e3\u51fd\u6570\u88ab\u79f0\u4e3a\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc<\/span> (Physics-Informed Neural Network, PINN)<\/span> \u3002\u4f8b\u5982\u5728\u8ba1\u7b97\u673a\u89c6\u89c9\u548c\u56fe\u5f62\u5b66\u4e2d\uff0c\u9690\u5f0f\u795e\u7ecf\u7f51\u7edc<\/span>\uff08Implicit Neural Networks\uff09<\/span>\u88ab\u7528\u6765\u8868\u793a\u5355\u4e00\u901f\u5ea6\u573a\u6216 3D \u573a\u666f\u3002<\/span><\/section>\n

<\/span><\/section>\n
PINN\u53ca\u5176\u53d8\u4f53\u5728\u89e3\u51b3\u7a33\u6001PDE\u65b9\u9762\u5df2\u7ecf\u663e\u793a\u51fa\u6548\u679c\u3002\u7136\u800c\uff0c\u5bf9\u4e8e\u65f6\u53d8PDE\u7b49\u95ee\u9898\uff0cPINN\u7684\u4f18\u5316\u5341\u5206\u5177\u6709\u6311\u6218\u6027\uff0c\u4e0d\u80fd\u8f7b\u6613\u5730\u4f7f\u7528\u6807\u51c6\u7684\u57fa\u4e8e\u68af\u5ea6\u6cd5\u6c42\u51fa\u5176\u89e3\u3002\u800c\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7b97\u5b50<\/span> (Physics-Informed Neural Operators, PINO) <\/span>\u5e94\u8fd0\u800c\u751f\u3002PINO \u9664\u4e86PDE\u4fe1\u606f\u5916\uff0c\u8fd8\u7ed3\u5408\u4e86\u8bad\u7ec3\u6570\u636e\uff0c\u4f7f\u5f97\u4f18\u5316\u666f\u89c2<\/span> (Optimization Landscape) <\/span>\u66f4\u52a0\u6613\u4e8e\u5904\u7406\uff0c\u5e76\u80fd\u591f\u5b66\u4e60\u590d\u6742\u65f6\u53d8PDE\u7684\u89e3\u7b97\u5b50\u3002\u6b64\u5916\uff0c\u4e0e\u7eaf\u7cb9\u7684\u6570\u636e\u9a71\u52a8\u795e\u7ecf\u7b97\u5b50\u76f8\u6bd4\uff0cPINO \u5177\u6709\u66f4\u4f18\u8d8a\u7684\u6cdb\u5316\u548c\u5916\u63a8\u80fd\u529b\uff0c\u4e14\u51cf\u5c11\u4e86\u8bad\u7ec3\u6570\u636e\u7684\u9700\u6c42\uff0c\u53ef\u4ee5\u5c06\u4f4e\u5206\u8fa8\u7387\u7684\u8bad\u7ec3\u6570\u636e\u4e0e\u9ad8\u5206\u8fa8\u7387\u7684\u7269\u7406\u7ea6\u675f\u76f8\u7ed3\u5408\uff0c\u4ece\u800c\u5f97\u5230\u5e95\u5c42\u7b97\u5b50\u7684\u66f4\u597d\u8fd1\u4f3c\uff0c\u5b9e\u73b0\u7cbe\u786e\u7684\u96f6\u6837\u672c\u8d85\u5206\u8fa8\u7387\u3002\u6b64\u5916\uff0c\u4e5f\u53ef\u4ee5\u5728\u5229\u7528\u5176\u4ed6\u7b97\u5b50\u8fdb\u884c\u8bad\u7ec3\u540e\uff0c\u901a\u8fc7\u5728\u6d4b\u8bd5<\/span> (test) <\/span>\u65f6\u5bf9\u6a21\u578b\u8fdb\u884c\u5fae\u8c03\uff0c\u4ee5\u6700\u5c0f\u5316 PDE \u635f\u5931\u51fd\u6570\uff0c\u8fdb\u4e00\u6b65\u63d0\u9ad8PINO\u7684\u5728\u6d4b\u8bd5\u65f6\u7684\u51c6\u786e\u6027\u3002\u8fd9\u4e2a\u5fae\u8c03\u6b65\u9aa4\u4e0e PINN \u4f18\u5316\u8fc7\u7a0b\u76f8\u540c\uff0c\u4f46\u662f\u533a\u522b\u5728\u4e8e\u91c7\u7528\u9884\u8bad\u7ec3\u7684 PINO \u6a21\u578b\u8fdb\u884c\u521d\u59cb\u5316\uff0c\u5e76\u8fdb\u4e00\u6b65\u5fae\u8c03\u5b83\uff0c\u800c\u4e0d\u662f\u50cf PINN \u4e2d\u90a3\u6837\u4ece\u968f\u673a\u521d\u59cb\u5316\u5f00\u59cb\u4f18\u5316\u3002\u56e0\u6b64\uff0c\u5b83\u53ef\u4ee5\u514b\u670d PINN \u4e2d\u7684\u4f18\u5316\u56f0\u96be\u3002<\/span><\/section>\n

<\/section>\n
5. \u751f\u6210\u6027\u795e\u7ecf\u7b97\u5b50<\/span><\/strong><\/section>\n

<\/span><\/section>\n
\u4ee5\u4e0a\u63d0\u5230\u7684\u795e\u7ecf\u7b97\u5b50\u5747\u7528\u5728\u8fd1\u4f3c\u51fd\u6570\u7a7a\u95f4\u4e4b\u95f4\u7684\u786e\u5b9a\u6027\u6620\u5c04\u4e0a\uff0c\u4f46\u8fd8\u6709\u4e00\u7c7b\u751f\u6210\u6027\u795e\u7ecf\u7b97\u5b50\uff0c\u5c06\u786e\u5b9a\u6027\u6620\u5c04\u6269\u5c55\u5230\u51fd\u6570\u7a7a\u95f4\u4e0a\u7684\u6982\u7387\u6620\u5c04\u3002\u8fd9\u4e9b\u7b97\u5b50\u7684\u57fa\u7840\u539f\u7406\u662f\u5c06\u751f\u6210\u6027\u5bf9\u6297\u7f51\u7edc<\/span>\uff08generative adversarial networks\uff09<\/span>\u3001\u6269\u6563\u6a21\u578b<\/span>\uff08diffusion models\uff09<\/span>\u6216\u53d8\u5206\u81ea\u7f16\u7801\u5668<\/span>\uff08variational autoencoders\uff09<\/span>\u6269\u5c55\u5230\u51fd\u6570\u7a7a\u95f4\u3002\u503c\u5f97\u5173\u6ce8\u7684\u662f\uff0c\u6269\u6563\u795e\u7ecf\u7b97\u5b50<\/span>\uff08diffusion neural operator\uff09<\/span>\u4ee5\u4e00\u4e2a\u9ad8\u65af\u968f\u673a\u573a\u4f5c\u4e3a\u8f93\u5165\uff0c\u5e76\u80fd\u751f\u6210\u4efb\u4f55\u6307\u5b9a\u5206\u8fa8\u7387\u7684\u6837\u672c\u3002\u751f\u6210\u6027\u795e\u7ecf\u7b97\u5b50\u5df2\u7ecf\u5e94\u7528\u4e8e\u8bb8\u591a\u79d1\u5b66\u95ee\u9898\u4e2d\uff0c\u7528\u4e8e\u586b\u8865\u96be\u4ee5\u83b7\u53d6\u7684\u771f\u5b9e\u4e16\u754c\u73b0\u8c61\u6570\u636e\u3002\u5176\u4e2d\u5305\u62ec\u706b\u5c71\u548c\u5730\u9707\u6d3b\u52a8\u3001\u7eb3\u7ef4-\u65af\u6258\u514b\u65af\u65b9\u7a0b\u7684\u8f93\u5165\u53c2\u6570\u7279\u6027\uff0c\u4ee5\u53ca\u65e0\u5206\u8fa8\u7387\u7684\u89c6\u89c9\u4f20\u611f\u5668\u3002<\/span><\/section>\n

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

\u56fe3 \u795e\u7ecf\u7b97\u5b50\uff08\u4ee5FNO\u548cPINO\uff09\u4e3a\u4f8b\u5728\u8f93\u51fa\u5206\u8fa8\u7387\uff08a\uff09\u4e0a\u7684\u4f18\u52bf\u53ca\u5176\u79bb\u6563\u5316\u6536\u655b\uff08b\uff09\u7684\u6027\u8d28\u3002a,x\u8f74\u662f\u5085\u91cc\u53f6\u6ce2\u6570\uff0cy\u8f74\u662f\u6bcf\u4e2a\u8c31\u7684\u80fd\u91cf\u3002\u4ee5\u8ba1\u7b97\u6d41\u4f53\u8fd0\u52a8\u7684Kolmogorov\u6d41\u52a8\u4e3a\u4f8b\uff0c\u5085\u91cc\u53f6\u795e\u7ecf\u7b97\u5b50\uff08FNO\uff09\u53ef\u4ee5\u4f7f\u7528\u4ec5\u6709\u7684\u6709\u9650\u5206\u8fa8\u7387\u8bad\u7ec3\u6570\u636e\u5916\u63a8\u5230\u672a\u89c1\u8fc7\u7684\u9891\u7387\u3002\u5782\u76f4\u865a\u7ebf\u5de6\u4fa7\u7684\u4f4e\u9891\u662f\u8bad\u7ec3\u9891\u7387\u5206\u8fa8\u7387\uff0c\u6a21\u578b\u5916\u63a8\u5230\u865a\u7ebf\u53f3\u4fa7\u7684\u66f4\u9ad8\u9891\u7387\u3002\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7b97\u5b50\uff08PINO\uff09\u540c\u65f6\u4f7f\u7528\u8bad\u7ec3\u6570\u636e\u548c\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u4f5c\u4e3a\u635f\u5931\u51fd\u6570\uff0c\u5e76\u4e14\u53ef\u4ee5\u5b8c\u7f8e\u6062\u590d\u771f\u5b9e\u9891\u8c31\u3002\u7ecf\u8fc7\u8bad\u7ec3\u7684UNet\uff08\u4e00\u79cd\u6d41\u884c\u7684\u795e\u7ecf\u7f51\u7edc\uff09\u52a0\u4e0a\u4e09\u7ebf\u6027\u63d2\u503c\uff08NN + \u63d2\u503c\uff09\u5728\u8bad\u7ec3\u6570\u636e\u5206\u8fa8\u7387\u4e4b\u5916\u7684\u66f4\u9ad8\u9891\u7387\u4e0a\u6709\u4e25\u91cd\u7684\u5931\u771f\u3002<\/span><\/p>\n

b, \u795e\u7ecf\u7b97\u5b50\u662f\u79bb\u6563\u5316\u6536\u655b\u7684\uff0c\u8fd9\u610f\u5473\u7740\u968f\u7740\u79bb\u6563\u5316\u7684\u7ec6\u5316\uff0c\u6a21\u578b\u4f1a\u6536\u655b\u5230\u76ee\u6807\u8fde\u7eed\u7b97\u5b50\u3002x\u8f74\u662f\u6d4b\u8bd5\u6570\u636e\u7684\u5206\u8fa8\u7387\uff0cy\u8f74\u662f\u8be5\u7ed9\u5b9a\u5206\u8fa8\u7387\u4e0b\u7684\u6d4b\u8bd5\u8bef\u5dee\u3002\u8fd9\u91cc\u4ee5\u8ba1\u7b97\u591a\u5b54\u4ecb\u8d28\u4e2d\u6d41\u4f53\u8fd0\u52a8\u7684Darcy\u65b9\u7a0b\u4e3a\u4f8b\u3002\u6bcf\u79cd\u67b6\u6784\u2014\u2014UNet\u3001FNO\u548c\u56fe\u795e\u7ecf\u7b97\u5b50\uff08GNO\uff09\u2014\u2014\u90fd\u5728\u7ed9\u5b9a\u7684\u5206\u8fa8\u7387\u4e0b\u8fdb\u884c\u8bad\u7ec3\uff0c\u5e76\u5728\u76f8\u540c\u7684\u5206\u8fa8\u7387\u4e0b\u8fdb\u884c\u6d4b\u8bd5\uff08\u6ca1\u6709\u8d85\u5206\u8fa8\u7387\uff09\u3002\u968f\u7740\u5206\u8fa8\u7387\u7684\u63d0\u9ad8\uff0cFNO\u548cGNO\u6709\u4e00\u81f4\u7684\u8bef\u5dee\uff0c\u4f46UNet\u7684\u8bef\u5dee\u4f1a\u589e\u52a0\uff0c\u56e0\u4e3a\u5b83\u7684\u611f\u53d7\u91ce\u5927\u5c0f\u968f\u7740\u5206\u8fa8\u7387\u7684\u53d8\u5316\u800c\u53d8\u5316\u3002\u8fd9\u663e\u793a\u51fa\u795e\u7ecf\u7b97\u5b50\u76f8\u6bd4\u666e\u901a\u795e\u7ecf\u7f51\u7edc\u5177\u6709\u79bb\u6563\u5316\u6536\u655b\u7684\u4f18\u52bf\u3002\uff5c\u56fe\u7247\u6765\u6e90\uff1a\u539f\u6587\u56fe1<\/span><\/p>\n


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<\/svg><\/section>\n<\/section>\n<\/section>\n
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3. \u795e\u7ecf\u7b97\u5b50\u7684\u5e94\u7528<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


<\/p>\n

\u795e\u7ecf\u7b97\u5b50\u5728\u591a\u4e2a\u9886\u57df\u4e2d\u5c55\u73b0\u4e86\u663e\u8457\u7684\u4f18\u52bf\uff0c\u4f8b\u5982\uff1a<\/span><\/section>\n

<\/span><\/section>\n
    \n
  1. \n

    \u5929\u6c14\u9884\u62a5<\/strong>\uff1a\u795e\u7ecf\u7b97\u5b50\u5728\u4e2d\u671f\u5929\u6c14\u9884\u62a5\u65b9\u9762\u6bd4\u76ee\u524d\u7684\u6570\u503c\u5929\u6c14\u6a21\u578b\u5feb\u6570\u4e07\u500d\uff0c\u5e76\u4e14\u9996\u6b21\u5b9e\u73b0\u4e86\u51c6\u786e\u7684\u9ad8\u5206\u8fa8\u7387<\/span>\uff080.25\u5ea6\uff09<\/span>\u5929\u6c14\u9884\u62a5\u3002\u8fd9\u79cd\u901f\u5ea6\u63d0\u5347\u4f7f\u5f97\u5bf9\u6781\u7aef\u5929\u6c14\u4e8b\u4ef6<\/span>\uff08\u5982\u98d3\u98ce\u548c\u70ed\u6d6a\uff09<\/span>\u7684\u98ce\u9669\u8bc4\u4f30\u53d8\u5f97\u66f4\u52a0\u51c6\u786e\uff0c\u56e0\u4e3a\u8fd9\u4e9b\u4e8b\u4ef6\u9700\u8981\u591a\u6b21\u8fd0\u884c\u5929\u6c14\u6a21\u578b\u4ee5\u8fdb\u884c\u4e0d\u786e\u5b9a\u6027\u91cf\u5316[3]\u3002<\/span><\/p>\n<\/li>\n

  2. \n

    \u78b3\u6355\u83b7\u4e0e\u50a8\u5b58<\/strong>\uff1a\u5728\u78b3\u6355\u83b7\u4e0e\u5b58\u50a8<\/span> (Carbon Capture and Storage, CCS) <\/span>\u5e94\u7528\u4e2d\uff0c\u5d4c\u5957\u7684\u5085\u91cc\u53f6\u795e\u7ecf\u7b97\u5b50<\/span> (FNO) <\/span>\u6a21\u578b\u6bd4\u5f53\u4e0b\u6570\u503c\u6a21\u62df\u5668\u5feb\u51e0\u5341\u4e07\u500d\uff0c\u8fd9\u4f7f\u5f97\u5927\u89c4\u6a21\u8bc4\u4f30\u5730\u8d28CO2<\/sub>\u50a8\u5b58\u5e93\u6210\u4e3a\u53ef\u80fd\u3002\u8fd9\u79cd\u52a0\u901f\u4f7f\u5f97\u5bf9\u6700\u5927\u538b\u529b\u7d2f\u79ef\u548c\u4e8c\u6c27\u5316\u78b3\u7fbd\u6d41\u8db3\u8ff9\u7684\u6982\u7387\u8bc4\u4f30\u53ef\u4ee5\u5728\u4ec5 2.8 \u79d2\u5185\u5b8c\u6210\uff0c\u800c\u6a21\u62df\u5668\u5219\u9700\u8981\u8fd1 2 \u5e74\u7684\u65f6\u95f4[4]\u3002<\/span><\/p>\n<\/li>\n

  3. \n

    \u6c42\u89e3\u9006\u95ee\u9898\uff1a<\/strong>\u5728\u9006\u95ee\u9898\u6216\u8ba1\u7b97\u673a\u9006\u5411\u8bbe\u8ba1\u4e2d\uff0c\u901a\u5e38\u9700\u8981\u5728\u7ed9\u5b9a\u524d\u5411\u6a21\u578b\u7684\u60c5\u51b5\u4e0b\uff0c\u5bf9\u7ed9\u5b9a\u7684\u4e00\u7ec4\u53c2\u6570\u8fdb\u884c\u4f18\u5316\uff0c\u4e3a\u8fed\u4ee3\u8bbe\u8ba1\u6539\u8fdb\u63d0\u4f9b\u6307\u5bfc\uff0c\u6216\u8005\u4ece\u5934\u5f00\u59cb\u751f\u6210\u4f18\u5316\u8bbe\u8ba1[5]\u3002\u800c\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\u901a\u8fc7\u4f7f\u7528\u795e\u7ecf\u7b97\u5b50\u6a21\u578b\uff0c\u53ef\u4ee5\u4e3a\u4e00\u79cd\u65b0\u7684\u533b\u7597\u5bfc\u7ba1\u8bbe\u8ba1\u51fa\u4f18\u5316\u65b9\u6848\uff0c\u5c06\u7ec6\u83cc\u6c61\u67d3\u51cf\u5c11\u4e24\u4e2a\u6570\u91cf\u7ea7\u3002\u795e\u7ecf\u7b97\u5b50\u6a21\u578b\u80fd\u591f\u51c6\u786e\u6a21\u62df\u5728\u4efb\u610f\u5f62\u72b6\u7ba1\u9053\u6d41\u52a8\u7684\u6d41\u4f53\u4e2d\u7684\u7ec6\u83cc\u5bc6\u5ea6\uff0c\u56e0\u800c\u53ef\u4ee5\u7528\u4e8e\u4f18\u5316\u5bfc\u7ba1\u5185\u7684\u8bbe\u8ba1\u5f62\u72b6\uff0c\u4ee5\u9632\u6b62\u7ec6\u83cc\u5411\u4e0a\u6e38\u8fdb\u5165\u4eba\u4f53[6]\u3002<\/span><\/p>\n<\/li>\n

  4. \n

    \u52a8\u529b\u7cfb\u7edf\u957f\u671f\u7edf\u8ba1\u7279\u6027\u5b66\u4e60<\/strong>\uff1a\u795e\u7ecf\u7b97\u5b50\u5df2\u88ab\u7528\u4e8e\u5b66\u4e60\u6df7\u6c8c\u7cfb\u7edf\u7684\u5438\u5f15\u5b50\u548c\u68c0\u6d4b\u975e\u5e73\u7a33\u7cfb\u7edf\u4e2d\u7684\u4e34\u754c\u70b9\u3002\u57fa\u4e8e\u51fd\u6570\u7a7a\u95f4\u4e2d\u7684\u751f\u6210\u6027\u5bf9\u6297\u7f51\u7edc\u6216\u6269\u6563\u6a21\u578b\u7684\u795e\u7ecf\u7b97\u5b50\uff0c\u53ef\u7528\u4e8e\u6a21\u62df\u968f\u673a\u81ea\u7136\u73b0\u8c61<\/span>\uff08\u5982\u706b\u5c71\u548c\u6c14\u5019\u6d3b\u52a8\uff09<\/span>\u548c\u968f\u673a\u5fae\u5206\u65b9\u7a0b\u3002<\/span><\/p>\n<\/li>\n

  5. \n
    \u5176\u4ed6\u5e94\u7528<\/strong>\uff1a\u795e\u7ecf\u7b97\u5b50\u8fd8\u88ab\u7528\u4e8e\u6d41\u4f53\u52a8\u529b\u5b66\u30013D \u5de5\u4e1a\u7ea7\u6c7d\u8f66\u7a7a\u6c14\u52a8\u529b\u5b66\u3001\u57ce\u5e02\u5fae\u6c14\u5019\u5efa\u6a21\u3001\u6750\u6599\u5f62\u53d8\u3001\u8ba1\u7b97\u5149\u523b\u3001\u5149\u58f0\u6210\u50cf\u548c\u7535\u78c1\u573a\u6a21\u62df\u7b49\u9886\u57df\u3002<\/span><\/section>\n<\/li>\n<\/ol>\n

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

    <\/section>\n

    \n
    \n
    \n
    \n
    \n

    <\/section>\n

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

    4. \u603b\u7ed3\u4e0e\u5c55\u671b<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

    \n

    <\/section>\n

    <\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/h3>\n


    <\/p>\n

    \u795e\u7ecf\u7b97\u5b50\u65e8\u5728\u4ee5\u7f51\u683c\u6536\u655b\u7684\u65b9\u5f0f\u76f4\u63a5\u5b66\u4e60\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u89e3\u7b97\u5b50\u3002\u76f8\u6bd4\u4e8e\u6807\u51c6\u795e\u7ecf\u7f51\u7edc\uff0c\u795e\u7ecf\u7b97\u5b50\u66f4\u9002\u5408\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u3002\u795e\u7ecf\u7b97\u5b50\u80fd\u591f\u76f4\u63a5\u4ece\u89c2\u6d4b\u6570\u636e\u4e2d\u5b66\u4e60\uff0c\u4e0d\u53d7\u6570\u503c\u65b9\u6cd5\u4e2d\u7684\u5efa\u6a21\u8bef\u5dee\u5f71\u54cd\u3002\u5f53\u5b83\u88ab\u7528\u4e8e\u9006\u95ee\u9898\uff0c\u5982\u4f18\u5316\u8bbe\u8ba1\u751f\u6210\u65f6\uff0c\u5b83\u8868\u73b0\u51fa\u7c7b\u4f3c\u4e00\u822c\u79d1\u7814\u5de5\u4f5c\u8005\u7684\u521b\u65b0\u80fd\u529b\u3002\u795e\u7ecf\u7b97\u5b50\u7684\u5e94\u7528\u4e5f\u80fd\u7b80\u5316\uff0c\u964d\u4f4e\u73b0\u4ee3\u79d1\u7814\u5de5\u4e1a\u7684\u5de8\u5927\u8de8\u9886\u57df\u58c1\u5792\u548c\u5b66\u4e60\u6210\u672c\uff0c\u56e0\u4e3a\u8fd0\u884c\u8bad\u7ec3\u6709\u7d20\u7684\u795e\u7ecf\u7b97\u5b50\u4e0d\u9700\u8981\u4f20\u7edf\u6c42\u89e3\u5668\u6240\u9700\u7684\u6df1\u539a\u9886\u57df\u4e13\u4e1a\u77e5\u8bc6\u3002\u5f53\u7136\uff0c\u4f5c\u4e3a\u4e00\u79cd\u73b0\u4ee3\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\uff0c\u795e\u7ecf\u7b97\u5b50\u4e5f\u5b58\u5728\u8fc7\u62df\u5408 <\/span>(overfitting) <\/span>\u95ee\u9898\u3002\u5728\u6700\u5c11\u6570\u636e\u8bbe\u7f6e\u60c5\u51b5\u4e0b\uff0c\u5b83\u4eec\u53ef\u80fd\u4f1a\u8fc7\u62df\u5408\u8bad\u7ec3\u6570\u636e\uff1b\u5728\u975e\u5e38\u4f4e\u5206\u8fa8\u7387\u7684\u573a\u666f\u4e2d\uff0c\u5b83\u4eec\u53ef\u80fd\u4f1a\u8fc7\u62df\u5408\u5206\u8fa8\u7387\u7684\u4e0d\u8db3\u3002\u5728\u8fd9\u4e9b\u60c5\u51b5\u4e0b\uff0c\u5b83\u4eec\u53ef\u80fd\u7f3a\u4e4f\u5bf9\u8d85\u51fa\u5e73\u5747\u7684\u5206\u5e03\u5916\u6570\u636e\u5904\u7406\u7684\u6cdb\u5316\u80fd\u529b\u3002\u8fd9\u4e9b\u95ee\u9898\u53ef\u4ee5\u901a\u8fc7\u9886\u57df\u77e5\u8bc6\u7684\u96c6\u6210\u8bbe\u8ba1\u3001\u66f4\u591a\u6570\u636e\u3001\u66f4\u9ad8\u4fdd\u771f\u5ea6\u7684\u6570\u636e\u548c\u7269\u7406\u65b9\u7a0b\u6765\u89e3\u51b3\u3002<\/span><\/section>\n

    <\/span><\/section>\n
    \u795e\u7ecf\u7b97\u5b50\u7684\u8fd9\u4e9b\u57fa\u7840\u6027\u8d28\u544a\u8bc9\u6211\u4eec\u4e00\u4e2a\u597d\u6d88\u606f\u2014\u2014\u90a3\u5c31\u662f\u77ed\u671f\u5185AI\u8fd8\u65e0\u6cd5\u62a2\u8d70\u5927\u90e8\u5206\u79d1\u5b66\u5bb6\u7684\u5c97\u4f4d\uff0c\u56e0\u4e3a\u4e00\u5207\u7ed3\u679c\u7684\u53ef\u9760\u6027\u4ecd\u6709\u8d56\u4e8e\u4eba\u7684\u5224\u65ad\u3002\u4f46\u672c\u6587\u4f5c\u8005\u76f8\u4fe1\uff0c\u795e\u7ecf\u7b97\u5b50\u4e3a\u6570\u636e\u6a21\u62df\u548c\u6d41\u7a0b\u8bbe\u8ba1\u63d0\u4f9b\u4e86\u4e00\u79cd\u53d8\u9769\u6027\u7684\u65b9\u6cd5\uff0c\u4f7f\u7814\u7a76\u548c\u5f00\u53d1\u80fd\u591f\u5feb\u901f\u8fdb\u884c\u3002AI\u6216\u8bb8\u8fd8\u672a\u80fd\u4e3b\u5bfc\u79d1\u7814\uff0c\u4f46\u6216\u8bb8\u8fd9\u662f\u6211\u4eec\u7f3a\u4e4f\u201c\u79d1\u5b66\u65b9\u6cd5\u6570\u636e\u96c6\u201d\u9020\u6210\u7684\u3002\u795e\u7ecf\u7b97\u5b50\u4f53\u73b0\u4e86\u83b7\u53d6\u8fde\u7eed\u51fd\u6570\u7684\u5f3a\u5927\u8ba1\u7b97\u80fd\u529b\uff0c\u4e5f\u8bb8\u5728\u4e0d\u8fdc\u7684\u672a\u6765\uff0c\u6211\u4eec\u5c31\u80fd\u4f53\u4f1a\u5230\u5b83\u5728\u52a0\u901f\u4eba\u7c7b\u8ba4\u77e5\u7269\u8d28\u4e16\u754c\u7684\u8fdb\u7a0b\u3002<\/span><\/section>\n

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

    <\/strong><\/span><\/section>\n
    \u53c2\u8003\u6587\u732e<\/span><\/strong><\/span><\/section>\n
    [1]Z. Li et al.<\/em>, \u201cFourier Neural Operator for Parametric Partial Differential Equations,\u201d arXiv:2010.08895 [cs, math]<\/em>, Oct. 2020, Available: <\/span>arXiv:2010.08895v3<\/span><\/section>\n
    [2]Li, Zongyi, et al. \u201cFourier Neural Operator with Learned Deformations for PDEs on General Geometries.\u201d ArXiv.org, 11 July 2022, arxiv.org\/abs\/2207.05209.<\/span><\/section>\n
    [3]T. Kurth, \u201cFourCastNet: Accelerating Global High-Resolution Weather Forecasting Using Adaptive Fourier Neural Operators\u201d, 2023, pp. 1\u201311. doi: 10.1145\/3592979.3593412.<\/span><\/section>\n
    [4]G. Wen, Z. Li, Q. Long, Kamyar Azizzadenesheli, Anima Anandkumar, and S. Benson, \u201cReal-time high-resolution CO(2) geological storage prediction using nested Fourier neural operators,\u201d Energy & Environmental Science<\/em>, vol. 16, no. 4, pp. 1732\u20131741, Jan. 2023, doi: <\/span>10.1039\/d2ee04204e.<\/span><\/section>\n
    [5]H. Sun, Y. Yang, K. Azizzadenesheli, R. W. Clayton, and Z. E. Ross, \u201cAccelerating Time-Reversal Imaging with Neural Operators for Real-time Earthquake Locations,\u201d arXiv:2210.06636 [physics]<\/em>, Oct. 2022, Available: <\/span>arXiv:2210.06636v1<\/span><\/section>\n
    [6]T. Zhou et al.<\/em>, \u201cAI-aided geometric design of anti-infection catheters,\u201d Science Advances<\/em>, vol. 10, no. 1, Jan. 2024, doi: <\/span>10.1126\/sciadv.adj1741.<\/span><\/section>\n
    \n
    \n


    <\/p>\n


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

    \n
    \n
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    \n

    AI+Science \u8bfb\u4e66\u4f1a<\/strong><\/p>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

    \n


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

    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

    <\/section>\n
    \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\u5df2\u5b8c\u7ed3\uff0c\u73b0\u5728\u62a5\u540d\u53ef\u52a0\u5165\u793e\u7fa4\u5e76\u89e3\u9501\u56de\u653e\u89c6\u9891\u6743\u9650\u3002<\/span><\/span><\/section>\n

    <\/section>\n

    <\/span><\/a><\/p>\n


    <\/p>\n

    \u8be6\u60c5\u8bf7\u89c1\uff1a<\/span>
    <\/section>\n
    \u4eba\u5de5\u667a\u80fd\u548c\u79d1\u5b66\u53d1\u73b0\u76f8\u4e92\u8d4b\u80fd\u7684\u65b0\u8303\u5f0f\uff1aAI+Science \u8bfb\u4e66\u4f1a\u542f\u52a8<\/a><\/span><\/section>\n
    <\/strong><\/a><\/section>\n


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

    <\/strong><\/section>\n
    \u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/strong><\/strong><\/section>\n
    <\/strong><\/section>\n
    1. <\/strong>\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<\/a><\/strong><\/span><\/strong><\/strong><\/section>\n
    2. <\/strong>\u524d\u6cbf\u8fdb\u5c55\uff1aKoopman\u795e\u7ecf\u7b97\u5b50\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b<\/a><\/strong><\/span><\/section>\n
    3. <\/strong>\u5982\u4f55\u53d1\u73b0 AI+Science \u4e2d\u7684\u4e0b\u4e00\u4e2a AlphaFold \u548c ChatGPT\uff1f<\/a><\/strong><\/span><\/section>\n
    <\/strong><\/strong><\/span><\/section>\n
    <\/strong><\/strong><\/section>\n
    4. <\/strong><\/span>\u5f20\u6c5f\uff1a\u7b2c\u4e09\u4ee3\u4eba\u5de5\u667a\u80fd\u6280\u672f\u57fa\u7840\u2014\u2014\u4ece\u53ef\u5fae\u5206\u7f16\u7a0b\u5230\u56e0\u679c\u63a8\u7406 | \u96c6\u667a\u5b66\u56ed\u5168\u65b0\u8bfe\u7a0b<\/a><\/strong><\/strong><\/span><\/section>\n
    <\/strong>5.<\/strong> \u9f99\u5e74\u5927\u8fd0\u8d77\uff0c\u5b66\u4e60\u6b63\u5f53\u65f6\uff01\u89e3\u9501\u96c6\u667a\u5168\u7ad9\u5185\u5bb9\uff0c\u5f00\u542f\u65b0\u5e74\u5b66\u4e60\u8ba1\u5212<\/a><\/strong><\/strong><\/strong><\/strong><\/span><\/strong><\/strong><\/section>\n

    6. <\/strong>\u52a0\u5165\u96c6\u667a\uff0c\u4e00\u8d77\u590d\u6742\uff01<\/strong><\/span><\/a><\/p>\n


    <\/p>\n


    <\/p>\n

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