{"id":52509,"date":"2024-09-04T21:41:12","date_gmt":"2024-09-04T13:41:12","guid":{"rendered":"https:\/\/swarma.org\/?p=52509"},"modified":"2024-09-04T21:41:12","modified_gmt":"2024-09-04T13:41:12","slug":"%e5%89%8d%e6%b2%bf%e8%bf%9b%e5%b1%95%ef%bc%9akoopman%e7%a5%9e%e7%bb%8f%e7%ae%97%e5%ad%90%e6%b1%82%e8%a7%a3%e5%81%8f%e5%be%ae%e5%88%86%e6%96%b9%e7%a8%8b","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=52509","title":{"rendered":"\u524d\u6cbf\u8fdb\u5c55\uff1aKoopman\u795e\u7ecf\u7b97\u5b50\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b"},"content":{"rendered":"
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\u8bba\u6587\u9898\u76ee\uff1a<\/span><\/section>\n
Koopman neural operator as a mesh-free solver of non-linear partial differential equations<\/span><\/section>\n
\u8bba\u6587\u94fe\u63a5\uff1a<\/span><\/section>\n
https:\/\/doi.org\/10.1016\/j.jcp.2024.113194<\/span><\/section>\n
KoopmanLab\u7684\u8bba\u6587\uff1a<\/span><\/section>\n
https:\/\/doi.org\/10.1063\/5.0157763<\/span><\/section>\n
KoopmanLab\u7684\u4ee3\u7801\uff1a<\/span><\/section>\n
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1. \u80cc\u666f<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u7b97\u5b50\u5b66\u4e60\u662f\u7814\u7a76\u6df1\u5ea6\u5b66\u4e60\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u91cd\u8981\u65b9\u6cd5\uff0c\u5176\u6838\u5fc3\u662f\u4f7f\u7528\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\u6765\u5b66\u4e60\u7528\u4e8e\u63cf\u8ff0\u65e0\u9650\u7ef4Banach\u7a7a\u95f4\u7684\u7b97\u5b50\uff0c\u800c\u504f\u5fae\u5206\u65b9\u7a0b\u6c42\u89e3\u53ef\u4ee5\u5728\u6570\u5b66\u4e0a\u8f6c\u5316\u4e3a\u7b97\u5b50\u5b66\u4e60\u95ee\u9898\u3002\u4e0d\u540c\u4e8e\u6df1\u5ea6\u5b66\u4e60\u7684\u5176\u4ed6\u95ee\u9898\uff0c\u7b97\u5b50\u5b66\u4e60\u6a21\u578b\u9700\u8981\u5177\u5907\u5728Banach\u7a7a\u95f4\u4e2d\u6cdb\u5316\u7684\u80fd\u529b\uff0c\u5373\u8bad\u7ec3\u597d\u7684\u6a21\u578b\u9700\u8981\u80fd\u6cdb\u5316\u4e0d\u540c\u7684\u521d\u59cb\u6761\u4ef6\u3001\u7269\u7406\u53c2\u6570\u6216\u5206\u8fa8\u7387\u7f51\u683c\u7b49\u3002\u7136\u800c\uff0c\u5982\u4f55\u4f7f\u7528\u6709\u9650\u7684\u6570\u636e\u548c\u6709\u9650\u53c2\u6570\u7684\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\u53bb\u5b66\u4e60\u63cf\u8ff0\u65e0\u9650\u7ef4\u6620\u5c04\u7684\u7b97\u5b50<\/strong>\uff0c\u662f\u8fd9\u4e00\u9886\u57df\u9700\u8981\u89e3\u51b3\u7684\u91cd\u8981\u95ee\u9898\u3002\u5bf9\u4e8e\u8fd9\u4e00\u4efb\u52a1\uff0c\u524d\u4eba\u7684\u7814\u7a76\u8fdb\u884c\u4e86\u8bf8\u591a\u63a2\u7d22\uff0c\u6bd4\u5982DeepONet\u6a21\u578b\u57fa\u4e8e\u901a\u7528\u8fd1\u4f3c\u5b9a\u7406<\/span>\uff08Universal approximation <\/span>theorem\uff09<\/span>\u63a2\u8ba8\u4e86\u5982\u4f55\u5bf9\u51fd\u6570\u7a7a\u95f4\u91c7\u6837\u6765\u4f7f\u6a21\u578b\u6536\u655b\u4e8e\u76ee\u6807\u7b97\u5b50\uff0c\u5085\u91cc\u53f6\u795e\u7ecf\u7b97\u5b50<\/span>\uff08Fourier Neural Operator\uff09<\/span>\u7b49\u6a21\u578b\u63a2\u7a76\u4e86\u5982\u4f55\u4f7f\u7528\u795e\u7ecf\u7f51\u7edc\u5c06\u683c\u6797\u51fd\u6570\u6cd5\u6269\u5c55\u5230\u6c42\u89e3\u975e\u7ebf\u6027\u504f\u5fae\u5206\u65b9\u7a0b\u3002<\/span>\u672c\u6587\u5c06\u4ece\u52a8\u529b\u7cfb\u7edf\u3001Koopman\u7406\u8bba\u548c\u7b97\u5b50\u5b66\u4e60\u89c6\u89d2\u63d0\u51fa\u4e00\u79cd\u65b0\u7684\u504f\u5fae\u5206\u65b9\u7a0b\u6c42\u89e3\u6846\u67b6\uff0c\u4f7f\u7528Koopman\u795e\u7ecf\u7b97\u5b50\u5b66\u4e60\u975e\u7ebf\u6027\u7684\u89c2\u6d4b\u51fd\u6570\u3001Koopman\u4e0d\u53d8\u5b50\u7a7a\u95f4\u548cKoopman\u7b97\u5b50\u7684\u7ebf\u6027\u8868\u5f81\u77e9\u9635\uff0c\u7528\u6765\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u548c\u9884\u6d4b\u771f\u5b9e\u4e16\u754c\u7684\u52a8\u529b\u6f14\u5316\u3002<\/span><\/section>\n

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2. Koopman \u7406\u8bba\u548c\u52a8\u529b\u7cfb\u7edf\u89c6\u89d2\u4e0b\u7684<\/strong><\/span><\/strong><\/strong><\/p>\n

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\u4e3a\u4e86\u68c0\u9a8c\u7406\u8bba\u7684\u6b63\u786e\u6027\uff0c\u6211\u4eec\u6d4b\u8bd5\u4e86Navier-Stokes\u65b9\u7a0b<\/span>\uff08\u4e09\u4e2a\u4e0d\u540c\u7684\u7c98\u6027\u7cfb\u6570\uff09<\/span>\u3001\u6d45\u6c34\u65b9\u7a0b\u3001\u745e\u5229-\u8d1d\u7eb3\u5fb7\u5bf9\u6d41\u3001\u6d77\u6d0b\u897f\u8fb9\u754c\u6d41<\/span>\uff08\u9ed1\u6f6e\u548c\u6e7e\u6d41\uff09<\/span>\u548c\u6c34\u6c7d\u9065\u611f\u7b49\u573a\u666f\u4e0b\u7684\u6a21\u578b\u8868\u73b0\uff0c\u5176\u6548\u679c\u5982\u4e0b\u56fe\u6240\u793a\u3002<\/span><\/section>\n

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4. Koopman\u795e\u7ecf\u7b97\u5b50\u7684\u6027\u8d28<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u5206\u8fa8\u7387\u4e0d\u53d8\u6027<\/span><\/strong>\uff1aKoopman\u795e\u7ecf\u7b97\u5b50\u5177\u6709\u5206\u8fa8\u7387\u4e0d\u53d8\u6027\uff0c\u8fd9\u4e00\u6027\u8d28\u4e0d\u4ec5\u9700\u8981\u6a21\u578b\u5728\u63a8\u7406\u65f6\u80fd\u63a5\u53d7\u4e0d\u540c\u7684\u7f51\u683c\u5206\u8fa8\u7387\uff0c\u8fd8\u8981\u5728\u63a8\u7406\u65f6\u7684\u8bef\u5dee\u8868\u73b0\u4e0d\u968f\u7740\u7f51\u683c\u5206\u8fa8\u7387\u7684\u6539\u53d8\u800c\u6539\u53d8\u3002\u5982\u4e0b\u56fe\u6240\u793a\uff0cKoopman\u7b97\u5b50\u5728\u5efa\u6a21Burger’s\u65b9\u7a0b\u65f6\uff0c\u4f7f\u7528\u7f51\u683c\u4e0b\u7684\u6570\u503c\u89e3\u9a71\u52a8\u8bad\u7ec3\uff0c\u5728\u66f4\u9ad8\u5206\u8fa8\u7387\u7f51\u683c\u4e0b\u63a8\u7406\u65f6\u7684\u8bef\u5dee\u8868\u73b0\u3002\u672c\u6587\u8fd8\u63a2\u7a76\u4e86\u4f7f\u7528\u590d\u6742\u5377\u79ef\u7ed3\u6784\u5b9e\u73b0\u9ad8\u9891\u6355\u6349\u6a21\u5757\u5bf9\u5206\u8fa8\u7387\u4e0d\u53d8\u6027\u7834\u574f\u7684\u5b9e\u9a8c\u3002<\/span><\/section>\n

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Zero-shot\u9884\u6d4b<\/span><\/strong>\uff1aKoopman\u795e\u7ecf\u7b97\u5b50\u5177\u6709\u65f6\u95f4\u5916\u63a8\u548c\u5185\u63d2\u7684\u80fd\u529b\u3002\u56fex\u5c55\u793a\u4e86Koopman\u795e\u7ecf\u7b97\u5b50\u5728\u5efa\u6a21\u745e\u5229-\u8d1d\u7eb3\u5fb7\u5bf9\u6d41\u65f6\u7684zero-shot\u5b9e\u9a8c\u7684\u6548\u679c\uff0c\u5176\u4e2d(a-b)\u5c55\u793a\u4e86\u65f6\u95f4\u5185\u63d2\u7684\u5b9e\u9a8c\u7ed3\u679c\uff0c\u6a21\u578b\u5728\u8bad\u7ec3\u65f6\u5206\u522b\u4f7f\u7528\u76f8\u96942\u4e2a\u62164\u4e2a\u65f6\u95f4\u6b65\u7684\u7269\u7406\u573a\u76d1\u7763\u8bad\u7ec3\uff0c\u5bf9\u672a\u676540\u4e2a\u65f6\u95f4\u6b65\u7684\u63a8\u7406\uff0c\u4ee5\u6b64\u68c0\u9a8c\u65f6\u95f4\u6b65\u5185\u63d2\u7684\u6548\u679c\u3002(<\/span>c-d)<\/span>\u5c55\u793a\u4e86\u65f6\u95f4\u5916\u63a8\u7684\u5b9e\u9a8c\u7ed3\u679c\uff0c\u5206\u522b\u4f7f\u7528\u524d20\u4e2a\u548c10\u4e2a\u65f6\u95f4\u6b65\u8fdb\u884c\u76d1\u7763\uff0c\u5bf9\u672a\u676540\u4e2a\u65f6\u95f4\u6b65\u7684\u63a8\u7406\uff0c\u4ee5\u6b64\u68c0\u9a8c\u65f6\u95f4\u6b65\u5916\u63a8\u7684\u6548\u679c\u3002<\/span><\/section>\n

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5. \u603b\u7ed3<\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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\u7efc\u4e0a\uff0c\u6211\u4eec\u63d0\u51fa\u4e00\u79cd\u7528\u4e8e\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\u7684\u795e\u7ecf\u7b97\u5b50\u6a21\u578b\uff0c\u5177\u6709\u5efa\u6a21\u504f\u5fae\u5206\u65b9\u7a0b\u6216\u771f\u5b9e\u52a8\u529b\u7cfb\u7edf\u957f\u7a0b\u65f6\u95f4\u6f14\u5316\u7684\u80fd\u529b\uff0c\u540c\u65f6\u5177\u6709\u5206\u8fa8\u7387\u4e0d\u53d8\u6027\u7b49\u6027\u8d28\u3002\u8fd9\u4e00\u6027\u8d28\u4f7f\u5176\u5728\u6709\u6f5c\u529b\u5728\u4f4e\u5206\u8fa8\u7387\u6570\u636e\u4e0a\u8fdb\u884c\u8bad\u7ec3\uff0c\u5e76\u7528\u4e8e\u9ad8\u5206\u8fa8\u7387\u63a8\u7406\u3002\u8fd9\u4e9b\u7279\u6027\u5bf9\u672a\u6765\u7684\u591a\u5206\u8fa8\u7387\u8bad\u7ec3\u3001\u591a\u5c3a\u5ea6\u5efa\u6a21\u548c\u65e0\u7f51\u683c\u65b9\u6cd5\u7b49\u7814\u7a76\u65b9\u5411\u5c55\u73b0\u4e86\u66f4\u591a\u53ef\u6f5c\u5728\u7684\u80fd\u6027\u3002<\/span><\/section>\n

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\u9664\u6b64\u4e4b\u5916\uff0c\u6211\u4eec\u5f00\u53d1\u4e86\u540d\u4e3aKoopmanLab\u7684\u8ba1\u7b97\u5e93\uff0c\u5c06Koopman\u7b97\u5b50\u548c\u5176\u884d\u751f\u6a21\u578b\u5728GitHub\u5f00\u6e90\u3002<\/span><\/section>\n

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\u76f8\u5173\u6587\u732e<\/span><\/strong><\/span><\/section>\n
[1] Xiong W, Huang X, Zhang Z, Deng R, Sun P, Tian Y. Koopman neural operator as a mesh-free solver of non-linear partial differential equations. Journal of Computational Physics. 2024 Jun 13:113194.<\/span><\/section>\n
[2] Xiong W, Ma M, Huang X, Zhang Z, Sun P, Tian Y. Koopmanlab: machine learning for solving complex physics equations. APL Machine Learning. 2023 Sep 1;1(3).<\/span><\/section>\n
[3] Lu L, Jin P, Pang G, Zhang Z, Karniadakis GE. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence. 2021 Mar;3(3):218-29.<\/span><\/section>\n
[4] Kovachki N, Li Z, Liu B, Azizzadenesheli K, Bhattacharya K, Stuart A, Anandkumar A. Neural operator: Learning maps between function spaces with applications to PDEs. Journal of Machine Learning Research. 2023;24(89):1-97.<\/span><\/section>\n
[5] Koopman BO. Hamiltonian systems and transformation in Hilbert space. Proceedings of the National Academy of Sciences. 1931 May;17(5):315-8.<\/span><\/section>\n
 <\/span><\/section>\n

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AI By Complexity\u8bfb\u4e66\u4f1a<\/strong>
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\u5927\u6a21\u578b\u3001\u591a\u6a21\u6001\u3001\u591a\u667a\u80fd\u4f53\u5c42\u51fa\u4e0d\u7a77\uff0c\u5404\u79cd\u5404\u6837\u7684\u795e\u7ecf\u7f51\u7edc\u53d8\u4f53\u5728AI\u5927\u821e\u53f0\u5404\u663e\u8eab\u624b\u3002\u590d\u6742\u7cfb\u7edf\u9886\u57df\u5bf9\u4e8e\u6d8c\u73b0\u3001\u5c42\u7ea7\u3001\u9c81\u68d2\u6027\u3001\u975e\u7ebf\u6027\u3001\u6f14\u5316\u7b49\u95ee\u9898\u7684\u63a2\u7d22\u4e5f\u5728\u6301\u7eed\u63a8\u8fdb\u3002\u800c\u4f18\u79c0\u7684AI\u7cfb\u7edf\u3001\u521b\u65b0\u6027\u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u5f80\u5f80\u5728\u4e00\u5b9a\u7a0b\u5ea6\u4e0a\u5177\u5907\u4f18\u79c0\u590d\u6742\u7cfb\u7edf\u7684\u7279\u5f81\u3002\u56e0\u6b64\uff0c\u53d1\u5c55\u4e2d\u7684\u590d\u6742\u7cfb\u7edf\u7406\u8bba\u65b9\u6cd5\u5982\u4f55\u6307\u5bfc\u672a\u6765AI\u7684\u8bbe\u8ba1\uff0c\u6b63\u5728\u6210\u4e3a\u5907\u53d7\u5173\u6ce8\u7684\u95ee\u9898\u3002<\/span><\/p>\n


<\/p>\n

\u96c6\u667a\u4ff1\u4e50\u90e8\u8054\u5408\u52a0\u5229\u798f\u5c3c\u4e9a\u5927\u5b66\u5723\u8fed\u6208\u5206\u6821\u52a9\u7406\u6559\u6388\u5c24\u4ea6\u5e84\u3001\u5317\u4eac\u5e08\u8303\u5927\u5b66\u526f\u6559\u6388\u5218\u5b87\u3001\u5317\u4eac\u5e08\u8303\u5927\u5b66\u7cfb\u7edf\u79d1\u5b66\u5b66\u9662\u5728\u8bfb\u535a\u58eb\u5f20\u7ae0\u3001\u725f\u7267\u4e91\u548c\u5728\u8bfb\u7855\u58eb\u6768\u660e\u54f2\u3001\u6e05\u534e\u5927\u5b66\u5728\u8bfb\u535a\u58eb\u7530\u6d0b\u5171\u540c\u53d1\u8d77\u300cAI By Complexity\u300d\u8bfb\u4e66\u4f1a<\/a>\uff0c\u63a2\u7a76\u5982\u4f55\u5ea6\u91cf\u590d\u6742\u7cfb\u7edf\u7684\u201c\u597d\u574f\u201d\uff1f\u5982\u4f55\u7406\u89e3\u590d\u6742\u7cfb\u7edf\u7684\u673a\u5236\uff1f\u8fd9\u4e9b\u7406\u89e3\u662f\u5426\u53ef\u4ee5\u542f\u53d1\u6211\u4eec\u8bbe\u8ba1\u66f4\u597d\u7684AI\u6a21\u578b\uff1f\u5728\u672c\u8d28\u4e0a\u5e2e\u52a9\u6211\u4eec\u8bbe\u8ba1\u66f4\u597d\u7684AI\u7cfb\u7edf\u3002\u8bfb\u4e66\u4f1a\u4e8e6\u670810\u65e5\u5f00\u59cb\uff0c\u6bcf\u5468\u4e00\u665a\u4e0a20:00-22:00\u4e3e\u529e\u3002\u6b22\u8fce\u4ece\u4e8b\u76f8\u5173\u9886\u57df\u7814\u7a76\u3001\u5bf9AI+Complexity\u611f\u5174\u8da3\u7684\u670b\u53cb\u4eec\u62a5\u540d\u8bfb\u4e66\u4f1a\u4ea4\u6d41\uff01<\/span><\/p>\n


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\u8be6\u60c5\u8bf7\u89c1\uff1a<\/span><\/section>\n
AI by Complexity \u8bfb\u4e66\u4f1a\u542f\u52a8\uff1a\u590d\u6742\u6027\u600e\u6837\u91cf\u5316\u548c\u9a71\u52a8\u4e0b\u4e00\u4ee3AI\u7cfb\u7edf<\/span><\/a>
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\u63a8\u8350\u9605\u8bfb<\/span><\/strong><\/strong><\/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>Nat. Rev. Phys.\u901f\u9012\uff1a\u795e\u7ecf\u7b97\u5b50\u52a0\u901f\u79d1\u5b66\u4eff\u771f\u4e0e\u8bbe\u8ba1<\/a> <\/strong><\/span><\/strong><\/section>\n
3. <\/strong>\u6df1\u5ea6\u795e\u7ecf\u7f51\u7edc\u7684\u7edf\u8ba1\u7269\u7406\uff1a\u5bfb\u627e\u6700\u4f18\u521d\u59cb\u5316\u7406\u8bba<\/a><\/strong><\/span><\/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


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

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