{"id":5247,"date":"2024-12-04T12:05:19","date_gmt":"2024-12-04T04:05:19","guid":{"rendered":"https:\/\/nullthought.net\/?p=5247"},"modified":"2024-12-04T12:05:21","modified_gmt":"2024-12-04T04:05:21","slug":"%e9%80%9a%e8%bf%87%e5%a4%9a%e7%9b%ae%e6%a0%87%e4%bc%98%e5%8c%96%ef%bc%88multi-objective-optimization%ef%bc%89%e8%87%aa%e5%8a%a8%e5%8f%91%e7%8e%b0%e6%9c%80%e4%bd%b3%e5%85%83%e8%a7%a3%e7%ae%97%e5%99%a8","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=5247","title":{"rendered":"\u901a\u8fc7\u591a\u76ee\u6807\u4f18\u5316\uff08multi-objective optimization\uff09\u81ea\u52a8\u53d1\u73b0\u6700\u4f73\u5143\u89e3\u7b97\u5668\uff08meta-solvers\uff09"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2412.00063\" target=\"_blank\" rel=\"noreferrer noopener\">Automatic discovery of optimal meta-solvers via multi-objective optimization<\/a><\/strong>\u300a\u901a\u8fc7\u591a\u76ee\u6807\u4f18\u5316\u81ea\u52a8\u53d1\u73b0\u6700\u4f73\u5143\u89e3\u7b97\u5668\u300b\u4e3b\u8981\u8ba8\u8bba\u4e86\u4e00\u79cd\u65b0\u7684\u65b9\u6cd5\uff0c\u901a\u8fc7\u7ed3\u5408<strong><a href=\"https:\/\/nullthought.net\/?s=%E7%A5%9E%E7%BB%8F%E7%AE%97%E5%AD%90\" target=\"_blank\" rel=\"noreferrer noopener\">\u795e\u7ecf\u7b97\u5b50<\/a><\/strong>\uff08Neural Operators\uff09\u548c\u7ecf\u5178\u7684\u8fed\u4ee3\u6c42\u89e3\u65b9\u6cd5\uff0c\u81ea\u52a8\u53d1\u73b0\u6c42\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDE\uff09\u79bb\u6563\u5316\u6240\u4ea7\u751f\u7ebf\u6027\u7cfb\u7edf\u7684\u6700\u4f18\u5143\u89e3\u7b97\u5668\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aYoungkyu Lee, Shanqing Liu, Jerome Darbon, George Em Karniadakis\uff0c\u6765\u81eaBrown University\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">1. \u80cc\u666f\u548c\u52a8\u673a<\/h4>\n\n\n\n<p>\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDEs\uff09\u5e7f\u6cdb\u7528\u4e8e\u63cf\u8ff0\u81ea\u7136\u79d1\u5b66\u548c\u5de5\u7a0b\u5b66\u79d1\u4e2d\u7684\u5404\u79cd\u7269\u7406\u8fc7\u7a0b\uff0c\u4f8b\u5982\u70ed\u4f20\u5bfc\u3001\u6d41\u4f53\u6d41\u52a8\u3001\u7535\u78c1\u573a\u3001\u6750\u6599\u53d8\u5f62\u7b49\u3002\u5728\u8fd9\u4e9b\u5e94\u7528\u4e2d\uff0cPDE\u6c42\u89e3\u7684\u901f\u5ea6\u548c\u7cbe\u5ea6\u662f\u51b3\u5b9a\u79d1\u5b66\u8ba1\u7b97\u548c\u5de5\u7a0b\u8bbe\u8ba1\u6548\u7387\u7684\u5173\u952e\u56e0\u7d20\u4e4b\u4e00\u3002<\/p>\n\n\n\n<p>\u5728\u79d1\u5b66\u8ba1\u7b97\u4e2d\uff0c\u7279\u522b\u662f\u5728\u6d89\u53ca\u6570\u5341\u4ebf\u672a\u77e5\u91cf\u7684\u5927\u89c4\u6a21\u95ee\u9898\u4e2d\uff0c\u627e\u5230\u9ad8\u6548\u7684\u6c42\u89e3\u65b9\u6cd5\u4e00\u76f4\u662f\u4e00\u4e2a\u5de8\u5927\u7684\u6311\u6218\u3002\u6b64\u5916\uff0cPDE\u6c42\u89e3\u5728\u4e0d\u786e\u5b9a\u6027\u91cf\u5316\uff08Uncertainty Quantification\uff09\u7b49\u591a\u67e5\u8be2\u5e94\u7528\u4e2d\u4e5f\u626e\u6f14\u91cd\u8981\u89d2\u8272\uff0c\u8981\u6c42\u5728\u4e0d\u540c\u7684\u6761\u4ef6\uff08\u521d\u59cb\u6761\u4ef6\u3001\u8fb9\u754c\u6761\u4ef6\u3001\u6750\u6599\u5c5e\u6027\u7b49\uff09\u4e0b\u591a\u6b21\u91cd\u590d\u6c42\u89e3\u3002\u8fd9\u7c7b\u5e94\u7528\u5bf9\u8ba1\u7b97\u6027\u80fd\u63d0\u51fa\u4e86\u66f4\u9ad8\u7684\u8981\u6c42\u3002<\/p>\n\n\n\n<p>\u76ee\u524d\uff0c\u6700\u5e38\u7528\u7684\u6c42\u89e3PDE\u79bb\u6563\u5316\u65b9\u7a0b\u7ec4\u7684\u65b9\u6cd5\u5305\u62ec\u7ecf\u5178\u7684\u6709\u9650\u5dee\u5206\u6cd5\u3001\u6709\u9650\u5143\u6cd5\u548c\u8c31\u65b9\u6cd5\u7b49\u3002\u5728\u8fd9\u4e9b\u65b9\u6cd5\u4e2d\uff0c\u751f\u6210\u7684\u79bb\u6563\u5316\u7ebf\u6027\u7cfb\u7edf\u9700\u8981\u4f7f\u7528\u6570\u503c\u65b9\u6cd5\u6c42\u89e3\u3002\u65e9\u671f\uff0c\u8fd9\u4e9b\u7ebf\u6027\u7cfb\u7edf\u901a\u5e38\u4f7f\u7528\u7b80\u5355\u7684\u8fed\u4ee3\u65b9\u6cd5\u6c42\u89e3\uff0c\u5982Jacobi\u3001Gauss-Seidel\uff0c\u4ee5\u53ca\u5b83\u4eec\u7684\u53d8\u4f53SOR\uff08Successive Over-Relaxation\uff09\u548cSSOR\uff08Symmetric SOR\uff09\u65b9\u6cd5\u3002\u968f\u7740\u8ba1\u7b97\u89c4\u6a21\u7684\u589e\u5927\uff0cKrylov\u5b50\u7a7a\u95f4\u65b9\u6cd5\uff08\u5982\u5171\u8f6d\u68af\u5ea6\u6cd5Conjugate Gradient\uff0cGMRES\u548cBiCGStab\u7b49\uff09\u9010\u6e10\u6210\u4e3a\u6c42\u89e3\u5927\u89c4\u6a21\u7ebf\u6027\u7cfb\u7edf\u7684\u4e3b\u6d41\uff0c\u56e0\u4e3a\u8fd9\u4e9b\u65b9\u6cd5\u5177\u6709\u66f4\u4f4e\u7684\u8fed\u4ee3\u8ba1\u7b97\u6210\u672c\uff0c\u4e14\u9002\u5408\u5e76\u884c\u8ba1\u7b97\u73af\u5883\u3002<\/p>\n\n\n\n<p>\u8fd1\u5e74\u6765\uff0c\u79d1\u5b66\u673a\u5668\u5b66\u4e60\uff08Scientific Machine Learning\uff0cSciML\uff09\u5f97\u5230\u4e86\u5e7f\u6cdb\u5173\u6ce8\uff0c\u5c24\u5176\u662f\u7528\u4e8e\u89e3\u51b3\u7269\u7406\u548c\u5de5\u7a0b\u4e2d\u51fa\u73b0\u7684PDE\u6570\u503c\u6c42\u89e3\u95ee\u9898\u3002SciML\u7ed3\u5408\u4e86\u4f20\u7edf\u6570\u503c\u65b9\u6cd5\u4e0e\u73b0\u4ee3\u673a\u5668\u5b66\u4e60\u6280\u672f\uff0c\u63d0\u4f9b\u4e86\u4e00\u79cd\u65b0\u7684\u6c42\u89e3\u601d\u8def\u3002\u7279\u522b\u662f\uff0c\u6df1\u5ea6\u7b97\u5b50\u7f51\u7edc\uff08DeepONet\uff09\u3001<a href=\"https:\/\/nullthought.net\/?p=4266\" target=\"_blank\" rel=\"noreferrer noopener\">\u5085\u91cc\u53f6\u795e\u7ecf\u7b97\u5b50\uff08FNO\uff09<\/a>\u7b49\u795e\u7ecf\u7b97\u5b50\u5728PDE\u6c42\u89e3\u65b9\u9762\u5c55\u73b0\u4e86\u5f88\u5927\u6f5c\u529b\u3002\u7136\u800c\uff0c\u7531\u4e8e\u795e\u7ecf\u7f51\u7edc\u7684\u5149\u8c31\u504f\u7f6e\uff08spectral bias\uff09\u7279\u6027\uff0c\u795e\u7ecf\u7b97\u5b50\u5728\u7cbe\u7ec6\u5c3a\u5ea6\u4e0a\u8fd1\u4f3c\u7b97\u5b50\u65f6\u5b58\u5728\u6311\u6218\uff0c\u96be\u4ee5\u6709\u6548\u5904\u7406\u9ad8\u9891\u8bef\u5dee\u3002<\/p>\n\n\n\n<p>\u4e3a\u6b64\uff0c\u6df7\u5408\u9884\u6761\u4ef6\u7b56\u7565\u5e94\u8fd0\u800c\u751f\uff0c\u5373\u5229\u7528\u795e\u7ecf\u7b97\u5b50\u5b9a\u4e49\u8fed\u4ee3\u6c42\u89e3\u5668\u7684\u9884\u6761\u4ef6\u5668\uff0c\u800c\u4e0d\u662f\u76f4\u63a5\u4f5c\u4e3a\u89e3\u7b97\u5668\u3002\u8fd9\u4e9b\u6df7\u5408\u65b9\u6cd5\u5df2\u88ab\u8bc1\u660e\u5728\u51cf\u5c11\u8ba1\u7b97\u65f6\u95f4\u548c\u8fed\u4ee3\u6b21\u6570\u65b9\u9762\u975e\u5e38\u6709\u6548\uff0c\u4f46\u5982\u4f55\u81ea\u52a8\u627e\u5230\u6700\u4f18\u7684\u5143\u89e3\u7b97\u5668\u4ee5\u9002\u5e94\u4e0d\u540c\u7684\u76ee\u6807\u95ee\u9898\u4ecd\u7136\u662f\u4e00\u4e2a\u5c1a\u672a\u89e3\u51b3\u7684\u95ee\u9898\u3002\u672c\u6587\u7684\u4e3b\u8981\u76ee\u6807\u5c31\u662f\u901a\u8fc7\u591a\u76ee\u6807\u4f18\u5316\u6765\u81ea\u52a8\u53d1\u73b0\u6700\u4f18\u7684\u5143\u89e3\u7b97\u5668\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2. \u7814\u7a76\u8d21\u732e<\/h4>\n\n\n\n<p>\u8bba\u6587\u63d0\u51fa\u4e86\u4e24\u7c7b\u5143\u89e3\u7b97\u5668\uff0c\u5373relaxation-based\u65b9\u6cd5\u548cKrylov\u65b9\u6cd5\uff0c\u8fd9\u4e24\u7c7b\u5143\u89e3\u7b97\u5668\u90fd\u7ed3\u5408\u4e86\u795e\u7ecf\u7b97\u5b50\u548c\u7ecf\u5178\u6570\u503c\u89e3\u7b97\u65b9\u6cd5\u7684\u4f18\u52bf\u3002<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>relaxation-based\u65b9\u6cd5\u7684\u5143\u89e3\u7b97\u5668<\/strong>\uff1a\u901a\u8fc7\u5c06\u795e\u7ecf\u7b97\u5b50\u4e0e\u7b80\u5355\u7684\u8fed\u4ee3\u89e3\u7b97\u65b9\u6cd5\uff08\u5982Jacobi\u548cGauss-Seidel\uff09\u7ed3\u5408\uff0c\u8bbe\u8ba1\u4e86\u4e00\u79cd\u6df7\u5408\u6c42\u89e3\u7b56\u7565\u3002\u795e\u7ecf\u7b97\u5b50\u7684\u4f18\u52bf\u5728\u4e8e\u5b83\u80fd\u591f\u5f88\u597d\u5730\u8fd1\u4f3c\u8bef\u5dee\u7684\u4f4e\u9891\u90e8\u5206\uff0c\u800c\u8fed\u4ee3\u89e3\u7b97\u65b9\u6cd5\u5219\u53ef\u4ee5\u5904\u7406\u9ad8\u9891\u8bef\u5dee\uff0c\u8fbe\u5230\u5feb\u901f\u6536\u655b\u7684\u6548\u679c\u3002\u5177\u4f53\u800c\u8a00\uff0c\u8fd9\u4e9b\u795e\u7ecf\u7b97\u5b50\u4f7f\u7528\u4e86DeepONet\u7684\u4e3b\u5e72\u7f51\u7edc\u4f5c\u4e3a\u7c97\u5c3a\u5ea6\u7684\u9884\u6761\u4ef6\u5668\uff0c\u7528\u4e8e\u52a0\u901f\u6574\u4f53\u6536\u655b\u3002<\/li>\n\n\n\n<li><strong>\u57fa\u4e8eKrylov\u65b9\u6cd5\u7684\u5143\u89e3\u7b97\u5668<\/strong>\uff1a\u901a\u8fc7\u5c06\u795e\u7ecf\u7b97\u5b50\u4e0eKrylov\u5b50\u7a7a\u95f4\u65b9\u6cd5\u7ed3\u5408\uff08\u5982GMRES\u3001BiCGStab\uff09\uff0c\u8fdb\u4e00\u6b65\u63d0\u9ad8\u4e86\u6c42\u89e3\u7684\u6548\u7387\u3002\u5728\u8fd9\u79cd\u6df7\u5408\u65b9\u6cd5\u4e2d\uff0c\u795e\u7ecf\u7b97\u5b50\u88ab\u7528\u6765\u5b9a\u4e49\u4e00\u79cd\u7279\u6b8a\u7684\u9884\u6761\u4ef6\u5668\uff0c\u4ee5\u6b64\u63d0\u5347Krylov\u65b9\u6cd5\u7684\u6536\u655b\u6027\u80fd\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u8bba\u6587\u63d0\u51fa\u4e86\u4e00\u79cd\u591a\u76ee\u6807\u4f18\u5316\uff08MOO\uff09\u6846\u67b6\u6765\u7cfb\u7edf\u5316\u5730\u53d1\u73b0\u8fd9\u4e9b\u5143\u89e3\u7b97\u5668\u4e2d\u7684\u6700\u4f18\u8005\u3002\u591a\u76ee\u6807\u4f18\u5316\u8003\u8651\u4e86\u591a\u4e2a\u8bc4\u4ef7\u6307\u6807\uff0c\u5305\u62ec\u8ba1\u7b97\u65f6\u95f4\u3001\u76f8\u5bf9\u8bef\u5dee\u3001\u8fed\u4ee3\u6b21\u6570\u3001\u6536\u655b\u901f\u5ea6\u3001\u5185\u5b58\u4f7f\u7528\u548c\u8bad\u7ec3\u65f6\u95f4\u7b49\u3002\u6700\u4f18\u7684\u5143\u89e3\u7b97\u5668\u901a\u8fc7Pareto\u524d\u6cbf\u7684\u6982\u5ff5\u6765\u9009\u62e9\uff0c\u8fd9\u4e9b\u89e3\u7b97\u5668\u5728\u67d0\u4e9b\u6307\u6807\u4e0a\u8868\u73b0\u51fa\u8272\uff0c\u4f46\u5728\u5176\u5b83\u6307\u6807\u4e0a\u53ef\u80fd\u7a0d\u900a\u4e00\u7b79\u3002<\/p>\n\n\n\n<p>\u6b64\u5916\uff0c\u8bba\u6587\u5f15\u5165\u4e86\u504f\u597d\u51fd\u6570\u7684\u6982\u5ff5\uff0c\u5141\u8bb8\u7528\u6237\u6839\u636e\u5b9e\u9645\u9700\u6c42\uff08\u4f8b\u5982\u66f4\u5173\u6ce8\u901f\u5ea6\u8fd8\u662f\u66f4\u5173\u6ce8\u7cbe\u5ea6\uff09\u9009\u62e9\u7279\u5b9a\u573a\u666f\u4e0b\u7684\u6700\u4f18\u89e3\u7b97\u5668\u3002\u504f\u597d\u51fd\u6570\u5c06\u591a\u76ee\u6807\u4f18\u5316\u95ee\u9898\u91cd\u65b0\u5b9a\u4e49\u4e3a\u4e00\u4e2a\u7ecf\u5178\u7684\u5355\u76ee\u6807\u4f18\u5316\u95ee\u9898\uff0c\u901a\u8fc7\u4e00\u4e2a\u7ebf\u6027\u89c4\u5212\uff08LP\uff09\u7b97\u6cd5\u53ef\u4ee5\u627e\u5230\u7528\u6237\u504f\u597d\u7684\u6700\u4f18\u89e3\u7b97\u5668\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">3. \u65b9\u6cd5\u8bba<\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">3.1 \u53c2\u6570\u5316\u5143\u89e3\u7b97\u5668<\/h5>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>relaxation-based\u65b9\u6cd5\u7684\u5143\u89e3\u7b97\u5668<\/strong>\uff1a\u8fd9\u4e9b\u89e3\u7b97\u5668\u901a\u8fc7\u4e00\u4e2a\u56db\u7ef4\u5b50\u7a7a\u95f4\u6765\u53c2\u6570\u5316\uff0c\u5206\u522b\u8868\u793a\u795e\u7ecf\u7b97\u5b50\u7684\u9009\u62e9\u3001\u7ecf\u5178\u8fed\u4ee3\u89e3\u7b97\u5668\u7684\u9009\u62e9\u3001\u795e\u7ecf\u7b97\u5b50\u4e0e\u4f20\u7edf\u89e3\u7b97\u5668\u7684\u6bd4\u4f8b\u4ee5\u53ca\u591a\u91cd\u7f51\u683c\u7684\u5c42\u6570\u3002<\/li>\n\n\n\n<li><strong>\u57fa\u4e8eKrylov\u65b9\u6cd5\u7684\u5143\u89e3\u7b97\u5668<\/strong>\uff1a\u8fd9\u4e9b\u89e3\u7b97\u5668\u901a\u8fc7\u4e00\u4e2a\u4e94\u7ef4\u5b50\u7a7a\u95f4\u6765\u53c2\u6570\u5316\uff0c\u9664\u4e86\u9009\u62e9\u795e\u7ecf\u7b97\u5b50\u548cKrylov\u89e3\u7b97\u5668\uff0c\u8fd8\u5305\u62ecrelaxation\u65b9\u6cd5\u7684\u9009\u62e9\u3001relaxation\u6b65\u9aa4\u7684\u7b56\u7565\u4ee5\u53ca\u591a\u91cd\u7f51\u683c\u7684\u5c42\u6570\u3002<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">3.2 \u591a\u76ee\u6807\u4f18\u5316\u4e0ePareto\u6700\u4f18<\/h5>\n\n\n\n<p>\u5728\u591a\u76ee\u6807\u4f18\u5316\u7684\u6846\u67b6\u4e0b\uff0c\u6bcf\u4e2a\u5143\u89e3\u7b97\u5668\u7684\u6027\u80fd\u7528\u4e00\u4e2a\u5411\u91cf\u503c\u51fd\u6570\u6765\u8bc4\u4f30\uff0c\u8fd9\u4e2a\u5411\u91cf\u7684\u6bcf\u4e2a\u5206\u91cf\u4ee3\u8868\u67d0\u4e00\u7279\u5b9a\u6027\u80fd\u6307\u6807\u3002\u7531\u4e8e\u5728\u73b0\u5b9e\u4e2d\u5f88\u96be\u627e\u5230\u4e00\u4e2a\u89e3\u7b97\u5668\u5728\u6240\u6709\u6307\u6807\u4e0a\u90fd\u662f\u6700\u4f18\u7684\uff0c\u56e0\u6b64\u91c7\u7528Pareto\u6700\u4f18\u7684\u6982\u5ff5\u6765\u63cf\u8ff0\u8fd9\u4e9b\u89e3\u7b97\u5668\u3002Pareto\u6700\u4f18\u89e3\u662f\u6307\u4e0d\u5b58\u5728\u5176\u5b83\u89e3\u7b97\u5668\u5728\u6240\u6709\u6307\u6807\u4e0a\u90fd\u4f18\u4e8e\u5b83\u7684\u89e3\u7b97\u5668\u3002Pareto\u524d\u6cbf\u662f\u6307\u6240\u6709Pareto\u6700\u4f18\u89e3\u7684\u96c6\u5408\uff0c\u8fd9\u4e9b\u89e3\u7b97\u5668\u4ee3\u8868\u4e86\u5728\u4e0d\u540c\u6307\u6807\u7ec4\u5408\u4e0b\u53ef\u80fd\u7684\u6700\u4f18\u6027\u80fd\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.3 \u504f\u597d\u51fd\u6570\u7684\u53d1\u73b0<\/h5>\n\n\n\n<p>\u901a\u8fc7\u5f15\u5165\u504f\u597d\u51fd\u6570\uff0c\u53ef\u4ee5\u8ba9\u7528\u6237\u5728Pareto\u6700\u4f18\u89e3\u4e2d\u9009\u62e9\u6700\u7b26\u5408\u7279\u5b9a\u9700\u6c42\u7684\u89e3\u7b97\u5668\u3002\u504f\u597d\u51fd\u6570\u662f\u4e00\u4e2a\u9012\u589e\u51fd\u6570\uff0c\u7528\u4e8e\u5bf9\u5404\u4e2a\u6027\u80fd\u6307\u6807\u8fdb\u884c\u52a0\u6743\u548c\u9009\u62e9\u3002\u8fd9\u6837\uff0c\u4f18\u5316\u95ee\u9898\u5c31\u53d8\u6210\u4e86\u5728Pareto\u6700\u4f18\u89e3\u96c6\u5408\u4e2d\u6700\u5c0f\u5316\u504f\u597d\u51fd\u6570\u7684\u503c\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">4. \u6570\u503c\u5b9e\u9a8c<\/h4>\n\n\n\n<p>\u8bba\u6587\u901a\u8fc7\u6570\u503c\u5b9e\u9a8c\u9a8c\u8bc1\u4e86\u6240\u63d0\u65b9\u6cd5\u7684\u6709\u6548\u6027\uff0c\u5206\u522b\u6d4b\u8bd5\u4e86\u4e00\u7ef4\u3001\u4e8c\u7ef4\u548c\u4e09\u7ef4\u6cca\u677e\u65b9\u7a0b\uff08Poisson Equation\uff09\uff0c\u8fd9\u4e9b\u65b9\u7a0b\u5728\u79d1\u5b66\u548c\u5de5\u7a0b\u4e2d\u6709\u7740\u5e7f\u6cdb\u7684\u5e94\u7528\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4.1 relaxation-based\u65b9\u6cd5\u5143\u89e3\u7b97\u5668\u7684\u5b9e\u9a8c\u7ed3\u679c<\/h5>\n\n\n\n<p>\u5b9e\u9a8c\u8bbe\u8ba1\u4e86588\u79cd\u4e0d\u540c\u7ec4\u5408\u7684relaxation-based\u65b9\u6cd5\u5143\u89e3\u7b97\u5668\uff0c\u7528\u4e8e\u4e09\u7ef4\u6cca\u677e\u65b9\u7a0b\u7684\u6c42\u89e3\u3002\u901a\u8fc7\u5206\u6790\u4e0d\u540c\u89e3\u7b97\u5668\u5728\u591a\u76ee\u6807\u6307\u6807\u4e0b\u7684\u8868\u73b0\uff0c\u5171\u53d1\u73b0\u4e86119\u4e2aPareto\u6700\u4f18\u89e3\u7b97\u5668\u3002<\/p>\n\n\n\n<p>\u5728\u5bf9\u8fd9\u4e9b\u6700\u4f18\u89e3\u7b97\u5668\u8fdb\u884c\u5206\u6790\u65f6\uff0c\u53d1\u73b0\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>JacobiKAN\u548cChebyKAN\u662f\u6700\u5e38\u51fa\u73b0\u5728Pareto\u6700\u4f18\u89e3\u7b97\u5668\u4e2d\u7684\u795e\u7ecf\u7b97\u5b50\u3002<\/li>\n\n\n\n<li>SSOR\u662f\u6700\u5e38\u4f7f\u7528\u7684\u7ecf\u5178\u8fed\u4ee3\u89e3\u7b97\u5668\uff0c\u5360\u636e\u4e86\u5927\u90e8\u5206\u7684Pareto\u6700\u4f18\u89e3\u7b97\u5668\u3002<\/li>\n\n\n\n<li>\u4e24\u7ea7\u591a\u91cd\u7f51\u683c\u65b9\u6cd5\u5728\u5927\u591a\u6570\u60c5\u51b5\u4e0b\u90fd\u8868\u73b0\u5f97\u975e\u5e38\u4f18\u79c0\uff0c\u8868\u660e\u591a\u91cd\u7f51\u683c\u5728\u51cf\u5c11\u8ba1\u7b97\u65f6\u95f4\u548c\u52a0\u901f\u6536\u655b\u65b9\u9762\u8d77\u5230\u4e86\u5173\u952e\u4f5c\u7528\u3002<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">4.2 Krylov\u65b9\u6cd5\u5143\u89e3\u7b97\u5668\u7684\u5b9e\u9a8c\u7ed3\u679c<\/h5>\n\n\n\n<p>\u5b9e\u9a8c\u8bbe\u8ba1\u4e86900\u79cdKrylov\u65b9\u6cd5\u5143\u89e3\u7b97\u5668\uff0c\u7528\u4e8e\u4e09\u7ef4\u6cca\u677e\u65b9\u7a0b\u7684\u6c42\u89e3\uff0c\u53d1\u73b0\u4e86\u5176\u4e2d\u7684210\u4e2aPareto\u6700\u4f18\u89e3\u7b97\u5668\u3002<\/p>\n\n\n\n<p>\u5206\u6790\u8868\u660e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>FBiCGStab\u65b9\u6cd5\u662f\u6700\u4f18\u89e3\u7b97\u5668\u4e2d\u6700\u5e38\u51fa\u73b0\u7684Krylov\u65b9\u6cd5\uff0c\u5176\u6b21\u662fFGMRES\u65b9\u6cd5\u3002<\/li>\n\n\n\n<li>SSOR\u662f\u6700\u5e38\u4f7f\u7528\u7684relaxation-based\u65b9\u6cd5\uff0c\u7279\u522b\u662f\u5728\u7ed3\u5408\u591a\u7ea7\u7f51\u683c\u6280\u672f\u65f6\u8868\u73b0\u5c24\u4e3a\u7a81\u51fa\u3002<\/li>\n\n\n\n<li>\u591a\u7ea7\u7f51\u683c\u7684\u4e8c\u7ea7\u548c\u4e09\u7ea7\u65b9\u6cd5\u5bf9\u4e8e\u63d0\u9ad8\u6574\u4f53\u6027\u80fd\u81f3\u5173\u91cd\u8981\uff0c\u5c24\u5176\u662f\u5728\u5927\u89c4\u6a21\u4e09\u7ef4\u6c42\u89e3\u4e2d\uff0c\u8f83\u9ad8\u7ea7\u7684\u591a\u91cd\u7f51\u683c\u5f80\u5f80\u8868\u73b0\u66f4\u597d\u3002<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">4.3 \u504f\u597d\u51fd\u6570\u4e0b\u7684\u6700\u4f18\u89e3\u53d1\u73b0<\/h5>\n\n\n\n<p>\u8bba\u6587\u901a\u8fc7\u4e0d\u540c\u7684\u504f\u597d\u51fd\u6570\u6765\u9009\u62e9\u6700\u4f18\u7684\u5143\u89e3\u7b97\u5668\u3002\u4f8b\u5982\uff0c\u5f53\u7528\u6237\u66f4\u52a0\u6ce8\u91cd\u8ba1\u7b97\u65f6\u95f4\u65f6\uff0c\u9009\u62e9\u7684\u89e3\u7b97\u5668\u4e0e\u6ce8\u91cd\u8bef\u5dee\u6216\u8005\u5185\u5b58\u4f7f\u7528\u7684\u89e3\u7b97\u5668\u4f1a\u6709\u6240\u4e0d\u540c\u3002\u5b9e\u9a8c\u4e2d\u5c55\u793a\u4e86\u4e0d\u540c\u504f\u597d\u51fd\u6570\u4e0b\u7684\u524d3\u4e2a\u6700\u4f18\u89e3\u7b97\u5668\u7684\u5177\u4f53\u7ec4\u5408\uff0c\u5e76\u5206\u6790\u4e86\u8fd9\u4e9b\u89e3\u7b97\u5668\u5728\u8ba1\u7b97\u65f6\u95f4\u3001\u8bef\u5dee\u3001\u5185\u5b58\u4f7f\u7528\u7b49\u6307\u6807\u4e0a\u7684\u8868\u73b0\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">5. \u7ed3\u8bba\u4e0e\u672a\u6765\u5de5\u4f5c<\/h4>\n\n\n\n<p>\u8bba\u6587\u901a\u8fc7\u63d0\u51fa\u7ed3\u5408\u795e\u7ecf\u7b97\u5b50\u4e0e\u4f20\u7edf\u8fed\u4ee3\u6c42\u89e3\u65b9\u6cd5\u7684\u5143\u89e3\u7b97\u5668\uff0c\u5229\u7528\u591a\u76ee\u6807\u4f18\u5316\u548cPareto\u524d\u6cbf\u7684\u65b9\u6cd5\u6765\u7cfb\u7edf\u5316\u5730\u81ea\u52a8\u53d1\u73b0\u6700\u4f18\u89e3\u7b97\u5668\uff0c\u663e\u8457\u63d0\u9ad8\u4e86\u504f\u5fae\u5206\u65b9\u7a0b\u79bb\u6563\u5316\u7cfb\u7edf\u7684\u6c42\u89e3\u6548\u7387\u3002\u672a\u6765\uff0c\u8bba\u6587\u8ba1\u5212\u5c06\u8fd9\u79cd\u81ea\u52a8\u5316\u7684\u65b9\u6cd5\u6269\u5c55\u5230\u975e\u7ebf\u6027\u7cfb\u7edf\u6c42\u89e3\u4e2d\uff0c\u4ee5\u53ca\u65f6\u95f4\u76f8\u5173\u7684PDE\u7684\u6c42\u89e3\u95ee\u9898\u4e0a\uff0c\u4ee5\u5e94\u5bf9\u66f4\u591a\u79d1\u5b66\u4e0e\u5de5\u7a0b\u4e2d\u7684\u590d\u6742\u5e94\u7528\u573a\u666f\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8bba\u6587Automatic discovery of optimal meta-solvers via multi [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[8,36],"tags":[39,63,48,71],"class_list":{"0":"post-5247","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"hentry","6":"category-tech","7":"category-36","8":"tag-ai","10":"tag-48","11":"tag-physics"},"rttpg_featured_image_url":null,"rttpg_author":{"display_name":"NullThought","author_link":"https:\/\/nullthought.net\/?author=1"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/nullthought.net\/?cat=8\" rel=\"category\">Tech<\/a> <a href=\"https:\/\/nullthought.net\/?cat=36\" rel=\"category\">\u79d1\u5b66<\/a>","rttpg_excerpt":"\u8bba\u6587Automatic discovery of optimal meta-solvers via multi&hellip;","_links":{"self":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5247","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5247"}],"version-history":[{"count":1,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5247\/revisions"}],"predecessor-version":[{"id":5248,"href":"https:\/\/nullthought.net\/index.php?rest_route=\/wp\/v2\/posts\/5247\/revisions\/5248"}],"wp:attachment":[{"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5247"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5247"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/nullthought.net\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5247"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}