{"id":4896,"date":"2024-10-17T19:02:52","date_gmt":"2024-10-17T11:02:52","guid":{"rendered":"https:\/\/nullthought.net\/?p=4896"},"modified":"2024-12-05T14:07:12","modified_gmt":"2024-12-05T06:07:12","slug":"%e5%a4%9a%e5%a4%b4%e7%89%a9%e7%90%86%e4%bf%a1%e6%81%af%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9c%ef%bc%88multi-head-physics-informed-neural-networks%ef%bc%89","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=4896","title":{"rendered":"\u591a\u5934\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08Multi-Head Physics-Informed Neural Networks\uff09"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2301.02152\" target=\"_blank\" rel=\"noreferrer noopener\">L-HYDRA: Multi-Head Physics-Informed Neural Networks<\/a><\/strong>\uff08\u300aL-HYDRA: \u591a\u5934\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\u300b\uff09\u65e8\u5728\u5c06\u591a\u5934\u795e\u7ecf\u7f51\u7edc\uff08Multi-Head Neural Networks, MH-NNs\uff09\u5f15\u5165<strong><a href=\"https:\/\/nullthought.net\/?p=4287\" target=\"_blank\" rel=\"noreferrer noopener\">\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08Physics-Informed Neural Networks, PINNs\uff09<\/a><\/strong>\u6846\u67b6\u4e2d\uff0c\u4ee5\u589e\u5f3a\u5176\u5728\u79d1\u5b66\u673a\u5668\u5b66\u4e60\uff08SciML\uff09\u4e2d\u7684\u591a\u4efb\u52a1\u5b66\u4e60\uff08Multi-Task Learning, MTL\uff09\u3001\u751f\u6210\u5efa\u6a21\u548c\u5c11\u6837\u672c\u5b66\u4e60\u7b49\u65b9\u9762\u7684\u8868\u73b0\u3002\u6587\u7ae0\u4ece\u7406\u8bba\u548c\u5e94\u7528\u7684\u89d2\u5ea6\u5bf9MH-PINNs\u8fdb\u884c\u4e86\u8be6\u7ec6\u63cf\u8ff0\uff0c\u5e76\u901a\u8fc7\u4e00\u7cfb\u5217\u57fa\u51c6\u6d4b\u8bd5\u8bc1\u660e\u5176\u6709\u6548\u6027\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aZongren Zou\u548cGeorge Em Karniadakis\uff0c\u6765\u81eaBrown University\uff08\u5e03\u6717\u5927\u5b66\uff09\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"826\" height=\"362\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2024\/10\/image-13.png\" alt=\"\" class=\"wp-image-4897\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2024\/10\/image-13.png 826w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/10\/image-13-300x131.png 300w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/10\/image-13-768x337.png 768w\" sizes=\"auto, (max-width: 826px) 100vw, 826px\" \/><figcaption class=\"wp-element-caption\"><strong><a href=\"https:\/\/arxiv.org\/abs\/2301.02152\" target=\"_blank\" rel=\"noreferrer noopener\">L-HYDRA: Multi-Head Physics-Informed Neural Networks<\/a><\/strong><\/figcaption><\/figure>\n\n\n\n<p>\u4ee5\u4e0b\u4e3a\u8bba\u6587\u6982\u8981\u5185\u5bb9\uff1a<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e00\u3001\u7814\u7a76\u80cc\u666f<\/h4>\n\n\n\n<p>\u79d1\u5b66\u673a\u5668\u5b66\u4e60\u4e2d\uff0c\u8bb8\u591a\u79d1\u5b66\u95ee\u9898\u5728\u672c\u8d28\u4e0a\u5b58\u5728\u7740\u76f8\u4f3c\u6027\u6216\u8005\u5171\u4eab\u7684\u7269\u7406\u89c4\u5f8b\uff0c\u56e0\u6b64\u8de8\u4efb\u52a1\u5b66\u4e60\uff08learning across tasks\uff09\u9010\u6e10\u6210\u4e3a\u7814\u7a76\u7684\u91cd\u70b9\u3002\u901a\u8fc7\u5728\u591a\u4e2a\u76f8\u4f3c\u4efb\u52a1\u4e4b\u95f4\u5171\u4eab\u77e5\u8bc6\uff0c\u53ef\u4ee5\u5927\u5e45\u63d0\u9ad8\u6a21\u578b\u7684\u6cdb\u5316\u80fd\u529b\u548c\u5b66\u4e60\u6548\u7387\u3002\u4f8b\u5982\uff0c\u903c\u8fd1\u968f\u673a\u8fc7\u7a0b\u3001\u89e3\u504f\u5fae\u5206\u65b9\u7a0b\uff08Partial Differential Equations, PDEs\uff09\u3001\u4ee5\u53ca\u5904\u7406\u53c2\u6570\u5316\u7684PDEs\u7b49\u573a\u666f\u4e2d\uff0c\u4e0d\u540c\u4efb\u52a1\u4e4b\u95f4\u5e38\u5e38\u5b58\u5728\u5185\u5728\u7684\u8054\u7cfb\u3002\u56e0\u6b64\uff0c\u5982\u4f55\u6709\u6548\u5730\u6784\u5efa\u80fd\u591f\u5904\u7406\u591a\u4efb\u52a1\u7684\u795e\u7ecf\u7f51\u7edc\u6846\u67b6\u662f\u672c\u7814\u7a76\u7684\u91cd\u70b9\u3002<\/p>\n\n\n\n<p>\u591a\u5934\u795e\u7ecf\u7f51\u7edc\uff08MH-NNs\uff09\u662f\u4e00\u79cd\u786c\u53c2\u6570\u5171\u4eab\u7684\u591a\u4efb\u52a1\u5b66\u4e60\u65b9\u6cd5\uff0c\u5b83\u901a\u8fc7\u4e00\u4e2a\u5171\u4eab\u7684\u4e3b\u4f53\u90e8\u5206\uff08body\uff09\u548c\u591a\u4e2a\u4efb\u52a1\u7279\u5b9a\u7684\u5934\u90e8\uff08head\uff09\u5b9e\u73b0\u591a\u4e2a\u4efb\u52a1\u7684\u540c\u65f6\u5b66\u4e60\u3002\u5728\u672c\u7814\u7a76\u4e2d\uff0c\u4f5c\u8005\u5c06\u8fd9\u79cd\u601d\u60f3\u5f15\u5165\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINNs\uff09\u4e2d\uff0c\u5f62\u6210\u4e86\u591a\u5934\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08MH-PINNs\uff09\uff0c\u4ee5\u6b64\u6765\u5904\u7406\u591a\u4efb\u52a1\u5b66\u4e60\u3001\u751f\u6210\u5efa\u6a21\u3001\u4ee5\u53ca\u5c11\u6837\u672c\u5b66\u4e60\u95ee\u9898\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e8c\u3001\u65b9\u6cd5\u8bba\uff1aMH-PINNs\u7684\u8bbe\u8ba1\u4e0e\u5b9e\u73b0<\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">1. \u591a\u5934\u795e\u7ecf\u7f51\u7edc\u4e0e\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\u7684\u7ed3\u5408<\/h5>\n\n\n\n<p>MH-PINNs\u57fa\u4e8e\u4f20\u7edf\u7684MH-NNs\u7ed3\u6784\u8fdb\u884c\u6269\u5c55\uff0c\u5305\u542b\u4e00\u4e2a\u4e3b\u4f53\u90e8\u5206\u548c\u591a\u4e2a\u5934\u90e8\u3002\u4e3b\u4f53\u90e8\u5206\u7531\u591a\u4e2a\u975e\u7ebf\u6027\u5c42\u6784\u6210\uff0c\u800c\u6bcf\u4e2a\u5934\u90e8\u4e3a\u4e00\u4e2a\u7ebf\u6027\u8f93\u51fa\u5c42\uff0c\u5206\u522b\u5bf9\u5e94\u4e0d\u540c\u7684\u4efb\u52a1\u3002\u5177\u4f53\u800c\u8a00\uff0cMH-PINNs\u7684\u4e3b\u4f53\u90e8\u5206\u7528\u4e8e\u63d0\u53d6\u5171\u4eab\u7684\u57fa\u7840\u8868\u793a\uff0c\u8fd9\u4e9b\u8868\u793a\u53ef\u4ee5\u770b\u4f5c\u662f\u591a\u4e2a\u4efb\u52a1\u7684\u516c\u5171\u7279\u5f81\uff0c\u800c\u6bcf\u4e2a\u5934\u90e8\u5219\u8d1f\u8d23\u4e0d\u540c\u7684\u4efb\u52a1\u3002\u8fd9\u6837\u4e00\u6765\uff0c\u591a\u4e2a\u4efb\u52a1\u901a\u8fc7\u5171\u4eab\u4e3b\u4f53\u90e8\u5206\u7684\u8868\u793a\u5f97\u5230\u8054\u7cfb\uff0c\u540c\u65f6\u6bcf\u4e2a\u5934\u90e8\u4fdd\u6301\u5bf9\u7279\u5b9a\u4efb\u52a1\u7684\u72ec\u7acb\u6027\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4e2d\u63d0\u5230\uff0cMH-PINNs\u901a\u8fc7\u4ee5\u4e0b\u4e24\u79cd\u65b9\u5f0f\u5b9e\u73b0\u4e86\u5bf9\u591a\u4efb\u52a1\u7684\u8fde\u63a5\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u5171\u4eab\u57fa\u7840\u51fd\u6570<\/strong>\uff1a\u901a\u8fc7\u4e00\u4e2a\u5171\u4eab\u7684\u4e3b\u4f53\u90e8\u5206\uff08\u5373\u7f51\u7edc\u7684\u9690\u85cf\u5c42\uff09\u6765\u8fde\u63a5\u591a\u4e2a\u4efb\u52a1\uff0c\u6bcf\u4e2a\u4efb\u52a1\u4f7f\u7528\u72ec\u7acb\u7684\u5934\u90e8\u6765\u5b8c\u6210\u7279\u5b9a\u4efb\u52a1\u7684\u5b66\u4e60\u3002<\/li>\n\n\n\n<li><strong>\u5171\u4eab\u5206\u5e03\u4fe1\u606f<\/strong>\uff1a\u901a\u8fc7\u5229\u7528\u6b63\u6001\u6d41\uff08Normalizing Flows, NFs\uff09\u5bf9\u5934\u90e8\u8fdb\u884c\u5bc6\u5ea6\u4f30\u8ba1\uff0c\u4ece\u800c\u5f62\u6210\u4e00\u4e2a\u5171\u4eab\u7684\u5206\u5e03\u6a21\u578b\u3002\u8fd9\u79cd\u5206\u5e03\u4fe1\u606f\u7528\u4e8e\u751f\u6210\u5efa\u6a21\u4ee5\u53ca\u4e0d\u786e\u5b9a\u6027\u91cf\u5316\u7b49\u3002<\/li>\n<\/ol>\n\n\n\n<h5 class=\"wp-block-heading\">2. MH-PINNs\u7684\u5b66\u4e60\u8fc7\u7a0b<\/h5>\n\n\n\n<p>MH-PINNs\u7684\u5b66\u4e60\u8fc7\u7a0b\u53ef\u4ee5\u5206\u4e3a\u4e24\u4e2a\u9636\u6bb5\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u7b2c\u4e00\u9636\u6bb5<\/strong>\uff1a\u901a\u8fc7MH-NNs\u5bf9\u591a\u4e2a\u4efb\u52a1\u8fdb\u884c\u8054\u5408\u8bad\u7ec3\uff0c\u5229\u7528\u591a\u4efb\u52a1\u4e4b\u95f4\u7684\u5171\u4eab\u4fe1\u606f\u63d0\u9ad8\u6574\u4f53\u7684\u5b66\u4e60\u6548\u7387\u3002\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u6a21\u578b\u4e2d\u7684\u4e3b\u4f53\u90e8\u5206\u4f1a\u5b66\u4e60\u591a\u4e2a\u4efb\u52a1\u7684\u516c\u5171\u7279\u5f81\uff0c\u800c\u6bcf\u4e2a\u5934\u90e8\u5219\u72ec\u7acb\u5904\u7406\u5404\u81ea\u7684\u4efb\u52a1\u3002<\/li>\n\n\n\n<li><strong>\u7b2c\u4e8c\u9636\u6bb5<\/strong>\uff1a\u901a\u8fc7\u6b63\u6001\u6d41\uff08NFs\uff09\u5bf9\u6bcf\u4e2a\u4efb\u52a1\u5934\u90e8\u7684\u8f93\u51fa\u8fdb\u884c\u5bc6\u5ea6\u4f30\u8ba1\uff0c\u4ece\u800c\u5b9e\u73b0\u751f\u6210\u5efa\u6a21\u3002\u6b63\u6001\u6d41\u662f\u4e00\u79cd\u751f\u6210\u6a21\u578b\uff0c\u5b83\u65e2\u53ef\u4ee5\u7528\u4e8e\u5bc6\u5ea6\u4f30\u8ba1\uff0c\u4e5f\u53ef\u4ee5\u7528\u4e8e\u6837\u672c\u7684\u751f\u6210\u3002\u56e0\u6b64\uff0c\u4f5c\u8005\u9009\u62e9\u6b63\u6001\u6d41\u800c\u975e\u5176\u4ed6\u5e38\u89c1\u7684\u751f\u6210\u6a21\u578b\uff08\u5982\u751f\u6210\u5bf9\u6297\u7f51\u7edcGANs\u3001\u53d8\u5206\u81ea\u7f16\u7801\u5668VAEs\u7b49\uff09\uff0c\u4ee5\u4fbf\u5728\u751f\u6210\u5efa\u6a21\u548c\u5bc6\u5ea6\u4f30\u8ba1\u4e2d\u90fd\u5177\u6709\u66f4\u9ad8\u7684\u7075\u6d3b\u6027\u3002<\/li>\n<\/ol>\n\n\n\n<h5 class=\"wp-block-heading\">3. MH-PINNs\u7684\u5e94\u7528<\/h5>\n\n\n\n<p>MH-PINNs\u7684\u5e94\u7528\u573a\u666f\u6db5\u76d6\u4e86\u591a\u4efb\u52a1\u5b66\u4e60\u3001\u751f\u6210\u5efa\u6a21\u3001\u4ee5\u53ca\u5c11\u6837\u672c\u5b66\u4e60\u7b49\u3002\u5177\u4f53\u800c\u8a00\uff0cMH-PINNs\u53ef\u4ee5\u7528\u4e8e\u4ee5\u4e0b\u76ee\u7684\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u903c\u8fd1\u968f\u673a\u8fc7\u7a0b<\/strong>\uff1a\u901a\u8fc7\u7269\u7406\u4fe1\u606f\u751f\u6210\u65b9\u6cd5\uff0cMH-PINNs\u53ef\u4ee5\u6709\u6548\u5730\u903c\u8fd1\u5177\u6709\u968f\u673a\u7279\u6027\u7684\u7269\u7406\u8fc7\u7a0b\u3002<\/li>\n\n\n\n<li><strong>\u5c11\u6837\u672c\u5b66\u4e60\uff08Few-shot Learning\uff09<\/strong>\uff1aMH-PINNs\u53ef\u4ee5\u4e3a\u4e0b\u6e38\u4efb\u52a1\uff08\u5982\u5143\u5b66\u4e60\u548c\u8fc1\u79fb\u5b66\u4e60\uff09\u63d0\u4f9b\u6709\u7528\u7684\u5148\u9a8c\u77e5\u8bc6\uff0c\u4ece\u800c\u5728\u6570\u636e\u7a00\u7f3a\u7684\u60c5\u51b5\u4e0b\u63d0\u5347\u5b66\u4e60\u6548\u679c\u3002<\/li>\n\n\n\n<li><strong>\u4e0d\u786e\u5b9a\u6027\u91cf\u5316\uff08Uncertainty Quantification, UQ\uff09<\/strong>\uff1a\u901a\u8fc7\u5bf9\u5934\u90e8\u8fdb\u884c\u5bc6\u5ea6\u4f30\u8ba1\uff0cMH-PINNs\u53ef\u4ee5\u5b9e\u73b0\u5bf9\u6a21\u578b\u8f93\u51fa\u7684\u751f\u6210\u5efa\u6a21\u4e0e\u4e0d\u786e\u5b9a\u6027\u91cf\u5316\u3002<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e09\u3001\u5b9e\u9a8c\u9a8c\u8bc1\u4e0e\u7ed3\u679c\u5206\u6790<\/h4>\n\n\n\n<p>\u8bba\u6587\u901a\u8fc7\u4e94\u4e2a\u57fa\u51c6\u5b9e\u9a8c\u5bf9MH-PINNs\u7684\u6709\u6548\u6027\u8fdb\u884c\u4e86\u9a8c\u8bc1\uff0c\u6db5\u76d6\u4e86\u591a\u4efb\u52a1\u5b66\u4e60\u3001\u56de\u5f52\u95ee\u9898\u3001\u751f\u6210\u5efa\u6a21\u548c\u5c11\u6837\u672c\u5b66\u4e60\u7b49\u591a\u4e2a\u9886\u57df\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">1. \u5b9e\u9a8c\u8bbe\u7f6e<\/h5>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u51fd\u6570\u903c\u8fd1<\/strong>\uff1a\u7814\u7a76\u8005\u9996\u5148\u5728\u4e00\u4e2a\u51fd\u6570\u903c\u8fd1\u95ee\u9898\u4e0a\u9a8c\u8bc1\u4e86MH-PINNs\u7684\u591a\u4efb\u52a1\u5b66\u4e60\u548c\u751f\u6210\u5efa\u6a21\u80fd\u529b\u3002\u51fd\u6570\u5f62\u5f0f\u4e3a\u968f\u673a\u8fc7\u7a0b\uff0c\u901a\u8fc7MH-NNs\u5bf9\u591a\u4e2a\u4e0d\u540c\u7684\u51fd\u6570\u8fdb\u884c\u8054\u5408\u8bad\u7ec3\uff0c\u6700\u7ec8\u5b9e\u73b0\u5bf9\u76ee\u6807\u51fd\u6570\u7684\u903c\u8fd1\u548c\u751f\u6210\u3002<\/li>\n\n\n\n<li><strong>\u975e\u7ebf\u6027\u5e38\u5fae\u5206\u65b9\u7a0b\u7cfb\u7edf<\/strong>\uff1a\u7814\u7a76\u8005\u5728\u4e00\u4e2a\u975e\u7ebf\u6027\u5e38\u5fae\u5206\u65b9\u7a0b\u7cfb\u7edf\u4e0a\u8fdb\u884c\u4e86\u6d4b\u8bd5\uff0c\u901a\u8fc7MH-PINNs\u5b66\u4e60\u5916\u529b\u9879\u7684\u968f\u673a\u8fc7\u7a0b\uff0c\u5e76\u6bd4\u8f83\u4e86\u4e0d\u540c\u751f\u6210\u6a21\u578b\uff08\u5982RealNVP\u3001MAF\u548cIAF\uff09\u7684\u6548\u679c\u3002<\/li>\n\n\n\n<li><strong>\u4e00\u7ef4\u975e\u7ebf\u6027\u53cd\u5e94\u6269\u6563\u65b9\u7a0b<\/strong>\uff1a\u5bf9\u4e00\u7ef4\u7684\u975e\u7ebf\u6027\u53cd\u5e94\u6269\u6563\u65b9\u7a0b\u8fdb\u884c\u6d4b\u8bd5\uff0c\u9a8c\u8bc1\u4e86MH-PINNs\u5728\u751f\u6210\u5efa\u6a21\u548c\u4e0d\u786e\u5b9a\u6027\u91cf\u5316\u4e2d\u7684\u6709\u6548\u6027\u3002<\/li>\n\n\n\n<li><strong>\u4e8c\u7ef4Allen-Cahn\u65b9\u7a0b<\/strong>\u548c<strong>\u4e8c\u7ef4Helmholtz\u65b9\u7a0b<\/strong>\uff1a\u901a\u8fc7\u8fd9\u4e9b\u8f83\u4e3a\u590d\u6742\u7684\u7269\u7406\u573a\u666f\uff0c\u9a8c\u8bc1\u4e86MH-PINNs\u5728\u903c\u8fd1\u590d\u6742\u591a\u7ef4\u968f\u673a\u8fc7\u7a0b\u4e2d\u7684\u8868\u73b0\uff0c\u4ee5\u53ca\u5728\u4e0d\u786e\u5b9a\u6027\u91cf\u5316\u548c\u5c11\u6837\u672c\u5b66\u4e60\u4e2d\u7684\u4f18\u8d8a\u6027\u3002<\/li>\n<\/ul>\n\n\n\n<h5 class=\"wp-block-heading\">2. \u5b9e\u9a8c\u7ed3\u679c<\/h5>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5728\u591a\u4efb\u52a1\u5b66\u4e60\uff08MTL\uff09\u4e2d\uff0cMH-PINNs\u901a\u8fc7\u5171\u4eab\u7684\u4e3b\u4f53\u90e8\u5206\u5b9e\u73b0\u4e86\u591a\u4e2a\u4efb\u52a1\u7684\u8054\u5408\u5b66\u4e60\uff0c\u5728\u51fd\u6570\u903c\u8fd1\u548c\u65b9\u7a0b\u6c42\u89e3\u4e2d\u5c55\u73b0\u4e86\u5f88\u597d\u7684\u6cdb\u5316\u6027\u80fd\u3002<\/li>\n\n\n\n<li>\u5728\u751f\u6210\u5efa\u6a21\u65b9\u9762\uff0c\u5b9e\u9a8c\u7ed3\u679c\u8868\u660eMH-PINNs\u7ed3\u5408\u6b63\u6001\u6d41\uff08NFs\uff09\u80fd\u591f\u6709\u6548\u5730\u903c\u8fd1\u76ee\u6807\u5206\u5e03\uff0c\u5e76\u751f\u6210\u7b26\u5408\u968f\u673a\u8fc7\u7a0b\u7279\u6027\u7684\u6837\u672c\u3002<\/li>\n\n\n\n<li>\u5728\u5c11\u6837\u672c\u5b66\u4e60\u4e2d\uff0cMH-PINNs\u5229\u7528\u4ece\u591a\u4efb\u52a1\u5b66\u4e60\u4e2d\u83b7\u5f97\u7684\u5171\u4eab\u57fa\u7840\u51fd\u6570\u548c\u751f\u6210\u5efa\u6a21\u4e2d\u7684\u5148\u9a8c\u77e5\u8bc6\uff0c\u5927\u5927\u63d0\u9ad8\u4e86\u5728\u6570\u636e\u4e0d\u8db3\u60c5\u51b5\u4e0b\u7684\u5b66\u4e60\u6548\u679c\uff0c\u5e76\u901a\u8fc7\u8d1d\u53f6\u65af\u63a8\u65ad\u5b9e\u73b0\u4e86\u5bf9\u4e0d\u786e\u5b9a\u6027\u7684\u5b9a\u91cf\u63cf\u8ff0\u3002<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">\u56db\u3001\u5bf9\u6bd4\u5206\u6790<\/h4>\n\n\n\n<p>\u8bba\u6587\u8fd8\u5bf9MH-PINNs\u4e0e\u5176\u4ed6\u76f8\u5173\u65b9\u6cd5\u8fdb\u884c\u4e86\u5bf9\u6bd4\u5206\u6790\uff0c\u7279\u522b\u662f\u4e0e\u4f20\u7edf\u7684\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINNs\uff09\u3001\u751f\u6210\u5bf9\u6297\u7f51\u7edc\uff08GANs\uff09\u548c\u7269\u7406\u4fe1\u606f\u751f\u6210\u5bf9\u6297\u7f51\u7edc\uff08PI-GANs\uff09\u7b49\u8fdb\u884c\u4e86\u8be6\u7ec6\u6bd4\u8f83\u3002\u7ed3\u679c\u8868\u660e\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u4e0e\u4f20\u7edf\u7684PINNs\u76f8\u6bd4\uff0cMH-PINNs\u901a\u8fc7\u591a\u4efb\u52a1\u5b66\u4e60\u548c\u5171\u4eab\u57fa\u7840\u51fd\u6570\uff0c\u63d0\u9ad8\u4e86\u5bf9\u76f8\u4f3c\u4efb\u52a1\u7684\u5b66\u4e60\u6548\u7387\u548c\u9884\u6d4b\u51c6\u786e\u6027\u3002<\/li>\n\n\n\n<li>\u4e0ePI-GANs\u76f8\u6bd4\uff0cMH-PINNs\u7531\u4e8e\u91c7\u7528\u4e86\u4e24\u9636\u6bb5\u7684\u5b66\u4e60\u65b9\u6cd5\uff0c\u6781\u5927\u5730\u63d0\u9ad8\u4e86\u751f\u6210\u5efa\u6a21\u7684\u7075\u6d3b\u6027\u548c\u8bad\u7ec3\u7684\u7a33\u5b9a\u6027\uff0c\u5c24\u5176\u662f\u5728\u7269\u7406\u4fe1\u606f\u5b66\u4e60\u548c\u751f\u6210\u5efa\u6a21\u4e4b\u95f4\u8fdb\u884c\u4e86\u89e3\u8026\uff0c\u4f7f\u5f97\u8bad\u7ec3\u8fc7\u7a0b\u66f4\u52a0\u7b80\u6d01\u3001\u6613\u4e8e\u63a7\u5236\u3002<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e94\u3001\u5e94\u7528\u524d\u666f\u4e0e\u6311\u6218<\/h4>\n\n\n\n<p>MH-PINNs\u4e3a\u79d1\u5b66\u673a\u5668\u5b66\u4e60\u4e2d\u7684\u591a\u4efb\u52a1\u5b66\u4e60\u3001\u751f\u6210\u5efa\u6a21\u548c\u5c11\u6837\u672c\u5b66\u4e60\u63d0\u4f9b\u4e86\u4e00\u79cd\u6709\u6548\u7684\u5de5\u5177\uff0c\u5176\u5e94\u7528\u524d\u666f\u5e7f\u6cdb\uff0c\u5c24\u5176\u5728\u4ee5\u4e0b\u51e0\u4e2a\u65b9\u9762\uff1a<\/p>\n\n\n\n<ol 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