{"id":5265,"date":"2024-12-05T14:02:56","date_gmt":"2024-12-05T06:02:56","guid":{"rendered":"https:\/\/nullthought.net\/?p=5265"},"modified":"2024-12-05T14:02:58","modified_gmt":"2024-12-05T06:02:58","slug":"%e7%bb%93%e5%90%88%e5%8d%b7%e7%a7%af%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9c%ef%bc%88cnn%ef%bc%89%e7%9a%84%e7%89%b9%e5%be%81%e6%8f%90%e5%8f%96%e8%83%bd%e5%8a%9b%e5%92%8c%e7%89%a9%e7%90%86%e4%bf%a1","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=5265","title":{"rendered":"\u7ed3\u5408\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff08CNN\uff09\u7684\u7279\u5f81\u63d0\u53d6\u80fd\u529b\u548c\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINNs\uff09\u7684\u7269\u7406\u89c4\u5f8b\u5efa\u6a21\u80fd\u529b\uff0c\u5b9e\u73b0\u5bf9\u6db2\u6ef4\u649e\u51fb\u548c\u4e24\u76f8\u754c\u9762\u6f14\u53d8\u7684\u4e09\u7ef4\u9ad8\u7cbe\u5ea6\u91cd\u5efa"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2411.15949\" target=\"_blank\" rel=\"noreferrer noopener\">PINNs4Drops: Convolutional feature-enhanced physics-informed neural networks for reconstructing two-phase flows<\/a><\/strong>\uff08\u201cPINNs4Drops: \u7528\u5377\u79ef\u7279\u5f81\u589e\u5f3a\u7684\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\u7528\u4e8e\u91cd\u5efa\u4e24\u76f8\u6d41\u52a8\u201d\uff09\u7684\u7814\u7a76\u805a\u7126\u4e8e\u89e3\u51b3\u4e24\u76f8\u6d41\u52a8\u7684\u4e09\u7ef4\u91cd\u5efa\u95ee\u9898\uff0c\u7279\u522b\u662f\u57fa\u4e8e\u5355\u89c6\u89d2\u6570\u636e\u7684\u91cd\u5efa\u6311\u6218\u3002\u5728\u5de5\u7a0b\u5e94\u7528\u4e2d\uff0c\u4e24\u76f8\u6d41\u52a8\u73b0\u8c61\uff08\u5982\u6db2\u6ef4\u5728\u6c14\u4f53\u73af\u5883\u4e2d\u7684\u8fd0\u52a8\uff09\u5e7f\u6cdb\u5b58\u5728\uff0c\u5e94\u7528\u573a\u666f\u5305\u62ec\u55b7\u96fe\u51b7\u5374\u3001\u71c3\u70e7\u3001\u6cb9\u6c34\u5206\u79bb\u3001\u71c3\u6599\u7535\u6c60\u7b49\u3002\u8fd9\u4e9b\u573a\u666f\u4e2d\uff0c\u754c\u9762\u548c\u5185\u90e8\u6d41\u52a8\u7684\u590d\u6742\u52a8\u6001\u4fe1\u606f\u5bf9\u4e8e\u6027\u80fd\u7684\u4f18\u5316\u548c\u63a7\u5236\u81f3\u5173\u91cd\u8981\u3002\u7136\u800c\uff0c\u4f20\u7edf\u7684\u5b9e\u9a8c\u548c\u6570\u503c\u6a21\u62df\u6280\u672f\u5728\u83b7\u53d6\u8fd9\u4e9b\u4fe1\u606f\u65f6\u9762\u4e34\u8bf8\u591a\u9650\u5236\uff0c\u7279\u522b\u662f\u53ea\u80fd\u8fdb\u884c\u4e8c\u7ef4\u7684\u89c2\u6d4b\u6216\u8ba1\u7b97\u6210\u672c\u8fc7\u9ad8\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u514b\u670d\u8fd9\u4e9b\u5c40\u9650\uff0c\u7814\u7a76\u8005\u63d0\u51fa\u4e86\u4e00\u79cd\u878d\u5408\u4e86\u7269\u7406\u63a7\u5236\u65b9\u7a0b\u548c\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u7684\u5377\u79ef\u7279\u5f81\u589e\u5f3a\u7684<strong><a href=\"https:\/\/nullthought.net\/?p=4287\" target=\"_blank\" rel=\"noreferrer noopener\">\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINNs\uff09<\/a><\/strong>\u65b9\u6cd5\u3002\u901a\u8fc7\u7ed3\u5408\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff08CNN\uff09\u7684\u7279\u5f81\u63d0\u53d6\u80fd\u529b\u548c\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINNs\uff09\u7684\u7269\u7406\u89c4\u5f8b\u5efa\u6a21\u80fd\u529b\uff0c\u7814\u7a76\u8005\u5b9e\u73b0\u4e86\u5bf9\u6db2\u6ef4\u649e\u51fb\u548c\u4e24\u76f8\u754c\u9762\u6f14\u53d8\u7684\u4e09\u7ef4\u9ad8\u7cbe\u5ea6\u91cd\u5efa\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aMaximilian Dreisbach, Elham Kiyani, Jochen Kriegseis, George Karniadakis, Alexander Stroh\uff0c\u6765\u81eaKarlsruhe Institute of Technology\u548cBrown University\u3002<\/p>\n\n\n\n<figure class=\"wp-block-image aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"561\" src=\"https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-4-1024x561.png\" alt=\"\" class=\"wp-image-5266\" srcset=\"https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-4-1024x561.png 1024w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-4-300x164.png 300w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-4-768x421.png 768w, https:\/\/nullthought.net\/wp-content\/uploads\/2024\/12\/image-4.png 1139w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><strong><a href=\"https:\/\/arxiv.org\/abs\/2411.15949\" target=\"_blank\" rel=\"noreferrer noopener\">PINNs4Drops: Convolutional feature-enhanced physics-informed neural networks for reconstructing two-phase flows<\/a><\/strong><\/figcaption><\/figure>\n\n\n\n<h4 class=\"wp-block-heading\">1. \u5f15\u8a00<\/h4>\n\n\n\n<p>\u5f15\u8a00\u90e8\u5206\u9996\u5148\u5f3a\u8c03\u4e86\u4e24\u76f8\u6d41\u52a8\u7684\u91cd\u8981\u6027\u548c\u6311\u6218\u6027\uff0c\u7279\u522b\u662f\u5728\u590d\u6742\u7684\u6c14\u6db2\u754c\u9762\u548c\u6d41\u573a\u91cd\u5efa\u65b9\u9762\u3002\u7814\u7a76\u8005\u6307\u51fa\uff0c\u4f20\u7edf\u7684\u6d4b\u91cf\u624b\u6bb5\u5982\u6697\u573a\u6210\u50cf\u548c\u7c92\u5b50\u56fe\u50cf\u6d4b\u901f\uff08PIV\uff09\u53ea\u80fd\u63d0\u4f9b\u4e8c\u7ef4\u6570\u636e\uff0c\u4e14\u9762\u4e34\u8bf8\u5982\u591a\u89d2\u5ea6\u89c2\u6d4b\u96be\u4ee5\u5b9e\u73b0\u3001\u5149\u5b66\u8bbf\u95ee\u53d7\u9650\u7b49\u95ee\u9898\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u7a81\u7834\u8fd9\u4e9b\u9650\u5236\uff0c\u8fd1\u5e74\u6765\uff0c\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u88ab\u5e94\u7528\u4e8e\u4ece\u5b9e\u9a8c\u6570\u636e\u4e2d\u63d0\u53d6\u4e30\u5bcc\u7684\u7279\u5f81\uff0c\u5c24\u5176\u662f\u5728\u5f71\u50cf\u6570\u636e\u7684\u4e0a\u4e0b\u6587\u7406\u89e3\u548c\u4e09\u7ef4\u7ed3\u6784\u91cd\u5efa\u65b9\u9762\u3002\u57fa\u4e8e\u8fd9\u4e00\u57fa\u7840\uff0cPINNs\u7684\u51fa\u73b0\u4e3a\u6df1\u5ea6\u5b66\u4e60\u65b9\u6cd5\u5e26\u6765\u4e86\u65b0\u7684\u53d1\u5c55\uff0c\u5373\u901a\u8fc7\u5c06\u7269\u7406\u63a7\u5236\u65b9\u7a0b\u76f4\u63a5\u878d\u5165\u7f51\u7edc\u7684\u635f\u5931\u51fd\u6570\u4e2d\uff0c\u7f51\u7edc\u4e0d\u4ec5\u4ec5\u662f\u62df\u5408\u6570\u636e\uff0c\u540c\u65f6\u4e5f\u9075\u5faa\u6d41\u4f53\u52a8\u529b\u5b66\u7684\u63a7\u5236\u89c4\u5f8b\uff0c\u4ece\u800c\u5728\u6570\u636e\u7f3a\u4e4f\u7684\u60c5\u51b5\u4e0b\u4ecd\u80fd\u505a\u51fa\u7cbe\u786e\u9884\u6d4b\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">2. \u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINNs\uff09\u4e0e\u5377\u79ef\u7279\u5f81\u589e\u5f3a<\/h4>\n\n\n\n<p>\u8bba\u6587\u63d0\u51fa\u4e86\u4e00\u79cd\u65b0\u578b\u7684\u5377\u79ef\u7279\u5f81\u589e\u5f3a\u7684\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\u6846\u67b6\uff08PINNs4Drops\uff09\uff0c\u7528\u4e8e\u4e09\u7ef4\u91cd\u5efa\u6c14\u6db2\u754c\u9762\u53ca\u5176\u76f8\u5e94\u7684\u901f\u5ea6\u573a\u548c\u538b\u529b\u573a\u3002PINNs\u7ed3\u5408\u4e86\u6df1\u5ea6\u5b66\u4e60\u548c\u4f20\u7edf\u7269\u7406\u65b9\u7a0b\uff08\u5982\u4e0d\u53ef\u538b\u7f29\u7684Navier-Stokes\u65b9\u7a0b\uff09\u7684\u4f18\u70b9\uff0c\u5176\u4e3b\u8981\u7279\u70b9\u662f\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u7269\u7406\u4fe1\u606f\u878d\u5408<\/strong>\uff1a\u901a\u8fc7\u5c06\u7269\u7406\u63a7\u5236\u65b9\u7a0b\u7eb3\u5165\u7f51\u7edc\u8bad\u7ec3\uff0cPINNs\u786e\u4fdd\u4e86\u7f51\u7edc\u9884\u6d4b\u7b26\u5408\u6d41\u4f53\u7684\u7269\u7406\u89c4\u5f8b\uff0c\u5373\u4f7f\u5728\u6570\u636e\u4e0d\u5b8c\u6574\u6216\u53d7\u566a\u58f0\u5f71\u54cd\u7684\u60c5\u51b5\u4e0b\uff0c\u4ecd\u53ef\u8fdb\u884c\u53ef\u9760\u7684\u9884\u6d4b\u3002<\/li>\n\n\n\n<li><strong>\u5377\u79ef\u7279\u5f81\u63d0\u53d6<\/strong>\uff1a\u91c7\u7528\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u6765\u5904\u7406\u8f93\u5165\u56fe\u50cf\uff0c\u5229\u7528\u5377\u79ef\u5c42\u63d0\u53d6\u5f71\u50cf\u4e2d\u7684\u7a7a\u95f4\u7ed3\u6784\u7279\u5f81\uff0c\u8fd9\u4e9b\u7279\u5f81\u4e3a\u6c14\u6db2\u754c\u9762\u548c\u5185\u90e8\u6d41\u573a\u7684\u91cd\u5efa\u63d0\u4f9b\u4e86\u66f4\u4e30\u5bcc\u7684\u4e0a\u4e0b\u6587\u4fe1\u606f\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u8bba\u6587\u4e2d\u63d0\u51fa\u7684\u6846\u67b6\u88ab\u79f0\u4e3a\u201c\u5377\u79ef\u7279\u5f81\u589e\u5f3a\u7684PINNs\u201d\uff0c\u5176\u5728\u4f20\u7edfPINNs\u7684\u57fa\u7840\u4e0a\u589e\u52a0\u4e86\u5377\u79ef\u7279\u5f81\u63d0\u53d6\u6a21\u5757\uff0c\u4f7f\u5f97\u7f51\u7edc\u53ef\u4ee5\u5728\u56fe\u50cf\u6570\u636e\u4e0a\u8fdb\u884c\u66f4\u6df1\u5165\u7684\u5b66\u4e60\uff0c\u4ece\u800c\u63d0\u9ad8\u5bf9\u6c14\u6db2\u754c\u9762\u590d\u6742\u51e0\u4f55\u5f62\u72b6\u7684\u91cd\u5efa\u80fd\u529b\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">3. \u65b9\u6cd5\u8be6\u89e3<\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">3.1 \u6570\u636e\u751f\u6210\u4e0e\u5b9e\u9a8c\u914d\u7f6e<\/h5>\n\n\n\n<p>\u672c\u7814\u7a76\u901a\u8fc7\u76f4\u63a5\u6570\u503c\u6a21\u62df\uff08DNS\uff09\u751f\u6210\u4e86\u7528\u4e8e\u8bad\u7ec3\u548c\u9a8c\u8bc1\u7684\u6570\u636e\u96c6\u3002\u7814\u7a76\u4e2d\u9996\u5148\u4f7f\u7528\u6570\u503c\u6a21\u62df\u5f97\u5230\u6db2\u6ef4\u649e\u51fb\u8fc7\u7a0b\u7684\u4e09\u7ef4\u51e0\u4f55\u548c\u6d41\u573a\u6570\u636e\uff0c\u7136\u540e\u57fa\u4e8e\u8fd9\u4e9b\u6570\u636e\u751f\u6210\u5408\u6210\u56fe\u50cf\u3002\u8fd9\u4e9b\u5408\u6210\u56fe\u50cf\u7684\u751f\u6210\u65b9\u6cd5\u5305\u62ec\u5728\u6e32\u67d3\u73af\u5883\u4e2d\u518d\u73b0\u5b9e\u9a8c\u4e2d\u7684\u5149\u5b66\u914d\u7f6e\uff0c\u901a\u8fc7\u7269\u7406\u7cbe\u786e\u7684\u5149\u7ebf\u8ffd\u8e2a\u6e32\u67d3\u751f\u6210\u5305\u542b\u53cd\u5c04\u70b9\u7684\u5f71\u50cf\uff0c\u6a21\u62df\u4e86\u6db2\u6ef4\u754c\u9762\u7684\u53cd\u5c04\u548c\u6298\u5c04\u5149\u5b66\u7279\u5f81\u3002<\/p>\n\n\n\n<p>\u5b9e\u9a8c\u91c7\u7528\u7684\u5f71\u50cf\u83b7\u53d6\u65b9\u6cd5\u4e3a<strong>\u5f69\u8272\u7f16\u7801\u6697\u573a\u6210\u50cf<\/strong>\u3002\u8fd9\u4e00\u65b9\u6cd5\u901a\u8fc7\u5728\u4e0d\u540c\u4f4d\u7f6e\u8bbe\u7f6e\u4e0d\u540c\u989c\u8272\u7684\u5149\u6e90\uff0c\u4f7f\u5f97\u5728\u6db2\u6ef4\u754c\u9762\u4e0a\u5f62\u6210\u4e0d\u540c\u989c\u8272\u7684\u53cd\u5c04\u70b9\uff0c\u4ece\u800c\u63d0\u4f9b\u989d\u5916\u7684\u4e09\u7ef4\u4fe1\u606f\u3002\u8fd9\u4e9b\u53cd\u5c04\u70b9\u7684\u989c\u8272\u548c\u4f4d\u7f6e\u80fd\u591f\u5e2e\u52a9\u7f51\u7edc\u7406\u89e3\u6db2\u6ef4\u754c\u9762\u7684\u51e0\u4f55\u4fe1\u606f\uff0c\u4e3a\u4e09\u7ef4\u91cd\u5efa\u63d0\u4f9b\u66f4\u4e30\u5bcc\u7684\u8f93\u5165\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">3.2 \u7f51\u7edc\u67b6\u6784<\/h5>\n\n\n\n<p>\u7f51\u7edc\u67b6\u6784\u7531\u4ee5\u4e0b\u51e0\u4e2a\u90e8\u5206\u7ec4\u6210\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u5377\u79ef\u7279\u5f81\u63d0\u53d6\u7f51\u7edc<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u8f93\u5165\u4e3a\u6697\u573a\u6210\u50cf\u83b7\u53d6\u7684\u56fe\u50cf\u3002<\/li>\n\n\n\n<li>\u4f7f\u7528\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff08CNN\uff09\u6765\u63d0\u53d6\u50cf\u7d20\u5bf9\u9f50\u7684\u7279\u5f81\u3002\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u901a\u8fc7\u9010\u5c42\u7684\u5377\u79ef\u64cd\u4f5c\uff0c\u80fd\u591f\u8bc6\u522b\u51fa\u56fe\u50cf\u4e2d\u7684\u5173\u952e\u6a21\u5f0f\u548c\u7279\u5f81\uff0c\u4f8b\u5982\u6db2\u6ef4\u754c\u9762\u7684\u8f6e\u5ed3\u548c\u5149\u5b66\u53cd\u5c04\u70b9\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u591a\u5c42\u611f\u77e5\u673a\uff08MLP\uff09<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u5c06\u63d0\u53d6\u5230\u7684\u5377\u79ef\u7279\u5f81\u4e0e\u7a7a\u95f4\uff08x, y, z\uff09\u548c\u65f6\u95f4\uff08t\uff09\u5750\u6807\u4e00\u8d77\u8f93\u5165\u5230\u591a\u5c42\u611f\u77e5\u673a\u3002<\/li>\n\n\n\n<li>\u591a\u5c42\u611f\u77e5\u673a\u8d1f\u8d23\u9884\u6d4b\u6c14\u6db2\u76f8\u5206\u5e03\uff08\u03d5\uff09\u3001\u4e09\u7ef4\u901f\u5ea6\u5206\u91cf\uff08u, v, w\uff09\u4ee5\u53ca\u538b\u529b\uff08p\uff09\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u7269\u7406\u4fe1\u606f\u7f51\u7edc<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u4f7f\u7528\u81ea\u52a8\u5fae\u5206\u7684\u65b9\u5f0f\u8ba1\u7b97\u9884\u6d4b\u8f93\u51fa\u76f8\u5bf9\u4e8e\u7a7a\u95f4\u548c\u65f6\u95f4\u5750\u6807\u7684\u5bfc\u6570\u3002<\/li>\n\n\n\n<li>\u901a\u8fc7\u8fd9\u4e9b\u5bfc\u6570\uff0c\u8ba1\u7b97\u7269\u7406\u63a7\u5236\u65b9\u7a0b\u7684\u6b8b\u5dee\uff0c\u5e76\u5c06\u5176\u4f5c\u4e3a\u7269\u7406\u4fe1\u606f\u635f\u5931\u9879\uff0c\u786e\u4fdd\u9884\u6d4b\u7b26\u5408\u4e0d\u53ef\u538b\u7f29Navier-Stokes\u65b9\u7a0b\u3001\u8fde\u7eed\u6027\u65b9\u7a0b\u4ee5\u53ca\u754c\u9762\u6f14\u5316\u65b9\u7a0b\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<h5 class=\"wp-block-heading\">3.3 \u7269\u7406\u63a7\u5236\u65b9\u7a0b\u4e0e\u635f\u5931\u51fd\u6570<\/h5>\n\n\n\n<p>\u4e3a\u4e86\u786e\u4fdd\u7f51\u7edc\u8f93\u51fa\u7b26\u5408\u6d41\u4f53\u7684\u7269\u7406\u89c4\u5f8b\uff0c\u7814\u7a76\u8005\u5c06\u4e0d\u53ef\u538b\u7f29\u7684\u5355\u573aNavier-Stokes\u65b9\u7a0b\u3001\u8fde\u7eed\u6027\u65b9\u7a0b\u548c\u754c\u9762\u6f14\u5316\u65b9\u7a0b\u5d4c\u5165\u5230\u7f51\u7edc\u7684\u635f\u5931\u51fd\u6570\u4e2d\u3002\u8fd9\u4e9b\u65b9\u7a0b\u5177\u4f53\u4e3a\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Navier-Stokes\u65b9\u7a0b<\/strong>\uff1a\u7528\u4e8e\u63cf\u8ff0\u901f\u5ea6\u573a\u548c\u538b\u529b\u573a\u7684\u6f14\u53d8\u3002<\/li>\n\n\n\n<li><strong>\u8fde\u7eed\u6027\u65b9\u7a0b<\/strong>\uff1a\u7528\u4e8e\u786e\u4fdd\u6d41\u4f53\u5728\u7a7a\u95f4\u4e2d\u7684\u5b88\u6052\u6027\uff0c\u5373\u4e0d\u53ef\u538b\u7f29\u6027\u3002<\/li>\n\n\n\n<li><strong>\u754c\u9762\u6f14\u5316\u65b9\u7a0b<\/strong>\uff1a\u91c7\u7528\u4f53\u79ef\u6db2\u4f53\u6cd5\uff08Volume of Fluid, VoF\uff09\u548c\u76f8\u573a\u6cd5\uff08Phase-field\uff09\u4e24\u79cd\u65b9\u5f0f\u63cf\u8ff0\u6c14\u6db2\u754c\u9762\u7684\u53d8\u5316\u3002\u4f53\u79ef\u6db2\u4f53\u6cd5\u901a\u8fc7\u4f53\u79ef\u5206\u6570\u03b1\u63cf\u8ff0\u6d41\u4f53\u754c\u9762\uff0c\u03b1\u53d6\u503c\u4e3a0\u52301\u4e4b\u95f4\uff0c\u8868\u793a\u4e0d\u540c\u76f8\u5728\u7a7a\u95f4\u5355\u5143\u4e2d\u7684\u5360\u6bd4\uff1b\u76f8\u573a\u6cd5\u5219\u901a\u8fc7\u76f8\u4f4d\u53d8\u91cfC\u8868\u793a\u754c\u9762\uff0cC\u7684\u503c\u5728-1\u548c1\u4e4b\u95f4\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u7269\u7406\u635f\u5931\u9879\u662f\u901a\u8fc7\u81ea\u52a8\u5fae\u5206\u8ba1\u7b97\u5f97\u5230\u7684\uff0c\u7f51\u7edc\u7684\u76ee\u6807\u662f\u6700\u5c0f\u5316\u8fd9\u4e9b\u7269\u7406\u635f\u5931\u9879\u4ee5\u53ca\u57fa\u4e8e\u5b9e\u9a8c\u6570\u636e\u7684\u76d1\u7763\u635f\u5931\u9879\uff0c\u4ee5\u5b9e\u73b0\u5bf9\u4e24\u76f8\u6d41\u52a8\u7684\u7cbe\u786e\u91cd\u5efa\u3002<\/p>\n\n\n\n<p>\u4e3a\u4e86\u63d0\u9ad8\u91cd\u5efa\u7684\u7cbe\u5ea6\uff0c\u8bba\u6587\u4e2d\u8fd8\u63d0\u5230\u4e86\u4e00\u4e9b\u6280\u672f\u7ec6\u8282\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u91c7\u6837\u65b9\u6cd5<\/strong>\uff1a\u9488\u5bf9\u6c14\u6db2\u754c\u9762\u7684\u4f4d\u7f6e\uff0c\u7814\u7a76\u8005\u5728\u754c\u9762\u9644\u8fd1\u8fdb\u884c\u4e86\u91c7\u6837\u70b9\u7684\u5bc6\u96c6\u5316\u5904\u7406\uff0c\u4ee5\u4fbf\u66f4\u597d\u5730\u6355\u6349\u754c\u9762\u9644\u8fd1\u7684\u590d\u6742\u7269\u7406\u7279\u6027\u3002<\/li>\n\n\n\n<li><strong>\u635f\u5931\u6743\u91cd\u52a8\u6001\u8c03\u6574<\/strong>\uff1a\u91c7\u7528\u56fa\u5b9a\u6743\u91cd\u548c\u81ea\u9002\u5e94\u6743\u91cd\u76f8\u7ed3\u5408\u7684\u65b9\u6cd5\uff0c\u52a8\u6001\u5730\u5e73\u8861\u7269\u7406\u635f\u5931\u4e0e\u6570\u636e\u9a71\u52a8\u7684\u635f\u5931\uff0c\u786e\u4fdd\u7f51\u7edc\u80fd\u591f\u6709\u6548\u5730\u540c\u65f6\u5b66\u4e60\u6570\u636e\u548c\u7269\u7406\u89c4\u5f8b\u3002<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">4. \u7ed3\u679c\u4e0e\u8ba8\u8bba<\/h4>\n\n\n\n<h5 class=\"wp-block-heading\">4.1 \u5408\u6210\u6570\u636e\u9a8c\u8bc1<\/h5>\n\n\n\n<p>\u7814\u7a76\u9996\u5148\u5728\u5408\u6210\u6570\u636e\u96c6\u4e0a\u9a8c\u8bc1\u4e86\u6240\u63d0\u51fa\u7684PINNs4Drops\u6a21\u578b\u7684\u6709\u6548\u6027\u3002\u5408\u6210\u6570\u636e\u662f\u57fa\u4e8e\u6570\u503c\u6a21\u62df\u7684\u7ed3\u679c\uff0c\u901a\u8fc7\u6e32\u67d3\u6280\u672f\u751f\u6210\u7684\uff0c\u56e0\u6b64\u65e2\u5305\u542b\u771f\u5b9e\u5b9e\u9a8c\u7684\u590d\u6742\u5149\u5b66\u7279\u6027\uff0c\u53c8\u5177\u6709\u6570\u503c\u6a21\u62df\u7684\u7cbe\u786e\u6027\u3002PINNs4Drops\u5bf9\u4e09\u7ef4\u6c14\u6db2\u754c\u9762\u7684\u91cd\u5efa\u5728\u4e09\u7ef4\u4ea4\u5e76\u6bd4\uff083D-IOU\uff09\u65b9\u9762\u53d6\u5f97\u4e86\u663e\u8457\u7684\u63d0\u5347\uff0c\u8fbe\u5230\u4e860.967\uff0c\u76f8\u6bd4\u4e8e\u57fa\u7ebf\u7684\u7eaf\u6570\u636e\u9a71\u52a8\u65b9\u6cd5\u63d0\u9ad8\u4e860.9%\u3002<\/p>\n\n\n\n<p>\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u4f7f\u7528\u4f53\u79ef\u6db2\u4f53\u6cd5\uff08VOF\uff09\u4f5c\u4e3a\u754c\u9762\u6f14\u5316\u65b9\u6cd5\u7684PINNs\u76f8\u8f83\u4e8e\u76f8\u573a\u6cd5\u5728\u754c\u9762\u91cd\u5efa\u7cbe\u5ea6\u4e0a\u8868\u73b0\u66f4\u4f18\u3002\u8fd9\u4e3b\u8981\u662f\u56e0\u4e3a\u76f8\u573a\u65b9\u6cd5\u5728\u5904\u7406\u9ad8\u5bc6\u5ea6\u6bd4\u65f6\u7a33\u5b9a\u6027\u8f83\u5dee\uff0c\u800cVOF\u65b9\u6cd5\u901a\u8fc7\u5f15\u5165\u4f53\u79ef\u5206\u6570\u7684\u5bf9\u6d41\u65b9\u7a0b\uff0c\u80fd\u591f\u6709\u6548\u5730\u907f\u514d\u9ad8\u5bc6\u5ea6\u6bd4\u4e0b\u7684\u6570\u503c\u53d1\u6563\u95ee\u9898\u3002<\/p>\n\n\n\n<h5 class=\"wp-block-heading\">4.2 \u4e09\u7ef4\u6d41\u573a\u9884\u6d4b<\/h5>\n\n\n\n<p>PINNs4Drops\u4e0d\u4ec5\u53ef\u4ee5\u91cd\u5efa\u6c14\u6db2\u754c\u9762\uff0c\u8fd8\u80fd\u591f\u9884\u6d4b\u6db2\u6ef4\u5185\u90e8\u548c\u5468\u56f4\u7684\u4e09\u7ef4\u901f\u5ea6\u573a\u548c\u538b\u529b\u573a\u3002\u5728\u901f\u5ea6\u573a\u7684\u9884\u6d4b\u4e2d\uff0c\u7f51\u7edc\u5bf9\u4e8e\u6db2\u6ef4\u5185\u90e8\u7684\u901f\u5ea6\u5206\u5e03\u7cbe\u5ea6\u8f83\u9ad8\uff0c\u800c\u5bf9\u4e8e\u6c14\u76f8\u533a\u57df\u7684\u9884\u6d4b\u7cbe\u5ea6\u7565\u4f4e\u3002\u9020\u6210\u8fd9\u79cd\u5dee\u5f02\u7684\u4e3b\u8981\u539f\u56e0\u662f\u6db2\u76f8\u533a\u57df\u7684\u7269\u7406\u635f\u5931\u9879\u8d21\u732e\u66f4\u5927\uff0c\u4ece\u800c\u66f4\u597d\u5730\u6307\u5bfc\u4e86\u6db2\u76f8\u5185\u7684\u6d41\u573a\u91cd\u5efa\u3002\u6b64\u5916\uff0c\u6c14\u6db2\u754c\u9762\u9644\u8fd1\u7684\u9884\u6d4b\u7cbe\u5ea6\u666e\u904d\u8f83\u9ad8\uff0c\u8fd9\u5f97\u76ca\u4e8e\u5728\u754c\u9762\u9644\u8fd1\u8fdb\u884c\u7684\u81ea\u9002\u5e94\u91c7\u6837\u548c\u6b8b\u5dee\u52a0\u6743\u7b56\u7565\u3002<\/p>\n\n\n\n<p>\u901a\u8fc7\u6bd4\u8f83\u4e0d\u540c\u6a21\u578b\u7684\u9884\u6d4b\u7ed3\u679c\uff0c\u53d1\u73b0VOF-PINNs\u5728\u9884\u6d4b\u7cbe\u5ea6\u65b9\u9762\u6574\u4f53\u4f18\u4e8e\u57fa\u7ebf\u7684\u7eaf\u6570\u636e\u9a71\u52a8\u6a21\u578b\u548c\u76f8\u573a\u6cd5\u7684PINNs\uff0c\u5c24\u5176\u662f\u5728\u901f\u5ea6\u548c\u538b\u529b\u573a\u7684\u9884\u6d4b\u4e0a\u5177\u6709\u66f4\u597d\u7684\u62d3\u6251\u4e00\u81f4\u6027\u548c\u6570\u503c\u7cbe\u5ea6\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">5. 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class=\"wp-block-heading\">6. \u7ed3\u8bba<\/h4>\n\n\n\n<p>\u672c\u7814\u7a76\u63d0\u51fa\u4e86\u4e00\u79cd\u5377\u79ef\u7279\u5f81\u589e\u5f3a\u7684\u7269\u7406\u4fe1\u606f\u795e\u7ecf\u7f51\u7edc\uff08PINNs4Drops\uff09\uff0c\u6210\u529f\u5b9e\u73b0\u4e86\u4ece\u5355\u89c6\u89d2\u6697\u573a\u6210\u50cf\u6570\u636e\u4e2d\u5bf9\u4e24\u76f8\u6d41\u52a8\u7684\u4e09\u7ef4\u91cd\u5efa\u548c\u6d41\u573a\u9884\u6d4b\u3002\u76f8\u6bd4\u4e8e\u4f20\u7edf\u7684\u5b9e\u9a8c\u6d4b\u91cf\u65b9\u6cd5\uff0c\u8be5\u65b9\u6cd5\u5728\u6570\u636e\u9700\u6c42\u3001\u8ba1\u7b97\u6210\u672c\u548c\u9884\u6d4b\u7cbe\u5ea6\u4e4b\u95f4\u627e\u5230\u4e86\u5e73\u8861\u3002\u5176\u4e3b\u8981\u8d21\u732e\u5728\u4e8e\uff1a<\/p>\n\n\n\n<ul 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