{"id":2410,"date":"2024-12-03T21:50:54","date_gmt":"2024-12-03T13:50:54","guid":{"rendered":"https:\/\/www.gnn.club\/?p=2410"},"modified":"2025-06-24T14:21:52","modified_gmt":"2025-06-24T06:21:52","slug":"egnn","status":"publish","type":"post","link":"http:\/\/gnn.club\/?p=2410","title":{"rendered":"EGNN"},"content":{"rendered":"\n<p>EGNN\uff08Equivariant Graph Neural Network\uff09\u662f\u4e00\u79cd\u7ecf\u5178\u7684\u5177\u6709\u4fdd\u6301\u7b49\u53d8\u6027\uff08Equivariance\uff09\u7684\u56fe\u795e\u7ecf\u7f51\u7edc\u3002\u8fd9\u610f\u5473\u7740\u5982\u679c\u8f93\u5165\u6570\u636e\u901a\u8fc7\u67d0\u79cd\u53d8\u6362\uff08\u5982\u65cb\u8f6c\u3001\u7ffb\u8f6c\u7b49\uff09\uff0c\u7f51\u7edc\u7684\u8f93\u51fa\u4e5f\u4f1a\u4ee5\u76f8\u540c\u7684\u65b9\u5f0f\u53d8\u6362\uff0c\u4ece\u800c\u4fdd\u8bc1\u8f93\u51fa\u4e0e\u8f93\u5165\u5728\u51e0\u4f55\u4e0a\u662f\u4e00\u81f4\u7684\u3002\u8fd9\u4e00\u7279\u6027\u4f7f\u5f97EGNN\u5728\u5904\u7406\u51e0\u4f55\u6570\u636e\u548c\u7269\u7406\u7cfb\u7edf\u6a21\u62df\u4e2d\u8868\u73b0\u51fa\u4f18\u5f02\u7684\u80fd\u529b\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1.EGNN\u7b49\u53d8\u6027\u7684\u5f15\u5165\u65b9\u5f0f<\/h3>\n\n\n\n<p>\u5728EGNN\u8fd9\u7bc7\u5de5\u4f5c\u4e2d\uff0c\u9996\u5148\u5bf9GNNs\u7684\u6846\u67b6\u8fdb\u884c\u4e86\u516c\u5f0f\u5316\u7684\u5b9a\u4e49\u3002\u5177\u4f53\u6765\u8bf4\u5176\u6846\u67b6\u5206\u6210\u4e09\u90e8\u5206\uff1a\u8fb9\u7279\u5f81\u7684\u5b9a\u4e49\uff08Edge\uff09\u3001\u8fb9\u7279\u5f81\u7684\u805a\u5408\uff08Agg\uff09\u548c\u8282\u70b9\u7279\u5f81\u7684\u66f4\u65b0\uff08Node\uff09\u3002 <\/p>\n\n\n\n<p>\u5bf9\u4e8e\u6734\u7d20\u7684GNN\u6a21\u578b\u800c\u8a00\uff0c\u8fb9\u7279\u5f81\uff08Edge\uff09\u7684\u516c\u5f0f\u53ef\u4ee5\u5b9a\u4e49\u4e3a\\(m_{ij} = \\phi_{e}\\left( h_{i}^{l}, h_{j}^{l}, a_{ij} \\right)\\)\uff0c\u7f51\u7edc\u4f1a\u4e3a\u6bcf\u5bf9\u76f8\u8fde\u7684\u8282\u70b9\uff08 \\(\\mathrm{i}\\) \u548c \\(\\mathrm{j}\\) \uff09\u8ba1\u7b97\u4e00\u4e2a\u6d88\u606f\u5411\u91cf\\(m_{ij}\\) \u3002\u8fd9\u4e2a\u8ba1\u7b97\u662f\u901a\u8fc7\u4e00\u4e2a\u7279\u5b9a\u7684\u51fd\u6570\uff08\u4e00\u822c\u662f\u795e\u7ecf\u7f51\u7edc\u5c42\u7684\u6620\u5c04\uff09\\(\\phi_{e}\\)\u5b8c\u6210\u7684\uff0c\u8be5\u51fd\u6570\u53d6\u51b3\u4e8e\u4e24\u4e2a\u8282\u70b9\u7684\u7279\u5f81\u5411\u91cf\\(\\boldsymbol{h}_i^l\\) \u548c\\(\\boldsymbol{h}_j^l\\) \uff0c\u4ee5\u53ca\u5b83\u4eec\u4e4b\u95f4\u7684\u8fb9\u7279\u5f81\\(a_{ij}\\) \u3002\u8fd9\u91cc\u7684\\(\\mathrm{l}\\) \u6307\u7684\u662f\u5f53\u524d\u5c42\uff0c\u6b64\u51fd\u6570\u662f\u8bbe\u8ba1\u6765\u6355\u6349\u8282\u70b9\u95f4\u7684\u76f4\u63a5\u5173\u7cfb\u548c\u4ea4\u4e92\u7684\uff0c\u53ef\u4ee5\u7406\u89e3\u4e3a\u4e00\u4e2a\u7f16\u7801\u8282\u70b9\u95f4\u5173\u7cfb\u7684\u65b9\u5f0f\u3002<\/p>\n\n\n\n<p>\u8fb9\u7279\u5f81\u7684\u805a\u5408\uff08Agg\uff09\u516c\u5f0f\u53ef\u4ee5\u5b9a\u4e49\u4e3a\\(m_i=\\sum_{j\\in\\mathcal{N}(i)}m_{ij}\\)\u3002\u5728\u805a\u5408\u6b65\u9aa4\u4e2d\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u8282\u70b9\\(\\mathrm{i}\\)  \uff0c\u5b83\u4ece\u5176\u6240\u6709\u90bb\u5c45\u8282\u70b9\\(\\mathrm{j}\\)\uff08\u8868\u793a\u4e3a\\(\\mathcal{N}(i)\\)\uff09\u6536\u96c6\u6d88\u606f\u5411\u91cf\\(m_{ij}\\)\uff0c\u5e76\u5c06\u8fd9\u4e9b\u6d88\u606f\u5411\u91cf\u6c42\u548c\uff0c\u5f97\u5230\\(m_{i}\\)\u3002\u8fd9\u4e00\u6b65\u7684\u76ee\u7684\u662f\u5c06\u4ece\u5404\u4e2a\u90bb\u5c45\u90a3\u91cc\u6536\u96c6\u5230\u7684\u4fe1\u606f\u6c47\u603b\u5230\u5355\u4e2a\u8282\u70b9\u4e0a\uff0c\u4ece\u800c\u4e3a\u4e0b\u4e00\u6b65\u7684\u8282\u70b9\u72b6\u6001\u66f4\u65b0\u63d0\u4f9b\u5fc5\u8981\u7684\u4fe1\u606f\u8f93\u5165\u3002<\/p>\n\n\n\n<p>\u8282\u70b9\u7279\u5f81\u7684\u66f4\u65b0\uff08Node\uff09\u516c\u5f0f\u53ef\u4ee5\u5b9a\u4e49\u4e3a\\(h_i^{l+1}=\\phi_h\\left(h_i^l,m_i\\right)\\)\u3002\u5728\u8282\u70b9\u66f4\u65b0\u6b65\u9aa4\u4e2d\uff0c\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u65b0\u7684\u9690\u85cf\u72b6\u6001\\(\\boldsymbol{h}_i^{l+1}\\)\u662f\u901a\u8fc7\u53e6\u4e00\u4e2a\u51fd\u6570\\(\\phi_{h}\\)\uff08\u4e00\u822c\u4e5f\u662f\u795e\u7ecf\u7f51\u7edc\uff09\u6765\u8ba1\u7b97\u7684\uff0c\u8be5\u51fd\u6570\u53d6\u51b3\u4e8e\u8282\u70b9\\(\\mathrm{i}\\)\u5f53\u524d\u7684\u9690\u85cf\u72b6\u6001\\(\\boldsymbol{h}_i^l\\) \u548c\u805a\u5408\u540e\u7684\u6d88\u606f\\(\\mathrm{i}\\)\u3002\u8fd9\u4e00\u6b65\u9aa4\u662f\u6a21\u578b\u5bf9\u6bcf\u4e2a\u8282\u70b9\u8fdb\u884c\u72b6\u6001\u66f4\u65b0\uff0c\u5b83\u4f7f\u5f97\u8282\u70b9\u80fd\u591f\u6839\u636e\u81ea\u5df1\u548c\u5468\u56f4\u90bb\u5c45\u7684\u4fe1\u606f\u8fdb\u884c\u81ea\u6211\u66f4\u65b0\u3002<\/p>\n\n\n\n<p>\u5728\u4e0a\u8ff0\u5b9a\u4e49\u7684\u56fe\u795e\u7ecf\u7f51\u7edc\uff08GNN\uff09\u7684\u6807\u51c6\u6846\u67b6\u4e2d\uff0c\u53ef\u4ee5\u53d1\u73b0\u5176\u8bbe\u8ba1\u5e76\u6ca1\u6709\u663e\u5f0f\u5730\u4fdd\u8bc1\u6a21\u578b\u8f93\u51fa\u5bf9\u8282\u70b9\u8f93\u5165\u7684\u51e0\u4f55\u53d8\u6362\u5177\u6709\u4e0d\u53d8\u6027\u6216\u7b49\u53d8\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5728Edge\u6b65\u9aa4\u4e2d\uff0c\\(\\phi_{e}\\)\u901a\u5e38\u662f\u4e2a\u666e\u901a\u7684\u795e\u7ecf\u7f51\u7edc\u5c42\uff0c\u5b83\u5904\u7406\u8f93\u5165\u7279\u5f81\u4f46\u4e0d\u4f1a\u76f4\u63a5\u8003\u8651\u8fd9\u4e9b\u7279\u5f81\u5982\u4f55\u5728\u7a7a\u95f4\u53d8\u6362\uff08\u5982\u65cb\u8f6c\u6216\u5e73\u79fb\uff09\u4e0b\u53d8\u5316\u3002\u4e5f\u5c31\u662f\u8bf4\uff0c\u5982\u679c\u56fe\u7684\u6574\u4f53\u7ed3\u6784\u7ecf\u5386\u4e86\u67d0\u79cd\u51e0\u4f55\u53d8\u6362\uff0c\\(m_{ij}\\) \u7684\u8ba1\u7b97\u65b9\u5f0f\u5e76\u6ca1\u6709\u76f4\u63a5\u673a\u5236\u6765\u4fdd\u8bc1\u76f8\u540c\u7684\u53d8\u6362\u53cd\u6620\u5728\u8f93\u51fa\u6d88\u606f\u4e2d\u3002\u540c\u7406\uff0c\u5728\u8282\u70b9\u7279\u5f81\u66f4\u65b0\u9636\u6bb5\u4e5f\u662f\u7c7b\u4f3c\u7684\u60c5\u51b5\u3002\u6700\u540e\uff0c\u5206\u6790\u5176\u805a\u5408\u6b65\u9aa4\uff0c\u4e00\u822c\u662f\u7b80\u5355\u7684\u7d2f\u52a0\u6240\u6709\u90bb\u5c45\u7684\u4fe1\u606f\u3002\u867d\u7136\u8fd9\u4e2a\u6b65\u9aa4\u5728\u6570\u5b66\u4e0a\u662f\u7f6e\u6362\u4e0d\u53d8\u7684\uff08\u5373\u8282\u70b9\u7684\u987a\u5e8f\u4e0d\u4f1a\u5f71\u54cd\u6c42\u548c\u7ed3\u679c\uff09\uff0c\u4f46\u8fd9\u5e76\u4e0d\u610f\u5473\u7740\u5b83\u5bf9\u51e0\u4f55\u53d8\u6362\u662f\u4e0d\u53d8\u7684\u6216\u7b49\u53d8\u7684\u3002\u805a\u5408\u64cd\u4f5c\u5ffd\u7565\u4e86\u8282\u70b9\u95f4\u7684\u7a7a\u95f4\u5173\u7cfb\u6216\u5176\u5728\u56fe\u4e2d\u7684\u7edd\u5bf9\u4f4d\u7f6e\uff0c\u56e0\u6b64\u5bf9\u4e8e\u56fe\u7684\u7a7a\u95f4\u53d8\u6362\u5e76\u4e0d\u654f\u611f\u3002<\/p>\n\n\n\n<p>\u7b49\u53d8\u6027\u548c\u4e0d\u53d8\u6027\u901a\u5e38\u9700\u8981\u901a\u8fc7\u989d\u5916\u7684\u7ed3\u6784\u8bbe\u8ba1\u6765\u5b9e\u73b0\uff0c\u5373\u7b97\u6cd5\u7684\u5f52\u7eb3\u504f\u7f6e\u3002\u4ee5\u4e4b\u524d\u8bb2\u89e3\u8fc7\u7684SchNet\u6a21\u578b\u4e3e\u4f8b\uff0c\u5f53\u4f7f\u7528\u4e0a\u8ff0EGNN\u5bf9GNNs\u7b97\u6cd5\u6846\u67b6\u7684\u5b9a\u4e49\u6765\u63cf\u8ff0SchNet\u65f6\uff0c\u4e0e\u4f20\u7edfGNN\u6700\u5927\u7684\u4e0d\u540c\u4f53\u73b0\u5728\u8fb9\u7279\u5f81\uff08Edge\uff09\u7684\u5b9a\u4e49\u6b65\u9aa4\u4e2d\uff0c\u516c\u5f0f\u53ef\u8868\u793a\u6210\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\boldsymbol{m}_{ij}=\\phi_{\\mathrm{cf}}\\left(\\left\\|\\boldsymbol{r}_{ij}^l\\right\\|\\right)\\phi_{\\mathrm{s}}\\left(\\boldsymbol{h}_j^l\\right)\\)\u00a0 \u00a0 \u00a0\uff081\uff09<\/p>\n\n\n<p>\u53ef\u4ee5\u770b\u51fa\uff0c\u6b64\u65f6\u8fb9\u7279\u5f81\u7684\u5b9a\u4e49\u4e0d\u4ec5\u4ec5\u662f\u7b80\u5355\u5730\u4f9d\u8d56\u4e8e\u8282\u70b9\u7684\u7279\u5f81\u5411\u91cf\uff0c\u800c\u662f\u901a\u8fc7\u4e24\u4e2a\u5206\u5f00\u7684\u51fd\u6570\\(\\phi_{\\mathrm{cf}}\\)\u548c\\(\\phi_{\\mathrm{s}}\\)\u6765\u5b9a\u4e49\u3002\u5177\u4f53\u6765\u8bf4\uff1a \\(\\phi_{\\mathrm{cf}}\\left( \\left\\lVert r_{ij}^l \\right\\rVert \\right)\\)\u662f\u4e00\u4e2a\u8fde\u7eed\u6ee4\u6ce2\u5377\u79ef \uff08Continuous Filter Convolution\uff09\uff0c\u5b83\u4ec5\u57fa\u4e8e\u8282\u70b9\\(\\mathrm{i}\\) \u548c\\(\\mathrm{j}\\)\u4e4b\u95f4\u7684\u8ddd\u79bb\\(\\left|r_{ij}^l\\right|\\)\u6765\u8ba1\u7b97\u5f97\u5230\u5377\u79ef\u64cd\u4f5c\u7684\u6743\u91cd\uff0c\u5373\u5377\u79ef\u6838\u7684\u6570\u503c\u3002\u8fd9\u4e9b\u6743\u91cd\u662f\u4e13\u95e8\u8bbe\u8ba1\u6765\u6355\u6349\u539f\u5b50\u95f4\u8ddd\u79bb\u5bf9\u76f8\u4e92\u4f5c\u7528\u5f3a\u5ea6\u7684\u5f71\u54cd\uff0c\u8fd9\u4e00\u70b9\u5728\u5904\u7406\u5206\u5b50\u7ed3\u6784\u65f6\u6781\u5176\u91cd\u8981\uff0c\u5b83\u5141\u8bb8\u6a21\u578b\u5bf9\u8282\u70b9\u4e4b\u95f4\u7684\u7269\u7406\u8ddd\u79bb\u8fdb\u884c\u654f\u611f\u54cd\u5e94\u3002\\(\\phi_\\mathrm{s}\\left(\\boldsymbol{h}_j^l\\right)\\)\u5219\u5904\u7406\u6765\u81ea\u8282\u70b9\\(\\mathrm{j}\\)\uff08\u76ee\u6807\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u90bb\u5c45\u8282\u70b9\uff09\u7684\u7279\u5f81\u5411\u91cf\uff0c\u901a\u8fc7\u795e\u7ecf\u7f51\u7edc\u5c42\u6620\u5c04\u5f97\u5230\u7684\u65b0\u7279\u5f81\u4e0e\u8ddd\u79bb\u6743\u91cd \\(\\phi_{\\mathrm{cf}}\\left( \\left\\lVert r_{ij}^l \\right\\rVert \\right)\\)\u76f8\u4e58\u5f97\u5230\u66f4\u65b0\u540e\u7684\u8fb9\u7279\u5f81\\(m_{ij}\\) \u3002<\/p>\n\n\n\n<p>\u901a\u8fc7\u5c06\u539f\u5b50\u95f4\u7684\u8ddd\u79bb\u4f5c\u4e3a\u8fb9\u7684\u4e00\u4e2a\u5173\u952e\u7279\u5f81\uff0cSchNet\u6a21\u578b\u80fd\u591f\u6709\u6548\u5e94\u5bf9\u5316\u5b66\u7269\u8d28\u5206\u6790\u4e2d\u7684\u4e00\u4e2a\u4e3b\u8981\u6311\u6218\uff1a\u51c6\u786e\u6a21\u62df\u539f\u5b50\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u529b\u3002\u8fd9\u4e9b\u76f8\u4e92\u4f5c\u7528\u529b\u901a\u5e38\u5f3a\u70c8\u4f9d\u8d56\u4e8e\u539f\u5b50\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\u3002\u66f4\u91cd\u8981\u7684\u662f\uff0c\u7531\u4e8e\u8fd9\u4e9b\u8ddd\u79bb\u4fe1\u606f\u672c\u8d28\u4e0a\u662f\u6807\u91cf\u503c\uff0c\u5b83\u4eec\u5728\u5e73\u79fb\u3001\u65cb\u8f6c\u6216\u53cd\u5c04\u7b49\u51e0\u4f55\u53d8\u6362\u4e0b\u4fdd\u6301\u4e0d\u53d8\u3002\u56e0\u6b64\uff0c\u4e3aSchNet\u6a21\u578b\u63d0\u4f9b\u4e86\u65cb\u8f6c\u4e0d\u53d8\u6027\u7684\u5f52\u7eb3\u504f\u7f6e\uff0c\u4f7f\u5176\u5728\u5904\u7406\u51e0\u4f55\u53d8\u6362\u65f6\u5177\u6709\u7a33\u5b9a\u7684\u8868\u73b0\u3002<\/p>\n\n\n\n<p>\u7c7b\u4f3c\u7684\uff0cEGNN\u4e5f\u53ef\u4ee5\u4f7f\u7528\u4e0a\u8ff0GNNs\u6846\u67b6\u8fdb\u884c\u5b9a\u4e49\uff0c\u5982\u88681\u6240\u793a\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"428\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241203224143474-1024x428.png\" alt=\"\" class=\"wp-image-2435\" srcset=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241203224143474-1024x428.png 1024w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241203224143474-300x125.png 300w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241203224143474-768x321.png 768w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241203224143474.png 1064w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u88681 EGNN\u6846\u67b6\u8868\u793a<\/figcaption><\/figure><\/div>\n\n\n<p>\u6839\u636e\u88681\uff0c\u5177\u4f53\u89e3\u91ca\u4e00\u4e0bEGNN\u7684\u516c\u5f0f\u8868\u793a\u3002\u9996\u5148\uff0c\u516c\u5f0f\\(\\boldsymbol{m}_{ij} = \\phi_e\\left( \\boldsymbol{h}_i^{l}, \\boldsymbol{h}_j^{l}, \\left\\lVert \\boldsymbol{r}_{ij}^{l} \\right\\rVert^2, a_{ij} \\right)\\)\u63cf\u8ff0\u4e86\u5982\u4f55\u8ba1\u7b97\u8282\u70b9\\(\\mathrm{i}\\)\u548c\u8282\u70b9\\(\\mathrm{j}\\)\u4e4b\u95f4\u7684\u8fb9\u7684\u7279\u5f81\\(m_{ij}\\)\u3002\u5176\u4e2d, \\(\\phi_{e}\\)\u662f\u4e00\u4e2a\u53ef\u5b66\u4e60\u7684\u7f51\u7edc\u5c42\uff0c\\(\\boldsymbol{h}_i^l\\)\u548c\\(\\boldsymbol{h}_j^l\\)\u5206\u522b\u662f\u8282\u70b9\\(\\mathrm{i}\\)\u548c\u8282\u70b9\\(\\mathrm{j}\\)\u5728\u7b2c\\(\\mathrm{l}\\) \u5c42\u7684\u7279\u5f81\u8868\u793a, \\(\\left\\lVert \\boldsymbol{r}_{ij}^l \\right\\rVert^2\\)\u662f\u8282\u70b9\\(\\mathrm{i}\\)\u548c\u8282\u70b9\\(\\mathrm{j}\\)\u4e4b\u95f4\u7684\u8ddd\u79bb\u7684\u5e73\u65b9\uff08\u901a\u5e38\u662f\u6b27\u6c0f\u8ddd\u79bb\uff09\uff0c\\(a_{ij}\\) \u662f\u8fb9\u7684\u7279\u5f81\u6216\u7c7b\u578b\u3002\u8fd9\u4e9b\u4fe1\u606f\u4f1a\u62fc\u63a5\u8d77\u6765\u4e00\u8d77\u4f5c\u4e3a\u7f51\u7edc\u5c42\\(\\phi_{e}\\)\u7684\u8f93\u5165\u3002\u76f8\u8f83\u4e8e\u6807\u51c6\u7684GNN\uff0cEGNN\u591a\u8003\u8651\u4e86\u8ddd\u79bb\u4fe1\u606f \\(\\left\\lVert \\boldsymbol{r}_{ij}^l \\right\\rVert^2\\)\uff1b\u76f8\u8f83\u4e8eSchNet\uff0c\u5176\u8ddd\u79bb\u4fe1\u606f\u7684\u8003\u8651\u975e\u5e38\u7b80\u5355\u4e0e\u76f4\u63a5\uff0c\u5e76\u6ca1\u6709\u7ecf\u8fc7\u5f84\u5411\u51fd\u6570\u7684\u5904\u7406\u3002\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0cEGNN\u5728\u8fb9\u7279\u5f81\u5b9a\u4e49\u7684\u9636\u6bb5\u6bd4\u6807\u51c6GNN\u548cSchNet\u90fd\u591a\u4e86\u4e00\u79cd\u65b0\u7684\u8f93\u5165\u4fe1\u606f\uff1a\u5411\u91cf\u4fe1\u606f\\(\\hat{m}_{ij}\\)\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c31\u662f\u7b80\u5355\u7684\u5c06\u8282\u70b9\\(\\mathrm{i}\\)\u5230\u8282\u70b9\\(\\mathrm{j}\\)\u7684\u5411\u91cf\\(r_{ij}^{l}\\)\u76f4\u63a5\u4e0e\u7ecf\u8fc7\u795e\u7ecf\u7f51\u7edc\u5c42\\(\\phi_{x}\\)\u5904\u7406\u540e\u7684\u8fb9\u7279\u5f81\\(m_{ij}\\)\u8fdb\u884c\u76f8\u4e58\u64cd\u4f5c\uff0c\u8fd9\u79cd\u5c06\u5411\u91cf\u4fe1\u606f\u4e0e\u8fb9\u7279\u5f81\u7ed3\u5408\u7684\u65b9\u6cd5\u80fd\u591f\u6355\u6349\u8282\u70b9\u4e4b\u95f4\u7684\u5b9a\u5411\u5173\u7cfb\uff0c\u589e\u5f3a\u4e86\u5bf9\u56fe\u4e2d\u8fb9\u7684\u65b9\u5411\u6027\u548c\u7a7a\u95f4\u7ed3\u6784\u7684\u7406\u89e3\uff0c\u4ece\u800c\u5728\u5404\u79cd\u9700\u8981\u7cbe\u786e\u7a7a\u95f4\u7406\u89e3\u7684\u5e94\u7528\u4e2d\uff0c\u63d0\u4f9b\u4e86\u6bd4\u4f20\u7edfGNN\u548cSchNet\u66f4\u4e3a\u5f3a\u5927\u7684\u6027\u80fd\u3002<\/p>\n\n\n\n<p>\u4fe1\u606f\u805a\u5408\u9636\u6bb5\u6bd4\u8f83\u7b80\u5355\uff0c\u516c\u5f0f\\(m_i=\\sum_{j\\in N(i)}m_{ij}\\)\u63cf\u8ff0\u4e86\u5982\u4f55\u5c06\u8282\u70b9\u7684\u6240\u6709\u90bb\u5c45\u8282\u70b9\u53d1\u6765\u7684\u6d88\u606f\u805a\u5408\u5728\u4e00\u8d77\uff0c\u5f97\u5230\\(m_{i}\\)\u3002\u7c7b\u4f3c\u5730\uff0c\u516c\u5f0f\\(\\hat{\\boldsymbol{m}}_i=C\\sum_{j\\neq i}\\hat{\\boldsymbol{m}}_{ij}\\)\u662f\u5c06\u6240\u6709\u65b9\u5411\u5316\u7684\u6d88\u606f\u805a\u5408\u5728\u4e00\u8d77\uff0c\u5f97\u5230\u4fee\u6b63\u7684\u805a\u5408\u6d88\u606f\\(\\hat{m}_i\\)\uff0c\u5176\u4e2d\\(\\mathrm{C}\\)\u662f\u4e00\u4e2a\u5f52\u4e00\u5316\u5e38\u6570\uff0c\u7528\u6765\u8c03\u6574\u805a\u5408\u7684\u8303\u56f4\u3002<\/p>\n\n\n\n<p>\u6700\u540e\uff0c\u5728\u8282\u70b9\u66f4\u65b0\u9636\u6bb5\u3002\u516c\u5f0f\\(h_i^{l+1}=\\phi_h\\left(h_i^l,m_i\\right)\\)\u662f\u8282\u70b9\u7279\u5f81\u7684\u66f4\u65b0\u89c4\u5219\uff0c\u5176\u4e2d\\(\\phi_{h}\\)\u662f\u4e00\u4e2a\u53ef\u5b66\u4e60\u7684\u795e\u7ecf\u7f51\u7edc\u5c42\uff0c\u7528\u5f53\u524d\u5c42\u7684\u8282\u70b9\u7279\u5f81\\(h_i^{l}\\)\u548c\u805a\u5408\u540e\u7684\u8fb9\u7279\u5f81\\(m_{i}\\)\u6765\u66f4\u65b0\u4e0b\u4e00\u5c42\u7684\u8282\u70b9\u7279\u5f81\\(h_i^{l+1}\\)\u3002\u800c\u516c\u5f0f\\(x_i^{l+1}=x_i^l+\\hat{m}_i\\)\u5219\u662fEGNN\u4e2d\u72ec\u7279\u7684\u6b65\u9aa4\uff0c\u4e5f\u662f\u6574\u4e2a\u5de5\u4f5c\u7684\u7cbe\u9ad3\u3002\u56e0\u4e3a\u5b83\u8003\u8651\u4e86\u8282\u70b9\u7684\u7a7a\u95f4\u4f4d\u7f6e\uff0c\u4f7f\u5f97\u7f51\u7edc\u80fd\u591f\u4fdd\u6301\u7b49\u53d8\u6027\u3002\u5177\u4f53\u6765\u8bf4\uff0c\\(x_{i}^{l}\\)\u8868\u793a\u5728\u7b2c\\(\\mathrm{l}\\)\u5c42\u65f6\uff0c\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u4f4d\u7f6e\u5750\u6807\u3002\\(\\hat{m}_i\\)\u662f\u4ece\u4e0e\u8282\u70b9\\(\\mathrm{i}\\)\u76f8\u8fde\u7684\u6240\u6709\u90bb\u5c45\u8282\u70b9\\(\\mathrm{j}\\)\u65b9\u5411\u5316\u7684\u805a\u5408\u4fe1\u606f\uff0c\u8868\u793a\u8282\u70b9\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\u4fe1\u606f\u3002 \\(x_i^{l+1}\\)\u8868\u793a\u5728\u7b2c\\(\\text{l+1}\\)\u5c42\u65f6\uff0c\u8282\u70b9\\(\\mathrm{j}\\)\u7684\u65b0\u4f4d\u7f6e\u5750\u6807\uff0c\u66f4\u65b0\u8282\u70b9\\(\\mathrm{j}\\)\u5750\u6807\u7684\u610f\u4e49\u5728\u4e8e\uff0c\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff08\u6bd4\u5982\u5206\u5b50\u52a8\u529b\u5b66\u6a21\u62df\uff09\uff0c\u8282\u70b9\u7684\u7269\u7406\u4f4d\u7f6e\u6216\u5750\u6807\u662f\u968f\u7740\u65f6\u95f4\u6216\u7f51\u7edc\u5c42\u6b21\u7684\u63a8\u8fdb\u800c\u4e0d\u65ad\u53d8\u5316\u7684\u3002\u8fd9\u4e2a\u516c\u5f0f\u901a\u8fc7\u6dfb\u52a0\u4fee\u6b63\u6d88\u606f\uff08\u53ef\u4ee5\u770b\u4f5c\u662f\u7a7a\u95f4\u53d8\u5316\u7684\u91cf\uff09\u5230\u5f53\u524d\u7684\u4f4d\u7f6e\u4e0a\uff0c\u4ece\u800c\u66f4\u65b0\u6bcf\u4e2a\u8282\u70b9\u7684\u4f4d\u7f6e\u3002\u8fd9\u6837\uff0c\u7f51\u7edc\u4e0d\u4ec5\u80fd\u591f\u66f4\u65b0\u8282\u70b9\u7684\u7279\u5f81\u8868\u793a\uff08\u5982\u7535\u8377\u3001\u80fd\u91cf\u72b6\u6001\u7b49\uff09\uff0c\u4e5f\u80fd\u591f\u66f4\u65b0\u8282\u70b9\u7684\u7a7a\u95f4\u4f4d\u7f6e\uff0c\u4ece\u800c\u4f7f\u5f97\u6a21\u578b\u80fd\u591f\u8003\u8651\u5230\u7ed3\u6784\u7684\u52a8\u6001\u53d8\u5316\u3002EGNN\u901a\u8fc7\u8fd9\u6837\u7684\u4f4d\u7f6e\u66f4\u65b0\u673a\u5236\uff0c\u4fdd\u8bc1\u4e86\u5f53\u6574\u4e2a\u56fe\u7ed3\u6784\u5728\u7a7a\u95f4\u4e2d\u8fdb\u884c\u53d8\u6362\uff08\u5982\u65cb\u8f6c\u3001\u53cd\u5c04\uff09\u65f6\uff0c\u8282\u70b9\u7684\u65b0\u4f4d\u7f6e\uff08 \\(x_i^{l+1}\\)\uff09\u5c06\u4ee5\u4e00\u81f4\u7684\u65b9\u5f0f\u53cd\u6620\u8fd9\u79cd\u53d8\u6362\uff0c\u8fd9\u4e5f\u662fEGNN\u5b9e\u73b0\u7b49\u53d8\u6027\u7684\u91cd\u8981\u65b9\u5f0f\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2.EGNN\u7b49\u53d8\u6027\u7684\u8bc1\u660e<\/h3>\n\n\n\n<p>\u7b49\u53d8\u6027\u662f\u6307\uff0c\u5f53\u8f93\u5165\u6570\u636e\u7ecf\u8fc7\u4e00\u4e2a\u53d8\u6362\uff08\u6bd4\u5982\u65cb\u8f6c\u3001\u7f29\u653e\u3001\u5e73\u79fb\uff09\u65f6\uff0c\u6a21\u578b\u7684\u8f93\u51fa\u4e5f\u4f1a\u4ee5\u76f8\u540c\u7684\u53d8\u6362\u6539\u53d8\u3002\u5728\u56fe\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u7279\u522b\u662f\u5728\u5904\u7406\u7a7a\u95f4\u6570\u636e\u65f6\uff0c\u7b49\u53d8\u6027\u662f\u4e00\u4e2a\u5173\u952e\u5c5e\u6027\uff0c\u56e0\u4e3a\u5b83\u5141\u8bb8\u7f51\u7edc\u6b63\u786e\u5730\u5904\u7406\u7a7a\u95f4\u5173\u7cfb\uff0c\u800c\u4e0d\u662f\u4ec5\u4ec5\u4f9d\u8d56\u4e8e\u8282\u70b9\u7684\u7279\u5f81\uff0c\u7528\u516c\u5f0f2\u8868\u793a\u7b49\u53d8\u6027\u5373\u4e3a\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(Qx^{l+1}+g,h^{l+1}=\\mathrm{EGCL}\\left(Qx^l+g,h^l\\right)\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \uff082\uff09<\/p>\n\n\n<p>\u5176\u4e2d\uff0cQ\u901a\u5e38\u662f\u4e00\u4e2a\u65cb\u8f6c\u77e9\u9635\u53d8\u6362\u77e9\u9635\uff0c g\u662f\u4e00\u4e2a\u5168\u5c40\u7684\u5e73\u79fb\u5411\u91cf\uff0c\u5b83\u4eec\u4f5c\u7528\u5728\u4f4d\u7f6e\u5750\u6807\\(x^{l}\\)\u4e0a\u53ef\u4ee5\u6a21\u62df\u8f93\u5165\u6570\u636e\u7684\u65cb\u8f6c\uff0c\u7f29\u653e\u548c\u5e73\u79fb\u53d8\u6362\u3002\\(h^{l}\\)\u662f\u8282\u70b9\u7684\u7279\u5f81\uff0c\u5f53\u8282\u70b9\u4f4d\u7f6e\u53d1\u751f\u53d8\u6362\u65f6\uff0c\\(h^{l}\\)\u4e5f\u9700\u8981\u4ee5\u4e00\u79cd\u4fdd\u6301\u7b49\u53d8\u6027\u7684\u65b9\u5f0f\u8fdb\u884c\u66f4\u65b0\u3002 EGCL\u8868\u793aEGNN\u7684\u7f51\u7edc\u5c42\uff0c\u5b83\u63a5\u6536\u53d8\u6362\u540e\u7684\u4f4d\u7f6e\u548c\u5f53\u524d\u7684\u8282\u70b9\u7279\u5f81\uff0c\u5e76\u8f93\u51fa\u66f4\u65b0\u540e\u7684\u4f4d\u7f6e\u548c\u7279\u5f81\u3002\u5982\u679cEGCL\u88ab\u8bbe\u8ba1\u4e3a\u7b49\u53d8\u7684\uff0c\u90a3\u4e48\u610f\u5473\u7740\u5982\u679c\u8f93\u5165\u53d1\u751f\u4e86\u65cb\u8f6c\uff0cEGCL\u7684\u8f93\u51fa\u4e5f\u4f1a\u53cd\u6620\u51fa\u76f8\u5e94\u7684\u65cb\u8f6c\u3002<\/p>\n\n\n\n<p>\u4e0b\u9762\uff0c\u5f00\u59cb\u8bc1\u660e\u4e0a\u9762\u7684EGNN\u5f0f2\u5177\u5907\u7b49\u53d8\u6027\uff0cEGNN\u7684\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(m_{ij}=\\phi_e\\left(h_i^l,h_j^l,\\left\\|x_i^l-x_j^l\\right\\|^2,a_{ij}\\right)\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \uff083\uff09<\/p>\n<p style=\"text-align: center;\">\\(\\boldsymbol{x}_i^{l+1}=\\boldsymbol{x}_i^l+C\\sum_{j\\neq i}\\left(\\boldsymbol{x}_i^l-\\boldsymbol{x}_j^l\\right)\\phi_x\\left(\\boldsymbol{m}_{ij}\\right)\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\uff084\uff09<\/p>\n<p style=\"text-align: center;\">\\(m_i=\\sum_{j\\neq i}m_{ij}\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\uff085\uff09<\/p>\n<p style=\"text-align: center;\">\\(h_i^{l+1}=\\phi_h\\left(h_i^l,m_i\\right)\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\uff086\uff09<\/p>\n\n\n<p>\u5bf9\u4e8e\u5f0f3\uff0c\u4e0d\u7ba1\u662f\u8ddd\u79bb\u4fe1\u606f\\(\\left|x_i^l-x_j^l\\right|^2\\)\uff0c\u8fd8\u662f\u8282\u70b9\u6216\u8fb9\u4fe1\u606f\\(h_i^l,h_j^l,a_{ij}\\)\u90fd\u4e0d\u4f1a\u56e0\u4e3a\u7a7a\u95f4\u7684\u6539\u53d8\u800c\u6539\u53d8\u3002\u6362\u53e5\u8bdd\u8bf4\uff1a\u8ddd\u79bb\u3001\u8282\u70b9\u6216\u8fb9\u4fe1\u606f\u4e0d\u5177\u5907\u65b9\u5411\u6027\uff0c\u4e0d\u4f1a\u56e0\u4e3a\u8f93\u5165\u6570\u636e\u53d1\u751f\u65cb\u8f6c\u3001\u7f29\u653e\u6216\u5e73\u79fb\u800c\u6539\u53d8\u81ea\u8eab\u7684\u503c\u3002\u56e0\u6b64\u5f0f3\u5728\u5bf9\u8282\u70b9\u4fe1\u606f\u8fdb\u884c\u8fed\u4ee3\u66f4\u65b0\u65f6\uff0c\u5b9e\u9645\u4e0a\u662f\u4e0d\u53d8\u6027\u8d28\u7684\uff0c\u800c\u4e0d\u662f\u7b49\u53d8\u6027\u8d28\u3002EGNN\u7684\u7b49\u53d8\u6027\u8d28\u4e3b\u8981\u4f53\u73b0\u5728\u5f0f4\u5bf9\u4f4d\u7f6e\u4fe1\u606f\u7684\u66f4\u65b0\u4e0a\uff0c\u5047\u8bbe\\(x^{i}\\)\u53d1\u751f\u65cb\u8f6c\u548c\u5e73\u79fb\u7684\u7a7a\u95f4\u53d8\u6362\\(Qx_i^{l+1}+g\\)\uff0c\u6839\u636e\u5f0f4\u53ef\u4ee5\u5f97\u5230\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\boldsymbol{Q}\\boldsymbol{x}_i^{l+1}+\\boldsymbol{g}=\\boldsymbol{Q}\\boldsymbol{x}_i^l+\\boldsymbol{g}+C\\sum_{j\\neq i}\\left(\\boldsymbol{Q}\\boldsymbol{x}_i^l+\\boldsymbol{g}-\\left[\\boldsymbol{Q}\\boldsymbol{x}_j^l+\\boldsymbol{g}\\right]\\right)\\phi_x\\left(\\boldsymbol{m}_{i,j}\\right)\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \uff087\uff09<\/p>\n\n\n<p>\u5bf9\u5f0f4\u8fdb\u884c\u63a8\u5bfc\uff0c\u53ef\u5f97\u5230\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\begin{aligned}<br \/>&amp; \\boldsymbol{Q}\\boldsymbol{x}_i^l+\\boldsymbol{g}+C\\sum_{j\\neq i}\\left(\\boldsymbol{Q}\\boldsymbol{x}_i^l+\\boldsymbol{g}-\\boldsymbol{Q}\\boldsymbol{x}_j^l-\\boldsymbol{g}\\right)\\phi_x\\left(\\boldsymbol{m}_{i,j}\\right) \\\\<br \/>&amp; =\\boldsymbol{Q}\\boldsymbol{x}_i^l+\\boldsymbol{g}+\\boldsymbol{Q}C\\sum_{j\\neq i}\\left(\\boldsymbol{x}_i^l-\\boldsymbol{x}_j^l\\right)\\phi_x\\left(\\boldsymbol{m}_{i,j}\\right) \\\\<br \/>&amp; =\\boldsymbol{Q}\\left(\\boldsymbol{x}_i^l+C\\sum_{j\\neq i}\\left(\\boldsymbol{x}_i^l-\\boldsymbol{x}_j^l\\right)\\phi_x\\left(\\boldsymbol{m}_{i,j}\\right)\\right)+\\boldsymbol{g} \\\\<br \/>&amp; =\\boldsymbol{Q}\\boldsymbol{x}_i^{l+1}+\\boldsymbol{g}<br \/>\\end{aligned}\\)\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 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Graph Neural Network\uff09\u662f\u4e00\u79cd\u7ecf\u5178\u7684\u5177\u6709\u4fdd\u6301\u7b49\u53d8\u6027\uff08Equ [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":2435,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[27,23],"tags":[],"class_list":["post-2410","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-27","category-23"],"_links":{"self":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2410","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2410"}],"version-history":[{"count":37,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2410\/revisions"}],"predecessor-version":[{"id":2454,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2410\/revisions\/2454"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/media\/2435"}],"wp:attachment":[{"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2410"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2410"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2410"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}