{"id":2547,"date":"2025-01-06T15:46:48","date_gmt":"2025-01-06T07:46:48","guid":{"rendered":"https:\/\/www.gnn.club\/?p=2547"},"modified":"2025-01-06T15:46:48","modified_gmt":"2025-01-06T07:46:48","slug":"segnn","status":"publish","type":"post","link":"http:\/\/gnn.club\/?p=2547","title":{"rendered":"SEGNN"},"content":{"rendered":"\n<p>Geometric and Physical Quantities Improve E(3) Equivariant Message Passing\u662f\u53e6\u4e00\u7bc7\u7814\u7a76\u5982\u4f55\u901a\u8fc7\u51e0\u4f55\u548c\u7269\u7406\u91cf\u6765\u589e\u5f3a<em>E(3)<\/em>\u7b49\u53d8\u6d88\u606f\u4f20\u9012\u7684\u8bba\u6587\u3002<em>E(3)<\/em> \u7b49\u53d8\u6027\u6307\u7684\u662f\u5728\u4e09\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d\uff0c\u6a21\u578b\u5728\u5e73\u79fb\u3001\u65cb\u8f6c\u548c\u53cd\u5c04\u53d8\u6362\u4e0b\u4fdd\u6301\u4e0d\u53d8\u6027\u3002\u5728\u5206\u5b50\u6a21\u62df\u548c\u5176\u4ed6\u4e09\u7ef4\u6570\u636e\u5e94\u7528\u4e2d\uff0c\u6355\u6349\u8f93\u5165\u6570\u636e\u7684\u5bf9\u79f0\u6027\uff08\u5982\u5e73\u79fb\u3001\u65cb\u8f6c\u548c\u53cd\u5c04\uff09\u662f\u81f3\u5173\u91cd\u8981\u7684\u3002SEGNN\u5b9e\u9645\u4e0a\u4e0eTFN\u975e\u5e38\u76f8\u4f3c\uff0c\u751a\u81f3\u5176\u6a21\u578b\u8ba1\u7b97\u4e0e\u7ed3\u6784\u66f4\u52a0\u7b80\u5355\uff0c\u4f46\u662fSEGNN\u5bf9\u4e8e\u4e00\u4e9b\u64cd\u4f5c\u7684\u672f\u8bed\u4f7f\u7528\u4e0e\u63cf\u8ff0\u66f4\u52a0\u4e13\u4e1a\u5408\u7406\uff0c\u53ef\u80fd\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\u63d0\u4f9b\u4e86\u66f4\u4e3a\u76f4\u89c2\u548c\u6613\u4e8e\u7406\u89e3\u7684\u7b49\u53d8\u6027\u5b9e\u73b0\u65b9\u6cd5\u3002\u53e6\u4e00\u65b9\u9762\uff0cTFN\u5f3a\u8c03\u4f7f\u7528\u5f20\u91cf\u573a\u6765\u6355\u6349\u51e0\u4f55\u5173\u7cfb\uff0c\u800cSEGNN\u5219\u66f4\u591a\u5730\u4f9d\u8d56\u4e8e\u56fe\u795e\u7ecf\u7f51\u7edc\u7684\u7ed3\u6784\u548c\u6d88\u606f\u4f20\u9012\u673a\u5236\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">1\u3001\u53ef\u5bfc\u5411\u7279\u5f81<\/h2>\n\n\n\n<p>\u4f7f\u7528\u53ef\u5bfc\u5411\u5411\u91cf\uff08Steerable features\uff09\u662f\u5728\u56fe\u795e\u7ecf\u7f51\u7edc\u4e2d\u5b9e\u73b0\u7b49\u53d8\u6027\u7684\u6709\u6548\u65b9\u6cd5\u3002\u8fd9\u91cc\u7684\u201c\u53ef\u5bfc\u5411\u6027\u201d\u6307\u7684\u662f\u662f\u4e00\u7c7b\u5728\u7279\u5b9a\u53d8\u6362\u4e0b\u5177\u6709\u7279\u5b9a\u884c\u4e3a\u7684\u7279\u5f81\u3002\u4f8b\u5982\uff0c\u5982\u679c\u4e00\u4e2a\u7ed3\u6784\u88ab\u65cb\u8f6c\uff0c\u90a3\u4e48\u5b83\u7684\u53ef\u5bfc\u5411\u7279\u5f81\u5c06\u4ee5\u5df2\u77e5\u7684\u65b9\u5f0f\u54cd\u5e94\u8fd9\u6b21\u65cb\u8f6c\u3002\u8fd9\u4e9b\u7279\u5f81\u901a\u5e38\u662f\u5728\u7f51\u7edc\u7684\u8bbe\u8ba1\u4e2d\u5f15\u5165\u5bf9\u79f0\u6027\u548c\u7b49\u53d8\u6027\u539f\u5219\uff0c\u4ee5\u786e\u4fdd\u5728\u53d8\u6362\uff08\u5982\u65cb\u8f6c\u3001\u5e73\u79fb\u6216\u53cd\u5c04\uff09\u4e0b\uff0c\u7279\u5f81\u80fd\u591f\u4fdd\u7559\u8f93\u5165\u7684\u5bf9\u79f0\u6027\u3002\u7b80\u5355\u8bf4\uff0c\u53ef\u5bfc\u5411\u7279\u5f81\u786e\u4fdd\u4e86\u7279\u5f81\u8868\u793a\u7684\u7b49\u53d8\u6027\uff0c\u5373\u5f53\u8f93\u5165\u53d1\u751f\u53d8\u6362\u65f6\uff0c\u7279\u5f81\u8868\u793a\u4e5f\u4f1a\u76f8\u5e94\u53d8\u6362\u3002<\/p>\n\n\n\n<p>\u5728SEGNN\u4e2d\uff0c\u53ef\u5bfc\u5411\u5411\u91cf\u7684\u5b9a\u4e49\u5982\u4e0b\uff1a\u5411\u91cf\\(\\tilde{h}\\)\u88ab\u5b9a\u4e49\u4e3a\u53ef\u5bfc\u5411\u7684\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u53ef\u4ee5\u901a\u8fc7\u77e9\u9635\\(\\boldsymbol{D}(g)\\)\u6765\u54cd\u5e94\u53d8\u6362\\(\\mathrm{g}\\)\u3002\u4f8b\u5982\uff0c\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\uff0c\u4e00\u4e2a\u6b27\u51e0\u91cc\u5f97\u5411\u91cf\u901a\u8fc7\u65cb\u8f6c\u77e9\u9635\\(\\mathrm{R}\\)\u8fdb\u884c\u53d8\u6362\uff0c\u4e3a\u4e86\u8868\u793a\u5411\u91cf\u5728\u53d8\u6362\u4e0b\u7684\u884c\u4e3a\uff0c\u77e9\u9635\\(\\boldsymbol{D}^{(l)}(g)\\)\u901a\u5e38\u662fWigner-D\uff0c\u6839\u636e\u7403\u8c10\u51fd\u6570\u7684\u5b9a\u4e49\uff0c\u8fd9\u4e9b\u77e9\u9635\u662f\\((2l+1)\\)\u7ef4\u7684\uff0c\u53ef\u4ee5\u4f5c\u7528\u4e8e\u5bf9\u5e94\u7ef4\u6570\u7684\u5411\u91cf\u7a7a\u95f4\u3002Wigner-D\u77e9\u9635\u5e2e\u52a9\u5b9e\u73b0\u4e86\u5411\u91cf\u5728\u65cb\u8f6c\u4e0b\u7684\u7b49\u53d8\u6027\uff0c\u5982\u56fe1\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=\"405\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241228121205557-1024x405.png\" alt=\"\" class=\"wp-image-2548\" srcset=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241228121205557-1024x405.png 1024w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241228121205557-300x119.png 300w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241228121205557-768x304.png 768w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241228121205557.png 1306w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56fe1 \u7279\u5f81\u66f4\u65b0<\/figcaption><\/figure><\/div>\n\n\n<p>\u56fe1(a)\u663e\u793a\u4e86\u4e00\u4e2a\u53ef\u5bfc\u5411\u5411\u91cf\\(\\tilde{\\boldsymbol{h}}^{(l)}\\)\u3002\u53ef\u5bfc\u5411\u5411\u91cf\u5728\u4e0d\u540c\\(\\mathrm{l}\\)\u9636\u7684\u7403\u8c10\u51fd\u6570\u5b50\u7a7a\u95f4\u4e2d\u88ab\u5c55\u5f00\uff0c\u8fd9\u91cc\\(l=0,1,2\\)\uff0c\u5206\u522b\u8868\u793a\u6807\u91cf\u7a7a\u95f4\u3001\u5411\u91cf\u7a7a\u95f4\u548c\u5f20\u91cf\u7a7a\u95f4\u3002\u6bcf\u4e2a\u5b50\u7a7a\u95f4\u901a\u8fc7\u57fa\u51fd\u6570\\(Y_m^{(l)}\\)\uff08\u7403\u8c10\u51fd\u6570\uff09\u8fdb\u884c\u8868\u793a\uff0c\u5411\u91cf\u7684\u4e0d\u540c\u5206\u91cf\\(\\left[h\\right]_l^m\\)\u4e0e\u5bf9\u5e94\u7684\u9636\u6570\u7684\u7403\u8c10\u51fd\u6570\u76f8\u4e58\uff0c\u5f62\u6210\u4e86\u8fd9\u4e2a\u5411\u91cf\u7684\u5b8c\u6574\u8868\u8fbe\u3002\u5177\u4f53\u6765\u8bf4\uff0c\\(Y^{(l)}(x)\\cdot Y_m^{(l)}(\\cdot)\\) \u7684\u8fd0\u7b97\u8868\u793a\u5728\u70b9\\(\\mathrm{x}\\)\u5904\u7684\u7403\u8c10\u51fd\u6570\u5c55\u5f00\u4e0e\u57fa\u51fd\u6570\\(Y_{m}^{(l)}(\\cdot)\\)\u7684\u7ec4\u5408\u3002\\(Y^{(l)}(x)\\)\u662f\u4e00\u4e2a\u5411\u91cf\uff0c\u5305\u542b\u6240\u6709\u78c1\u91cf\u5b50\u6570\\(\\mathrm{m}\\)\u5bf9\u5e94\u7684\u7403\u8c10\u51fd\u6570\u503c\uff0c\\(Y_{m}^{(l)}(\\cdot)\\)\u662f\u5177\u4f53\u7684\u4e00\u4e2a\u57fa\u51fd\u6570\uff0c\u8868\u793a\u5728\u67d0\u4e2a\u7279\u5b9a\u6a21\u5f0f\u4e0b\u7684\u503c\u3002\\(Y^{(l)}(x)\\)\u548c\\(Y_{m}^{(l)}(\\cdot)\\)\u7684\u4e58\u79ef\u8868\u793a\u4e00\u79cd\u5206\u89e3\u548c\u91cd\u6784\u65b9\u5f0f\uff0c\u5c06\u590d\u6742\u51fd\u6570\u8868\u793a\u4e3a\u4e00\u7ec4\u57fa\u51fd\u6570\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u8fd9\u79cd\u8868\u793a\u5728\u5904\u7406\u5177\u6709\u65cb\u8f6c\u5bf9\u79f0\u6027\u7684\u7cfb\u7edf\u65f6\u975e\u5e38\u6709\u6548\uff0c\u80fd\u591f\u65b9\u4fbf\u5730\u8fdb\u884c\u65cb\u8f6c\u548c\u5176\u4ed6\u5bf9\u79f0\u64cd\u4f5c\u7684\u8ba1\u7b97\u3002<\/p>\n\n\n\n<p>\u56fe1(b)\u63cf\u8ff0\u7684\u662f\u5c55\u793a\u4e86\u5e94\u7528\u65cb\u8f6c\u77e9\u9635\\(\\mathrm{R}\\)\u540e\u5411\u91cf\\(\\tilde{\\boldsymbol{h}}^{(l)}\\)\u7684\u53d8\u5316\u3002\u6bcf\u4e2a\u5b50\u7a7a\u95f4\u5206\u522b\u53d8\u6362\uff0c\u5411\u91cf\u7684\u6574\u4f53\u5f62\u6001\u968f\u7740\u65cb\u8f6c\u800c\u53d8\u5316\uff0c\u4f46\u5176\u57fa\u672c\u7ed3\u6784\u548c\u4fe1\u606f\u4fdd\u6301\u4e00\u81f4\uff0c\u4f53\u73b0\u4e86\u7b49\u53d8\u6027\u3002\u65cb\u8f6c\u77e9\u9635\\(\\mathrm{R}\\)\u7684\u672c\u8d28\u662fWinger-D\u77e9\u9635\uff0cWigner-D \u77e9\u9635\u662f\u5bf9\u7403\u8c10\u51fd\u6570\u5728\u65cb\u8f6c\u4e0b\u7684\u53d8\u6362\u8fdb\u884c\u63cf\u8ff0\uff0c\u4fdd\u8bc1\u4e86\u6bcf\u4e2a\u5206\u91cf\u5728\u65cb\u8f6c\u4e0b\u7684\u6b63\u786e\u53d8\u6362\u884c\u4e3a\u3002\u56e0\u6b64\uff0c\u5f53\u5e94\u7528\u65cb\u8f6c\u77e9\u9635\\(\\mathrm{R}\\)\u65f6\uff0c\u4e0d\u540c\u9636\u6570\\(\\mathrm{l}\\)\u7684\u5b50\u7a7a\u95f4\\(V_{l}\\)\u5185\u7684\u5411\u91cf\u90e8\u5206\u901a\u8fc7\u76f8\u5e94\u7684 Wigner-D \u77e9\u9635\\(D^{(l)}(R)\\)\u8fdb\u884c\u53d8\u6362\uff0c\u786e\u4fdd\u5411\u91cf\\(\\tilde{h}^{(l)}\\)\u7684\u6bcf\u4e2a\u7ec4\u6210\u90e8\u5206\u90fd\u6309\u7167\u7269\u7406\u7a7a\u95f4\u4e2d\u7684\u65cb\u8f6c\u8fdb\u884c\u76f8\u5e94\u7684\u8c03\u6574\uff0c\u4ece\u800c\u7ef4\u6301\u6574\u4f53\u5411\u91cf\u7684\u7269\u7406\u548c\u51e0\u4f55\u542b\u4e49\u3002<\/p>\n\n\n\n<p>\u8fd9\u4e9b\u662f\u901a\u8fc7\\(\\mathrm{l}\\)\u9636Wigner-D\u77e9\u9635\u53d8\u6362\u7684\u5411\u91cf\u7a7a\u95f4\uff0c\u79f0\u4e3a\u7c7b\u578b\\(\\mathrm{-l}\\)\u7684\u53ef\u5bfc\u5411\u5411\u91cf\u7a7a\u95f4\\(V_{l}\\)\u3002\u4e00\u822c\u7684\uff0c\u5c06\u4e0d\u540c\u7c7b\u578b\u7684\u53ef\u5bfc\u5411\u5411\u91cf\u7a7a\u95f4\uff0c\u4f8b\u5982\\(V_{0}\\)\u3001\\(V_{1}\\)\u548c\\(V_{2}\\)\u901a\u8fc7\u76f4\u548c\u7684\u65b9\u5f0f\uff08\u8868\u793a\u4e3a\\(V_{l}=V_{0}\\oplus V_{1}\\oplus V_{2}\\)\uff09\u8fdb\u884c\u8868\u793a\u3002\u76f4\u548c\uff08Direct Sum\uff09\u662f\u6570\u5b66\u4e2d\u4e00\u79cd\u7528\u4e8e\u7ec4\u5408\u591a\u4e2a\u5411\u91cf\u7a7a\u95f4\u7684\u65b9\u6cd5\u3002\u5f53\u6211\u4eec\u6709\u4e24\u4e2a\u6216\u591a\u4e2a\u5411\u91cf\u7a7a\u95f4\u65f6\uff0c\u53ef\u4ee5\u901a\u8fc7\u76f4\u548c\u5c06\u5b83\u4eec\u7ec4\u5408\u6210\u4e00\u4e2a\u66f4\u5927\u7684\u5411\u91cf\u7a7a\u95f4\u3002\u8fd9\u79cd\u7ec4\u5408\u65b9\u5f0f\u4fdd\u8bc1\u4e86\u65b0\u5411\u91cf\u7a7a\u95f4\u7684\u7ef4\u5ea6\u662f\u539f\u59cb\u5404\u5411\u91cf\u7a7a\u95f4\u7ef4\u5ea6\u7684\u603b\u548c\u3002\u76f4\u548c\u7684\u7ed3\u679c\u662f\u4e00\u4e2a\u66f4\u5927\u7684\u5411\u91cf\u7a7a\u95f4\uff0c\u53ef\u4ee5\u5c06\u5176\u7406\u89e3\u6210\u8ba1\u7b97\u673a\u9886\u57df\u5bf9\u4e0d\u540c\u7ef4\u5ea6\u7279\u5f81\u8fdb\u884c\u7684\u62fc\u63a5\u64cd\u4f5c\uff0c\u5176\u53d8\u6362\u884c\u4e3a\u9075\u5faa\u7ec4\u6210\u5b83\u7684\u5404\u5411\u91cf\u7a7a\u95f4\u7684\u53d8\u6362\u89c4\u5219\uff0c\u4f46\u6574\u4f53\u4e0a\u4f9d\u7136\u4fdd\u6301\u7ed3\u6784\u7684\u6574\u5408\u6027\u548c\u6570\u5b66\u7684\u4e00\u81f4\u6027\u3002\u8fd9\u79cd\u65b9\u6cd5\u5728\u5904\u7406\u9700\u8981\u8003\u8651\u591a\u79cd\u7c7b\u578b\u53d8\u6362\u7684\u95ee\u9898\u65f6\u7279\u522b\u6709\u7528\uff0c\u6bd4\u5982\u5728\u56fe\u795e\u7ecf\u7f51\u7edc\u4e2d\u5904\u7406\u5177\u6709\u590d\u6742\u5bf9\u79f0\u6027\u7684\u6570\u636e\u3002\u5047\u8bbe\\(V_l=V_0\\oplus V_1\\oplus V_2\\)\uff0c\u53ef\u5bfc\u5411\u7279\u5f81\\(V_{0}\\)\u3001\\(V_{1}\\)\u548c\\(V_{2}\\)\u7684\u7ef4\u5ea6\u4e3a[feature_dim, m_dim, m_dim]\uff0c\u4ee4feature_dim\u5206\u522b\u4e3a10\u30015\u30012\u3002\u6839\u636e\u7403\u8c10\u51fd\u6570\u7684\u5b9a\u4e49\uff1a\\(l=0\\)\u65f6\\(m=1\\)\uff0c\\(l=1\\)\u65f6\\(m=3\\)\uff0c\\(l=2\\)\u65f6\\(m=5\\)\uff0c\u56e0\u6b64\uff0c\\(V_{0}\\)\u7684\u7ef4\u5ea6\u4e3a[10, 1, 1]\uff0c\\(V_{1}\\)\u7684\u7ef4\u5ea6\u4e3a[5, 3, 3]\uff0c\\(V_{2}\\)\u7684\u7ef4\u5ea6\u4e3a[2, 5, 5]\uff0c\u5982\u56fe2\u6240\u793a\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-full is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"335\" height=\"335\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250103111927107.png\" alt=\"\" class=\"wp-image-2552\" style=\"width:337px;height:auto\" srcset=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250103111927107.png 335w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250103111927107-300x300.png 300w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250103111927107-150x150.png 150w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250103111927107-100x100.png 100w\" sizes=\"auto, (max-width: 335px) 100vw, 335px\" \/><figcaption class=\"wp-element-caption\">\u56fe2 \u53ef\u5bfc\u5411\u7279\u5f81\u56fe<\/figcaption><\/figure><\/div>\n\n\n<p>\u56fe2\u53ef\u4ee5\u79f0\u4e3a\u53ef\u5bfc\u5411\u7279\u5f81\u56fe\uff0c\\(V_{0}\\)\u5b50\u7a7a\u95f4\u5728\u77e9\u9635\u4e2d\uff0c\u5bf9\u5e94\u4e8e\u5de6\u4e0a\u89d2\u7684\u4e00\u4e2a\\(10\\times10\\)\u7684\u5757\uff0c\u5176\u5185\u90e8\u7ed3\u6784\u8f83\u4e3a\u7b80\u5355\uff0c\u56e0\u4e3a\u8fd9\u662f\u6807\u91cf\u573a\u7684\u8868\u793a\uff0c\u7ef4\u5ea6\u76f8\u5bf9\u8f83\u5c0f\u3002\\(V_{1}\\)\u5b50\u7a7a\u95f4\u5728\u77e9\u9635\u4e2d\u8868\u73b0\u4e3a\u4e2d\u95f4\u7684\u591a\u4e2a\\(5\\times5\\)\u7684\u5757\u3002\u7531\u4e8e\\(V_{1}\\)\u662f\u5411\u91cf\u573a\u7684\u8868\u793a\uff0c\u5176\u5185\u90e8\u7ed3\u6784\u7a0d\u5fae\u590d\u6742\u4e00\u4e9b\uff0c\u5c55\u793a\u4e86\u66f4\u591a\u7684\u53d8\u5316\u6a21\u5f0f\u3002\u7c7b\u4f3c\u7684\uff0c\u5728\u77e9\u9635\u7684\u53f3\u4e0b\u89d2\u662f\\(V_{2}\\)\u5b50\u7a7a\u95f4\u5bf9\u5e94\u7684\u5757\u3002\u4f5c\u4e3a\u66f4\u9ad8\u9636\u7684\u5f20\u91cf\u573a\u8868\u793a\uff0c\u5176\u7ef4\u5ea6\u548c\u7ed3\u6784\u66f4\u52a0\u590d\u6742\uff0c\u77e9\u9635\u4e2d\u7684\u8fd9\u4e00\u90e8\u5206\u5c55\u793a\u4e86\u66f4\u4e30\u5bcc\u7684\u6a21\u5f0f\u548c\u53d8\u6362\u3002\u5728\u56fe\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u5904\u7406\u590d\u6742\u5bf9\u79f0\u6027\u7684\u6570\u636e\u65f6\uff0c\u76f4\u548c\u77e9\u9635\u63d0\u4f9b\u4e86\u4e00\u79cd\u6709\u6548\u7684\u65b9\u6cd5\u6765\u5206\u89e3\u548c\u7ec4\u5408\u7279\u5f81\u3002\u8fd9\u79cd\u65b9\u6cd5\u53ef\u4ee5\u63d0\u5347\u7f51\u7edc\u5728\u6355\u6349\u548c\u5904\u7406\u591a\u7ef4\u5ea6\u3001\u591a\u7c7b\u578b\u7279\u5f81\u65f6\u7684\u80fd\u529b\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">2\u3001\u53ef\u5bfc\u5411\u591a\u5c42\u611f\u77e5\u673a<\/h2>\n\n\n\n<p>\u5728\u53ef\u5bfc\u5411\u7279\u5f81\u7684\u6846\u67b6\u4e0b\uff0c\u53ef\u4ee5\u8bbe\u8ba1\u591a\u5c42\u611f\u77e5\u5668\u6765\u64cd\u4f5c\u8fd9\u4e9b\u53ef\u5bfc\u5411\u7684\u5411\u91cf\u7a7a\u95f4\uff0c\u5b9e\u73b0\u5bf9\u4efb\u610f\u7c7b\u578b\u53ef\u5bfc\u5411\u5411\u91cf\u7a7a\u95f4\u7684\u53d8\u6362\u5904\u7406\u3002\u5bf9\u6bd4\u4f20\u7edf\u7684\u591a\u5c42\u611f\u77e5\u5668\uff0c\u6bcf\u5c42\u7684\u64cd\u4f5c\u901a\u5e38\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(y=Wx+b\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff0c\\(\\mathrm{x}\\)\u662f\u8f93\u5165\u5411\u91cf\uff0c\\(\\boldsymbol{W}\\)\u662f\u6743\u91cd\u77e9\u9635\uff0c\\(\\boldsymbol{b}\\)\u662f\u504f\u7f6e\u5411\u91cf\uff0c\u800c\\(\\boldsymbol{y}\\)\u662f\u8f93\u51fa\u5411\u91cf\u3002\u8fd9\u79cd\u7ed3\u6784\u5f88\u7b80\u5355\uff0c\u4e3b\u8981\u5173\u6ce8\u8f93\u5165\u548c\u8f93\u51fa\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\u53ca\u5176\u975e\u7ebf\u6027\u53d8\u6362\uff08\u901a\u8fc7\u6fc0\u6d3b\u51fd\u6570\uff0c\u5728\u516c\u5f0f\u4e2d\u88ab\u7b80\u5316\uff0c\u672a\u5728\u516c\u5f0f\u4e2d\u663e\u793a\uff09\u3002<\/p>\n\n\n\n<p>\u800c\u5728Steerable MLP\u4e2d\uff0c\u64cd\u4f5c\u4e0d\u4ec5\u6d89\u53ca\u7ebf\u6027\u53d8\u6362\uff0c\u8fd8\u8981\u4fdd\u6301\u5bf9\u7279\u5b9a\u7fa4\uff08\u5982\u65cb\u8f6c\u7fa4\\(SO(3)\\)\uff09\u7684\u7b49\u53d8\u6027\u3002\u8fd9\u610f\u5473\u7740\u7f51\u7edc\u5fc5\u987b\u5728\u53d8\u6362\u8f93\u5165\u65f6\u4fdd\u6301\u5176\u8f93\u51fa\u76f8\u5bf9\u4e8e\u8fd9\u4e9b\u53d8\u6362\u7684\u4e00\u81f4\u6027\u3002Steerable MLP\u901a\u5e38\u4f7f\u7528\u7403\u8c10\u51fd\u6570\u548cClebsch-Gordan\u7cfb\u6570\u6765\u5b9e\u73b0\u8fd9\u79cd\u7b49\u53d8\u6027\u3002<\/p>\n\n\n\n<p>\u5177\u4f53\u6765\u8bf4\uff0c\u5728Steerable MLP\u4e2d\uff0c\u8f93\u5165\u5411\u91cf\\(x\\in\\mathbb{R}^3\\)\u9996\u5148\u88ab\u8f6c\u6362\u6210\u7684\u53ef\u5bfc\u5411\u5411\u91cf\\(\\hat{a}^{(l)}\\)\u3002\u8fd9\u662f\u901a\u8fc7\u7403\u8c10\u51fd\u6570\u5d4c\u5165\u6765\u5b8c\u6210\u7684\uff0c\u5982\u4e0b\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\hat{a}^{(l)}=\\left(Y_{-l}^l(x),Y_{-l+1}^l(x),...,Y_l^l(x)\\right)^T\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff0c\\(Y_m^l(x)\\)\u662f\u5e94\u7528\u4e8e\\(\\mathrm{x}\\)\u7684\u7403\u8c10\u51fd\u6570\uff0c\u7528\u4e8e\u6355\u6349\\(\\mathrm{x}\\)\u5728\u7403\u9762\u4e0a\u7684\u65b9\u5411\u6027\u7279\u5f81\u3002\u8fd9\u4e2a\u5d4c\u5165\u7684\u5411\u91cf\\(\\hat{a}^{(l)}\\)\u4f5c\u4e3a\u6a21\u578b\u7b2c\u4e00\u5c42\u7684\u8f93\u5165\uff0c\u5373\\(\\hat{h}_0=\\hat{a}^{(l)}\\)\u4f5c\u4e3a\u7f51\u7edc\u7684\u521d\u59cb\u8f93\u5165\uff0c\u4e4b\u540e\u7f51\u7edc\u7684\u6bcf\u4e00\u5c42\u53ef\u4ee5\u901a\u8fc7\u4ee5\u4e0b\u65b9\u5f0f\u8ba1\u7b97\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\hat{\\boldsymbol{h}}_i=\\sigma{\\left(\\hat{\\boldsymbol{W}}_i\\hat{\\boldsymbol{h}}_{i-1}+\\boldsymbol{b}_i\\right)}\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff1a<\/p>\n\n\n\n<p>(1) \\(\\hat{\\boldsymbol{h}}_i\\)\u662f\u7b2c\\(\\mathrm{i}\\)\u5c42\u7684\u8f93\u51fa\u3002<\/p>\n\n\n\n<p>(2) \\(\\hat{W}_i\\)\u662f\u8c03\u6574\u4ee5\u4fdd\u6301\u7b49\u53d8\u6027\u7684\u6743\u91cd\u77e9\u9635\u3002<\/p>\n\n\n\n<p>(3) \\(\\boldsymbol{b}_{i}\\)\u662f\u504f\u7f6e\u5411\u91cf\u3002<\/p>\n\n\n\n<p>(4) \\(\\sigma\\)\u662f\u975e\u7ebf\u6027\u6fc0\u6d3b\u51fd\u6570\u3002<\/p>\n\n\n\n<p>\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u4fdd\u6301\u7b49\u53d8\u6027\u7684\u5173\u952e\u662f\u5982\u4f55\u5b9a\u4e49 \\(\\hat{W}_i\\)\u3002\u5bf9\u4e8eSteerable MLP\u6765\u8bf4\uff0c\\(\\hat{W}_i\\)\u4e0d\u4ec5\u4ec5\u662f\u5e38\u89c4\u7684\u7ebf\u6027\u6620\u5c04\uff0c\u800c\u662f\u9700\u8981\u6ee1\u8db3\u7b49\u53d8\u6027\u6761\u4ef6\u3002\u8fd9\u901a\u5e38\u610f\u5473\u7740\\(\\hat{W}_i\\)\u7684\u6784\u9020\u4f9d\u8d56\u4e8e\u5982Clebsch-Gordan\u7cfb\u6570\u8fd9\u6837\u7684\u7ed3\u6784\uff0c\u4ee5\u786e\u4fdd\u5728\u8fdb\u884c\u7ebf\u6027\u53d8\u6362\u65f6\u4fdd\u6301\u8f93\u5165\u5411\u91cf\u7684\u53d8\u6362\u7279\u6027\u3002CG\u7cfb\u6570\u7684\u4ecb\u7ecd\u8be6\u89c1TFN\u7f51\u7edc\uff0c\u7ed3\u5408CG\u7cfb\u6570\u7684Steerable MLP\u516c\u5f0f\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\hat{\\boldsymbol{h}}_i=\\sigma\\left(\\sum_{l_1,l_2}C_{l_1,l_2}^{l,m}\\hat{\\boldsymbol{W}}_{i-1}^{l_1,l_2}\\hat{\\boldsymbol{h}}_{i-1}^{(l_1)}\\otimes\\hat{\\boldsymbol{h}}_{i-1}^{(l_2)}+\\boldsymbol{b}_{i-1}\\right)\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff1a<\/p>\n\n\n\n<p>\uff081\uff09\\(C_{l_1,l_2}^{l,m}\\)\u662fClebsch-Gordan\u7cfb\u6570\uff0c\u7528\u4e8e\u8c03\u6574\u6765\u81ea\u524d\u4e00\u5c42\u7684\u4e0d\u540c\\(l_{1}\\)\u548c\\(l_{2}\\)\u7c7b\u578b\u7684\u53ef\u5bfc\u5411\u5411\u91cf\u7684\u7ec4\u5408\u3002<\/p>\n\n\n\n<p>\uff082\uff09\\(\\hat{W}_i^{l_1,l_2}\\)\u5bf9\u5e94\\(l_{1}\\),\\(l_{2}\\)\u7c7b\u578b\u8f93\u5165\u7684\u7279\u5b9a\u6743\u91cd\uff0c\u662f\u53ef\u5b66\u4e60\u7684\u53c2\u6570\u3002<\/p>\n\n\n\n<p>\uff083\uff09 \\(\u2297\\)\u8868\u793a\u9002\u5f53\u7684\u5f20\u91cf\u4e58\u79ef\uff0c\u4ee5\u4fdd\u6301\u53d8\u6362\u7684\u7b49\u53d8\u6027\u3002<\/p>\n\n\n\n<p>\u5bf9\u6bd4TFN\u6a21\u578b\u4e2d\u4f7f\u7528\u7684\u70b9\u5377\u79ef\u6765\u8bf4\uff0c\u70b9\u5377\u79ef\u662f\u5728\u7279\u5b9a\u7684\u90bb\u57df\u5185\u5bf9\u8f93\u5165\u7279\u5f81\u8fdb\u884c\u5c40\u90e8\u805a\u5408\u7684\u4e00\u79cd\u65b9\u5f0f\uff0c\u786e\u4fdd\u4e86\u8f93\u51fa\u5bf9\u65cb\u8f6c\u7684\u7b49\u53d8\u6027\u3002\u8fd9\u79cd\u65b9\u6cd5\u7279\u522b\u5f3a\u8c03\u4e86\u5728\u5c40\u90e8\u90bb\u57df\u5185\u7684\u7279\u5f81\u4ea4\u4e92\uff0c\u4ee5\u6b64\u6a21\u62df\u4f20\u7edf\u5377\u79ef\u7f51\u7edc\u4e2d\u7684\u5377\u79ef\u64cd\u4f5c\u3002\u5176\u4e2d\u7684\u5173\u952e\u662f\u5c40\u90e8\u6027\u539f\u5219\uff0c\u5373\u53ea\u8ba1\u7b97\u6bcf\u4e2a\u70b9\u53ca\u5176\u8fd1\u90bb\u7684\u4fe1\u606f\u3002\u76f8\u6bd4\u4e4b\u4e0b\uff0cSteerable MLP\u5728\u8ba1\u7b97\u65b9\u5f0f\u4e0a\u4e0e\u70b9\u5377\u79ef\u662f\u76f8\u540c\u7684\uff0c\u53ea\u662f\u8ba1\u7b97\u8303\u56f4\u4e0d\u4ec5\u9650\u4e8e\u5c40\u90e8\u90bb\u57df\u3002Steerable MLP\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u4e00\u79cd\u5728\u5168\u5c40\u7279\u5f81\u8868\u793a\u4e0a\u5b9e\u65bd\u65cb\u8f6c\u7b49\u53d8\u6027\u7684\u65b9\u6cd5\uff0c\u4e0e\u70b9\u5377\u79ef\u76f8\u6bd4\uff0c\u5176\u5e76\u4e0d\u5c40\u9650\u4e8e\u90bb\u57df\u5185\u7684\u8ba1\u7b97\u3002\u56e0\u6b64\uff0c\u5982\u679c\u6570\u636e\u7684\u5c5e\u6027\u7279\u5f81\u4e3b\u8981\u5728\u5c40\u90e8\u90bb\u57df\u4e2d\u4f53\u73b0\uff0c\u90a3\u4e48\u70b9\u5377\u79ef\u7684\u5904\u7406\u7ed3\u679c\u4f1a\u66f4\u597d\u3002\u53cd\u4e4b\uff0cSteerable MLP\u63d0\u4f9b\u4e86\u4e00\u79cd\u66f4\u4e3a\u7075\u6d3b\u7684\u65b9\u5f0f\u6765\u5728\u4efb\u4f55\u5c42\u6b21\u4e0a\u6574\u5408\u65cb\u8f6c\u7b49\u53d8\u6027\uff0c\u800c\u4e0d\u4ec5\u4ec5\u662f\u5728\u5c40\u90e8\u64cd\u4f5c\u4e2d\u3002<\/p>\n\n\n\n<p>\u6700\u540e\u7ed9\u51fa\\(\\sum_{l_{1},l_{2}}C_{l_{1},l_{2}}^{l,m}\\hat{\\boldsymbol{W}}_{i-1}^{l{1},l_{2}}\\hat{\\boldsymbol{h}}_{i-1}^{(l{1})}\\otimes\\hat{\\boldsymbol{h}}_{i-1}^{(l{2})}\\)\u66f4\u8be6\u7ec6\u7684\u516c\u5f0f\u89e3\u91ca\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\tilde{\\boldsymbol{h}}^{(l_{1})}\\otimes_{\\mathrm{CG}}\\tilde{\\boldsymbol{h}}^{(l_{2})}(l)=\\sum_{l_{1},l_{2}}C_{l_{1},l_{2}}^{l,m}\\hat{\\boldsymbol{W}}_{i-1}^{l_{1},l_{2}}\\hat{\\boldsymbol{h}}_{i-1}^{(l_{1})}\\otimes\\hat{\\boldsymbol{h}}_{i-1}^{(l_{2})}=w\\sum_{m_{1}=-l_{1}}^{l_{1}}\\sum_{m_{2}=-l_{2}}^{l_{2}}C\\left(l_{1},m_{1}\\right)\\left(l_{2},m_{2}\\right)(l,m)\\tilde{\\boldsymbol{h}}_{m_{1}}^{(l_{1})}\\tilde{\\boldsymbol{h}}_{m_{2}}^{(l_{2})}\\)<\/p>\n\n\n<p>\u5982\u4e0a\u8ff0\u516c\u5f0f\u63cf\u8ff0\uff0c\u5bf9\u4e8e\u7ed9\u5b9a\u7684\u4e24\u4e2a\u53ef\u5bfc\u5411\u5411\u91cf\\(\\tilde{\\boldsymbol{h}}(l_{1})\\)\u548c\\(\\tilde{\\boldsymbol{h}}(l_{2})\\)\uff0c\u9996\u5148\u9009\u5b9a\u5b83\u4eec\u5404\u81ea\u7684\u5206\u91cf\\(\\tilde{h}_{m_1}^{(l_1)}\\)\u548c\\(\\tilde{h}_{m_2}^{(l_2)}\\)\u3002\u8fd9\u4e9b\u5206\u91cf\u662f\u901a\u8fc7\\(m_{1}\\)\u548c\\(m_{2}\\)\u7684\u7d22\u5f15\u8bbf\u95ee\u7684\uff0c\u5176\u4e2d\\(m_{1}\\)\u4ece\\(-l_{1}\\)\u5230\\(l_{1}\\)\u53d8\u5316\uff0c\\(m_{2}\\)\u4ece\\(-l_{2}\\)\u5230\\(l_{2}\\)\u53d8\u5316\u3002\u5bf9\u4e8e\u6bcf\u4e00\u5bf9\\(m_{1}\\)\u548c\\(m_{2}\\)\uff0c\u8ba1\u7b97\u4e0e\u4e4b\u5bf9\u5e94\u7684CG\u7cfb\u6570\\(C\\left(l_1,m_1\\right)(l_2,m_2)(l,m)\\)\u3002\u8fd9\u4e9b\u7cfb\u6570\u662f\u9884\u5148\u8ba1\u7b97\u597d\u7684\uff0c\u901a\u5e38\u5b58\u50a8\u5728\u8868\u683c\u6216\u901a\u8fc7\u4e13\u95e8\u7684\u7b97\u6cd5\u5b9e\u65f6\u8ba1\u7b97\u3002\u5c06\u6bcf\u4e00\u5bf9\u5206\u91cf\\(\\tilde{\\boldsymbol{h}}_{m_1}^{(l_1)}\\)\u548c\\(\\tilde{h}_{m_2}^{(l_2)}\\)\u76f8\u4e58\uff0c\u5e76\u5c06\u7ed3\u679c\u4e58\u4ee5\u76f8\u5e94\u7684CG\u7cfb\u6570\\(C\\left(l_1,m_1\\right)(l_2,m_2)(l,m)\\)\u3002\u5bf9\u6240\u6709\\(m_{1}\\)\u548c\\(m_{2}\\)\u7684\u7ec4\u5408\u6267\u884c\u6c42\u548c\u64cd\u4f5c\uff0c\u5c06\u6bcf\u4e2a\u4e58\u79ef\u7d2f\u52a0\u3002\u52a0\u6743\u56e0\u5b50\\({\\mathcal{W}}\\)\u5e94\u7528\u4e8e\u6574\u4e2a\u6c42\u548c\u7ed3\u679c\uff0c\u7528\u4e8e\u8c03\u6574\u6574\u4f53\u5f20\u91cf\u79ef\u7684\u6bd4\u4f8b\u3002\u6700\u7ec8\u7ed3\u679c\\(\\tilde{\\boldsymbol{h}}^{(l_1)}\\otimes_{\\mathrm{CG}}\\tilde{\\boldsymbol{h}}^{(l_2)}(l)\\)\u662f\u4e00\u4e2a\u65b0\u7684\u7279\u5f81\u5411\u91cf\u3002\u5728\u51e0\u4f55\u56fe\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c \u548c \u53ef\u4ee5\u88ab\u89c6\u4e3a\u8282\u70b9\u7684\u7279\u5f81\u5411\u91cf\uff0c\u800c\u64cd\u4f5c\u7ed3\u679c \u53ef\u4ee5\u89c6\u4e3a\u8282\u70b9\u5728\u4fe1\u606f\u4f20\u9012\u540e\u7684\u65b0\u7279\u5f81\u5411\u91cf\uff0c\u8fd9\u53ef\u4ee5\u7528\u4e8e\u8282\u70b9\u72b6\u6001\u7684\u66f4\u65b0\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5728\u4fe1\u606f\u4f20\u9012\u9636\u6bb5\uff0c\u8fd9\u79cd\u8ba1\u7b97\u53ef\u4ee5\u63cf\u8ff0\u8282\u70b9\u95f4\u5982\u4f55\u4ea4\u4e92\uff0c\u7279\u522b\u662f\u5982\u4f55\u5c06\u90bb\u63a5\u8282\u70b9\u7684\u4fe1\u606f\u5408\u6210\u5230\u5f53\u524d\u8282\u70b9\u7684\u72b6\u6001\u4e2d\u3002\u5728\u8282\u70b9\\(\\mathrm{i}\\)\u548c\u5176\u90bb\u5c45\\(\\mathrm{j}\\)\u4e4b\u95f4\u7684\u4fe1\u606f\u4ea4\u4e92\u53ef\u80fd\u5c31\u662f\u901a\u8fc7\u8fd9\u6837\u7684CG\u7cfb\u6570\u8026\u5408\u6765\u5b9e\u73b0\u7684\uff0c\u8fd9\u6709\u52a9\u4e8e\u7f51\u7edc\u6355\u6349\u5230\u65cb\u8f6c\u548c\u5176\u4ed6\u5bf9\u79f0\u6027\u76f8\u5173\u7684\u7279\u6027\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">3\u3001\u57fa\u4e8e\u53ef\u5bfc\u5411\u7279\u5f81\u7684\u8282\u70b9\u4fe1\u606f\u8868\u793a<\/h2>\n\n\n\n<p>\u5728SEGNN\u4e2d\uff0c\u7279\u5f81\u6784\u5efa\u4fa7\u91cd\u4e8e\u7ed3\u5408\u8282\u70b9\u95f4\u7684\u51e0\u4f55\u4fe1\u606f\u4ee5\u53ca\u8282\u70b9\u672c\u8eab\u7684\u7279\u5f81\u3002SEGNN\u901a\u8fc7\u6574\u5408\u4e24\u4e2a\u8282\u70b9\u7684\u7279\u5f81\u5e76\u7ed3\u5408\u5b83\u4eec\u4e4b\u95f4\u7684\u8ddd\u79bb\u4fe1\u606f\u6765\u6784\u5efa\u8282\u70b9\u7684\u6d88\u606f\u5411\u91cf\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\tilde{\\boldsymbol{v}}_{ij}^{(L)}=\\tilde{\\boldsymbol{v}}_i^{(L)}\\oplus\\tilde{\\boldsymbol{v}}_j^{(L)}\\oplus\\left[\\left\\|\\widetilde{\\boldsymbol{x}}_{ij}\\right\\|^2\\right]\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff1a<\/p>\n\n\n\n<p>\uff081\uff09\\(\\tilde{\\nu}_i^{(L)}\\)\u548c\\(\\tilde{\\nu}_j^{(L)}\\)\u5206\u522b\u662f\u8282\u70b9\\(\\mathrm{i}\\)\u548c\u5176\u90bb\u5c45\u8282\u70b9\\(\\mathrm{j}\\)\u7684\u53ef\u5bfc\u5411\u7279\u5f81\u3002<\/p>\n\n\n\n<p>\uff082\uff09\\(\\tilde{x}_{ij}=\\tilde{x}_i-\\tilde{x}_j\\)\u662f\u8282\u70b9\\(\\mathrm{i}\\)\u548c\\(\\mathrm{j}\\)\u4e4b\u95f4\u7684\u5411\u91cf\u5dee\uff0c\u5373\u5f84\u5411\u5411\u91cf\u3002<\/p>\n\n\n\n<p>\uff083\uff09\\(\\left[\\left|\\tilde{x}_{ij}\\right|^2\\right]\\)\u662f\u8282\u70b9\u95f4\u8ddd\u79bb\u7684\u5e73\u65b9\uff0c\u4f5c\u4e3a\u4e00\u4e2a\u7279\u5f81\u52a0\u5165\u5230\u8282\u70b9\u5bf9\u7684\u7279\u5f81\u5411\u91cf\u4e2d\u3002<\/p>\n\n\n\n<p>\uff084\uff09\\(\u2295\\)\u8868\u793a\u6cbf\u7279\u5f81\u901a\u9053\u8fdb\u884c\u7684\u8fde\u63a5\u64cd\u4f5c\u3002<\/p>\n\n\n\n<p>\u800c\u5728TFN\u7684\u4ecb\u7ecd\u4e2d\uff0c\u5176\u53ef\u5bfc\u5411\u7279\u5f81\u7684\u6784\u5efa\u4e3b\u8981\u901a\u8fc7\u4ee5\u4e0b\u6b65\u9aa4\u5b8c\u6210\uff1a<\/p>\n\n\n\n<p>\uff081\uff09\u9009\u62e9\u4e2d\u5fc3\u70b9\uff1a\u786e\u5b9a\u7528\u4e8e\u805a\u5408\u4fe1\u606f\u7684\u4e2d\u5fc3\u70b9\u3002<\/p>\n\n\n\n<p>\uff082\uff09\u5411\u91cf\u8ba1\u7b97\uff1a\u5bf9\u6bcf\u4e2a\u90bb\u5c45\u70b9\uff0c\u8ba1\u7b97\u4ece\u4e2d\u5fc3\u70b9\u5230\u90bb\u5c45\u70b9\u7684\u5411\u91cf\\(\\mathbf{r}\\)\u3002<\/p>\n\n\n\n<p>\uff083\uff09\u5411\u91cf\u89c4\u8303\u5316\uff1a\u5c06\u5411\u91cf\u89c4\u8303\u5316\u5230\u5355\u4f4d\u5411\u91cf\\(\\hat{\\boldsymbol{r}}\\)\uff0c\u5e76\u8ba1\u7b97\u5176\u957f\u5ea6\\(\\mathrm{r}\\)\u3002<\/p>\n\n\n\n<p>\uff084\uff09\u5e94\u7528\u7403\u9762\u8c10\u51fd\u6570\uff1a\u4f7f\u7528\u7403\u9762\u8c10\u51fd\u6570\\(Y_{m}^{(l_{f})}(\\hat{r})\\)\u6355\u6349\u4e2d\u5fc3\u70b9\u4e0e\u90bb\u5c45\u4e4b\u95f4\u7684\u89d2\u5ea6\u5173\u7cfb\u3002<\/p>\n\n\n\n<p>\uff085\uff09\u7f16\u7801\u8ddd\u79bb\u4fe1\u606f\uff1a\u4f7f\u7528\u51fd\u6570\\(R_c^{l_f,l_i}(r)\\)\u6765\u7f16\u7801\u8ddd\u79bb\u4fe1\u606f\uff0c\u4f7f\u4e0d\u540c\u8ddd\u79bb\u7684\u70b9\u4ea7\u751f\u4e0d\u540c\u7684\u5f71\u54cd\u529b\u3002<\/p>\n\n\n\n<p>\uff086\uff09\u5b9a\u4e49\u5377\u79ef\u6838\uff1a\u6784\u9020\u5377\u79ef\u6838\\(F_{crm}^{\\left(l_f,l_i\\right)}(r)=R_c^{l_f,l_i}(r)\\boldsymbol{Y}_m^{\\left(l_f\\right)}(\\hat{\\boldsymbol{r}})\\)\uff0c\u7efc\u5408\u8ddd\u79bb\u4fe1\u606f\u548c\u89d2\u5ea6\u5173\u7cfb\uff0c\u4f7f\u7528\u8be5\u6838\u805a\u5408\u90bb\u5c45\u7684\u4fe1\u606f\uff0c\u5f62\u6210\u65b0\u7684\u4e2d\u5fc3\u70b9\u7279\u5f81\u5411\u91cf\u3002<\/p>\n\n\n\n<p>\u5bf9\u6bd4\u4e4b\u4e0b\uff0cSEGNN\u6784\u5efa\u7279\u5f81\u7684\u65b9\u5f0f\u66f4\u52a0\u6734\u7d20\uff0c\u901a\u8fc7\u76f4\u63a5\u8fde\u63a5\u8282\u70b9\u7279\u5f81\u548c\u51e0\u4f55\u8ddd\u79bb\u4fe1\u606f\u8fdb\u884c\u6d88\u606f\u4f20\u9012\uff0c\u8fd9\u79cd\u6734\u7d20\u7684\u7279\u5f81\u6784\u5efa\u65b9\u5f0f\u66f4\u52a0\u7684\u7075\u6d3b\u6613\u6269\u5c55\u3002<\/p>\n\n\n\n<p>\u7b2c\u4e00\u5c0f\u8282\u4e2d\u4ecb\u7ecd\u4e86\u4f7f\u7528Winger-D\u77e9\u9635\u548cCG\u7cfb\u6570\u8fdb\u884cSteerable MLPs\u7684\u7279\u5f81\u66f4\u65b0\uff0c\u4e0b\u9762\u5bf9\u5176\u7ec6\u8282\u8fdb\u884c\u89e3\u91ca\u8bf4\u660e\uff0c\u5982\u56fe3\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=\"378\" src=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250106150615912-1024x378.png\" alt=\"\" class=\"wp-image-2571\" srcset=\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250106150615912-1024x378.png 1024w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250106150615912-300x111.png 300w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250106150615912-768x283.png 768w, https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2025\/01\/20250106150615912.png 1103w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">\u56fe3 \u7279\u5f81\u66f4\u65b0<\/figcaption><\/figure><\/div>\n\n\n<p>\u56fe3(a)\u663e\u793a\u4e86\u4e00\u4e2a\u53ef\u5bfc\u5411\u5411\u91cf\\(\\tilde{h}^{(l)}\\)\uff0c\u8fd9\u79cd\u5411\u91cf\u5728\u4e0d\u540c\u7684\u5b50\u7a7a\u95f4\\(V_{l}\\)\u4e2d\u88ab\u5c55\u5f00\uff0c\u8fd9\u91cc\\(l=0,1,2\\)\u3002\u6bcf\u4e2a\u5b50\u7a7a\u95f4\u901a\u8fc7\u57fa\u51fd\u6570\\(Y_m^{(l)}\\)\uff08\u7403\u8c10\u51fd\u6570\uff09\u8fdb\u884c\u8868\u793a\uff0c\u7403\u8c10\u51fd\u6570\u7528\u4e8e\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u8868\u8fbe\u70b9\u7684\u65b9\u5411\u7279\u6027\u3002\u5b83\u4eec\u5bf9\u65cb\u8f6c\u5177\u6709\u826f\u597d\u7684\u6570\u5b66\u7279\u6027\uff0c\u4f7f\u5f97\u5728\u8fdb\u884c\u65cb\u8f6c\u64cd\u4f5c\u65f6\u80fd\u591f\u7cbe\u786e\u5730\u63a7\u5236\u548c\u9884\u6d4b\u5411\u91cf\u7684\u53d8\u5316\u3002\u5411\u91cf\u7684\u4e0d\u540c\u5206\u91cf\\([h]_l^m\\)\u4e0e\u5bf9\u5e94\u7684\u57fa\u51fd\u6570\u76f8\u4e58\uff0c\u5f62\u6210\u4e86\u8fd9\u4e2a\u5411\u91cf\u7684\u5b8c\u6574\u8868\u8fbe\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5411\u91cf\u7684\u4e0d\u540c\u5206\u91cf\\([h]_l^m\\)\u5b9e\u9645\u4e0a\u662f\u6807\u91cf\u7cfb\u6570\uff0c\u5b83\u4eec\u4e0e\u76f8\u5e94\u9636\u6570\u548c\u5e8f\u6570\u7684\u7403\u8c10\u51fd\u6570\\(Y_m^{(l)}\\)\u76f8\u4e58\u3002\u8fd9\u6837\uff0c\u53ef\u5bfc\u5411\u5411\u91cf\u7684\u6bcf\u4e2a\u7ec4\u6210\u90e8\u5206\u53ef\u4ee5\u5199\u4e3a\\([h]_l^mY_m^{(l)}\\)\u3002\u8fd9\u79cd\u5f62\u5f0f\u7684\u5411\u91cf\u8868\u793a\uff0c\u901a\u8fc7\u5c06\u6bcf\u4e2a\u5206\u91cf\u4e58\u4ee5\u5bf9\u5e94\u7684\u7403\u8c10\u51fd\u6570\uff0c\u4e0d\u4ec5\u4fdd\u7559\u4e86\u5411\u91cf\u7684\u539f\u59cb\u4fe1\u606f\uff0c\u800c\u4e14\u901a\u8fc7\u7403\u8c10\u51fd\u6570\u7684\u7279\u6027\uff0c\u8fd9\u4e9b\u5411\u91cf\u7684\u8868\u793a\u81ea\u7136\u5730\u9002\u5e94\u4e86\u65cb\u8f6c\u53d8\u6362\uff0c\u6574\u4e2a\u53ef\u5bfc\u5411\u5411\u91cf\\(\\tilde{h}^{(l)}\\)\u7684\u5b8c\u6574\u8868\u8fbe\u662f\u6240\u6709\u8fd9\u4e9b\u4e58\u79ef\u7684\u603b\u548c\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">4\u3001SEGNN\u7684\u6a21\u578b\u7ed3\u6784\u4e0e\u8ba1\u7b97\u65b9\u5f0f<\/h2>\n\n\n\n<p>SEGNN\u6a21\u578b\u53ef\u4ee5\u770b\u4f5cTFN\u6a21\u578b\u7684\u6539\u8fdb\uff0c\u4e3b\u8981\u6539\u8fdb\u70b9\u5728TFN\u7684\u70b9\u5377\u79ef\u64cd\u4f5c\u4e0a\uff0c\u4e0b\u9762\u5148\u603b\u7ed3\u4e00\u4e0bTFN\u6a21\u578b\uff0c\u518d\u5f15\u51faSEGNN\u7684\u6a21\u578b\u7ed3\u6784\u4e0e\u5176\u8ba1\u7b97\u65b9\u5f0f\u3002<\/p>\n\n\n\n<p>TFN\u7684\u6d88\u606f\u4f20\u9012\u8fc7\u7a0b\u53ef\u4ee5\u8868\u793a\u4e3a\u4ee5\u4e0b\u4e09\u4e2a\u6b65\u9aa4\u3002<\/p>\n\n\n\n<p>1\u3001\u4fe1\u606f\u4f20\u9012\u4e0e\u66f4\u65b0\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\boldsymbol{M}_{ij}^{(L)}=\\boldsymbol{Y}^{(L)}\\left(\\frac{\\boldsymbol{x}_{ij}}{\\|\\boldsymbol{x}_{ij}\\|}\\right)\\otimes_{\\boldsymbol{W}_{\\mathrm{cg}}}\\boldsymbol{V}_{j}^{(L)}\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff0c\\(x_{ij}=x_i-x_j\\)\u662f\u8282\u70b9\\(\\mathrm{i}\\)\u548c\u8282\u70b9\\(\\mathrm{j}\\)\u4e4b\u95f4\u7684\u5f84\u5411\u5411\u91cf\uff0c\\(\\boldsymbol{W}\\)\u4e2d\u7684\u5143\u7d20\u7531\u5f84\u5411\u591a\u5c42\u611f\u77e5\u673a\uff08MLP\uff09\\(f(\\begin{Vmatrix}x_{ij}\\end{Vmatrix})\\)\u6839\u636e\u8282\u70b9\u95f4\u7684\u8ddd\u79bb\\(\\left|x_{ij}\\right|\\)\u751f\u6210\uff0c\u8fd9\u91cc\\(x_{i}\\)\u56fa\u5b9a\u4e3a\u8f93\u5165\u6570\u636e\u7684\u521d\u59cb\u5750\u6807\u3002\u6ee4\u6ce2\u5668\u51fd\u6570\\(Y^{(L)}\\)\u7528\u4e8e\u8c03\u6574\u5f84\u5411\u5411\u91cf\u7684\u65b9\u5411\u7279\u6027\uff0c\\(\\otimes_{W_{\\mathrm{cg}}}\\)\u8868\u793a\u5e26\u6709\u6743\u91cd\\(W_{\\mathrm{cg}}\\)\u7684\u5f20\u91cf\u79ef\u64cd\u4f5c\u3002\u6bcf\u4e2a\u8282\u70b9\\(\\mathrm{j}\\)\u7684\u7279\u5f81\u5411\u91cf\u8868\u793a\u4e3a\\(V_{j}^{(L)}\\)\uff0c\u5e76\u4e14\u5728\u5377\u79ef\u64cd\u4f5c\u4e2d\u88ab\u7528\u4e8e\u66f4\u65b0\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u4fe1\u606f\uff0c\u6bcf\u4e2a\u8282\u70b9\u7684\u66f4\u65b0\u901a\u8fc7\u5305\u62ec\u805a\u5408\u5728\u5185\u7684\u4e00\u7cfb\u5217\u64cd\u4f5c\u6765\u5b9e\u73b0\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(U_i^{(L)}=V_i^{(L)}+\\sum_{j\\in\\mathcal{N}(i)}M_{ij}^{(L)}\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff0c\\(U_i^{(L)}\\)\u8868\u793a\u7b2c\\(\\text{L}\\)\u5c42\u4e2d\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u66f4\u65b0\u7279\u5f81\u5411\u91cf\uff1b\\(V_{i}^{(L)}\\)\u8868\u793a\u7b2c\\(\\text{L}\\)\u5c42\u4e2d\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u521d\u59cb\u7279\u5f81\u5411\u91cf\uff1b\\(\\mathcal{N}(i)\\)\u662f\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u90bb\u5c45\u96c6\u5408\u3002<\/p>\n\n\n\n<p>\u4e0a\u8ff0\u516c\u5f0f\u5c06\u6765\u81ea\u90bb\u5c45\u7684\u4fe1\u606f\\(\\begin{pmatrix}\\widetilde{\\boldsymbol{M}}_{ij}^{(L)}\\end{pmatrix}\\)\u805a\u5408\u5230\u5f53\u524d\u8282\u70b9\u7279\u5f81\\(\\tilde{V}_i^{(L)}\\)\u4e0a\u3002<\/p>\n\n\n\n<p>2\u3001\u81ea\u6211\u4ea4\u4e92\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\widetilde{V}_i^{(L)}=\\left\\{U^{(l)}W^{(l)}\\right\\}_{l\\in L}\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff0c\\(W^{(l)}\\)\u662f\u53ef\u5b66\u4e60\u7684\u901a\u9053\u6df7\u5408\u77e9\u9635\uff0c\u7528\u4e8e\u4e0d\u540c\u7c7b\u578b\\(\\mathrm{l}\\)\u7684\u7279\u5f81\u4e4b\u95f4\u7684\u7ebf\u6027\u8f6c\u6362\u3002<\/p>\n\n\n\n<p>3\u3001\u8282\u70b9\u975e\u7ebf\u6027\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\widetilde{\\boldsymbol{V}}_i^{(L)}=\\left\\{\\widetilde{\\boldsymbol{V}}_i^{(l)}\\sigma\\left(\\left\\|\\widetilde{\\boldsymbol{V}}_i^{(l)}\\right\\|_2+b^{(l)}\\right)\\right\\}_{l\\in L}\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff0c\\(\\sigma\\)\u662f\u6fc0\u6d3b\u51fd\u6570\uff08\u5982ReLU\u6216Sigmoid\uff09\uff0c\u7528\u4e8e\u5f15\u5165\u975e\u7ebf\u6027\u5e76\u5904\u7406\u5f52\u4e00\u5316\u540e\u7684\u7279\u5f81\u5411\u91cf\u7684\u6a21\u957f\uff0c\u975e\u7ebf\u6027\u7684\u8ba1\u7b97\u7ed3\u679c\\(\\sigma\\left(\\left|\\widetilde{\\boldsymbol{V}}_i^{(l)}\\right|_2+b^{(l)}\\right)\\)\u53ef\u4ee5\u770b\u4f5c\u7f29\u653e\u56e0\u5b50\uff0c\u5bf9\u8282\u70b9\\(\\widetilde{V}_i^{(l)}\\)\u8fdb\u884c\u7f29\u653e\u3002<\/p>\n\n\n\n<p>\u5bf9\u6bd4TFN\u6a21\u578b\uff0cSEGNN\u7684\u4e3b\u8981\u533a\u522b\u662f\u5728\u7b2c\u4e00\u6b65\uff0c\u5177\u4f53\u6765\u8bf4\u662f\u5f15\u5165\u4e86\u95e8\u63a7\u5411\u91cf\u5e76\u589e\u52a0\u7684\u5c42\u7ea7\u7ed3\u6784\uff0c\u95e8\u63a7\u673a\u5236\u666e\u904d\u8fd0\u7528\u5728\u957f\u77ed\u671f\u8bb0\u5fc6\u7f51\u7edcLSTM\u6a21\u578b\u4e2d\uff0c\u7528\u4e8e\u63a7\u5236\u4fe1\u606f\u6d41\uff0c\u7279\u522b\u662f\u5728\u5904\u7406\u590d\u6742\u8f93\u5165\u6a21\u5f0f\u6216\u9700\u8981\u7cbe\u7ec6\u7ba1\u7406\u4fe1\u606f\u4f20\u9012\u7684\u4efb\u52a1\u4e2d\u3002\u95e8\u63a7\u673a\u5236\u53ef\u4ee5\u52a8\u6001\u5730\u8c03\u8282\u7f51\u7edc\u4e2d\u4e0d\u540c\u90e8\u5206\u7684\u4fe1\u606f\u6743\u91cd\uff0c\u4ece\u800c\u5141\u8bb8\u7f51\u7edc\u5b66\u4e60\u5982\u4f55\u9009\u62e9\u548c\u5229\u7528\u6700\u6709\u7528\u7684\u4fe1\u606f\uff0c\u516c\u5f0f\u5982\u4e0b\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\boldsymbol{V}_{ij}^{(\\mathrm{L})},\\boldsymbol{g}_{ij}=\\boldsymbol{Y}^{(\\mathrm{L})}\\left(\\frac{\\boldsymbol{x}_{ij}}{\\left\\|\\boldsymbol{x}_{ij}\\right\\|}\\right)\\otimes_{\\mathrm{cg}}^{\\boldsymbol{W}}\\boldsymbol{V}_{j}^{(\\mathrm{L})}\\)<\/p>\n<p style=\"text-align: center;\">\\(\\boldsymbol{M}_{ij}^{(\\mathrm{L})}=\\mathrm{~Gate~}\\left(\\boldsymbol{V}_{ij}^{(\\mathrm{L})},\\mathrm{Swish}\\left(\\boldsymbol{g}_{ij}\\right)\\right)\\)<\/p>\n\n\n<p>\u5176\u4e2d\uff0c\\(\\mathbf{g}_{ij}\\)\u4f5c\u95e8\u63a7\u673a\u5236\u4e2d\u63a7\u5236\u4fe1\u606f\u6d41\u7684\u63a7\u5236\u56e0\u5b50\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5f53\u4f7f\u7528CG\u7cfb\u6570\u6765\u8fdb\u884c\u76ee\u6807\u70b9\\(\\mathrm{i}\\) \u548c\u5176\u90bb\u5c45\\(\\mathrm{j}\\)\u7684\u4fe1\u606f\u4ea4\u4e92\u65f6\uff0c\u8f93\u51fa\u5305\u62ec\u4e86\u4e0d\u540c\u9636\u6570\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\uff08irreps\uff09\u3002\u8fd9\u4e9birreps\u7684\u9636\u6570\\(\\mathrm{l}\\)\u8868\u793a\u5b83\u4eec\u7684\u65cb\u8f6c\u5bf9\u79f0\u7279\u6027\uff0c\u9636\u6570\u4e3a0\u7684irreps\u662f\u6807\u91cf\uff08\u6ca1\u6709\u65b9\u5411\u6027\uff09\uff0c\u5b83\u4eec\u4e0d\u53d8\u4e8e\u5750\u6807\u7cfb\u7684\u65cb\u8f6c\u3002\u66f4\u9ad8\u9636\u6570\u7684irreps\u5305\u542b\u65b9\u5411\u4fe1\u606f\uff0c\u5982\u5411\u91cf\uff08\\(l=1\\)\uff09\u3001\u5f20\u91cf\uff08\\(l=2\\)\uff09\u7b49\u3002\u901a\u8fc7CG\u5f20\u91cf\u4e58\u79ef\u5408\u6210\u7684\u4e0d\u540c\u9636\u6570\u7684irreps\u53ef\u7528\u4e8e\u7f16\u7801\u8282\u70b9\u95f4\u7684\u590d\u6742\u7a7a\u95f4\u5173\u7cfb\u3002\u8fd9\u91cc\u5c06\u9636\u6570\u4e3a0\u7684irreps \uff08\u5373\u6807\u91cf irreps\uff09\u7528\u4f5c\u95e8\u63a7\u673a\u5236\u4e2d\u7684\u8c03\u5236\u9879\\(\\boldsymbol{g}_{ij}\\)\u3002\u56e0\u4e3a\u5b83\u4eec\u662f\u6807\u91cf\uff0c\u4e0d\u5f15\u5165\u4e0e\u65b9\u5411\u76f8\u5173\u7684\u590d\u6742\u6027\u3002\u8fd9\u4e9b\u6807\u91cf\u901a\u8fc7\u6fc0\u6d3b\u51fd\u6570Swish\u8fdb\u884c\u5904\u7406\u5f97\u5230\u7684\u7ed3\u679c\u88ab\u79f0\u4e3a\u95e8\u63a7\u5411\u91cf\u3002Gate()\u51fd\u6570\u5177\u4f53\u5730\u5b9e\u73b0\u4e86\u5982\u4f55\u5c06\u95e8\u63a7\u5411\u91cf\\(\\mathrm{Swish}(\\boldsymbol{g}_{ij})\\)\u4e0e\u7279\u5f81\u5411\u91cf\\(V_{ij}^{(\\mathrm{L})}\\)\u76f8\u7ed3\u5408\uff0c\u6765\u51b3\u5b9a\u4fe1\u606f\u7684\u6d41\u52a8\u4e0e\u4fdd\u7559\u7684\u7a0b\u5ea6\u3002\u901a\u5e38\uff0c\u8fd9\u53ef\u4ee5\u662f\u4e00\u4e2a\u7b80\u5355\u7684\u4e58\u6cd5\u64cd\u4f5c\u3002<\/p>\n\n\n\n<p>\u53e6\u4e00\u65b9\u9762\uff0cSEGNN\u589e\u52a0\u4e86\u5c42\u7ea7\u7ed3\u6784\u6765\u6269\u5c55\u6a21\u578b\u7684\u62df\u5408\u80fd\u529b\uff0c\u5176\u66f4\u65b0\u516c\u5f0f\u4e3a\uff1a<\/p>\n\n\n<p style=\"text-align: center;\">\\(\\boldsymbol{V}_j^{(\\mathrm{L})},\\boldsymbol{g}_i=\\left(\\sum_{j\\in\\mathcal{N}_i}\\boldsymbol{Y}^{(\\mathrm{L})}\\left(\\frac{\\boldsymbol{x}_{ij}}{\\left\\|\\boldsymbol{x}_{ij}\\right\\|}\\right)\\right)\\otimes_{\\mathrm{cg}}^{\\boldsymbol{W}_2}\\left(\\boldsymbol{V}_i^{(\\mathrm{L})}+\\sum_{j\\in\\mathcal{N}_i}\\boldsymbol{M}_{ij}^\\mathrm{L}\\right)\\)<\/p>\n\n<p style=\"text-align: center;\">\\(V_i^{\\prime(\\mathrm{L})}=V_i^{(\\mathrm{L})}+\\mathrm{Gate}\\left(V_i^{(\\mathrm{L})},\\mathrm{Swish}(\\boldsymbol{g}_i)\\right)\\)<\/p>\n\n\n<p>\u5bf9\u6bd4\u4e4b\u4e0b\uff0c\\(V_{i}^{(\\mathcal{L})}+\\sum_{j\\in\\mathcal{N}{i}}\\boldsymbol{M}_{ij}^{\\mathcal{L}}\\)\u662fTFN\u548cSEGNN\u4e24\u4e2a\u6a21\u578b\u76f8\u540c\u7684\u5730\u65b9\uff0c\u90fd\u662f\u4ece\u90bb\u5c45\u8282\u70b9\u805a\u5408\u4fe1\u606f\u7684\u903b\u8f91\u3002\u5728SEGNN\u4e2d\uff0c\u5f15\u5165\u7684\u989d\u5916\u90e8\u5206\\(\\left(\\sum_{j\\in\\mathcal{N}{i}}\\boldsymbol{Y}^{(\\mathrm{L})}\\left(\\frac{\\boldsymbol{x}_{ij}}{\\left|\\boldsymbol{x}_{ij}\\right|}\\right)\\right)\\otimes{\\mathrm{cg}}^{\\boldsymbol{W}_{2}}\\)\u548c\\(V_i^{\\prime(\\mathrm{L})}=V_i^{(\\mathrm{L})}+\\mathrm{Gate}\\left(V_i^{(\\mathrm{L})},\\mathrm{Swish}(\\boldsymbol{g}_i)\\right)\\)\u5b9e\u9645\u4e0a\u662f\u5bf9\u805a\u5408\u540e\u7684\u4fe1\u606f\u518d\u7ecf\u8fc7\u4e00\u6b21SEGNN\u7684\u7b2c\u4e00\u6b65\uff0c\u5373CG\u7cfb\u6570\u7684\u7279\u5f81\u878d\u5408\u4e0e\u95e8\u63a7\u673a\u5236\u7684\u7279\u5f81\u9009\u62e9\u3002\u5806\u53e0\u5c42\u6b21\u7ed3\u6784\u53ef\u4ee5\u589e\u5f3a\u6a21\u578b\u7684\u8868\u8fbe\u80fd\u529b\uff0c\u4f7f\u5176\u80fd\u591f\u66f4\u597d\u5730\u62df\u5408\u590d\u6742\u7684\u6570\u636e\u6a21\u5f0f\uff0c\u89e3\u51b3\u66f4\u4e3a\u590d\u6742\u7684\u95ee\u9898\u3002\u901a\u8fc7\u5f15\u5165\u591a\u5c42\u7ed3\u6784\u548c\u95e8\u63a7\u673a\u5236\uff0cSEGNN 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