![\"\"](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241227224259246.png\")
\u5982\u56fe1\u6240\u793a\uff0c\u8003\u8651\u4e86\u4e00\u4e2a\u591a\u4f53\u7cfb\u7edf\u4e2d\u7684\u4e24\u4f53\u5b50\u7cfb\u7edf\\((x_i,x_j)\\)\uff0c\u8fd9\u4e2a\u7cfb\u7edf\u4e0d\u968f\u65f6\u95f4\u53d8\u5316\uff0c\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u4e00\u4e2a\u5177\u6709\u4e24\u4e2a\u8282\u70b9\u548c\u4e00\u6761\u8fb9\u7684\u56fe\u3002\u4e00\u822c\u800c\u8a00\uff0c\u4e24\u4e2a\u7c92\u5b50\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u51fd\u6570\u53ef\u4ee5\u5206\u89e3\u6210\u4e24\u90e8\u5206\uff1a\\(U\\left(x_{i},x_{j}\\right):=U_{\\mathrm{in}}\\left(\\parallel x_{i}-x_{j}\\parallel\\right)+U_{\\mathrm{ext}}\\left(x_{i},x_{j}\\right)\\) \uff0c\u5176\u4e2d\u7b2c\u4e00\u90e8\u5206\u662f\u76f8\u5bf9\u8ddd\u79bb\\(\\left|x_i-x_j\\right|\\)\u7684\u51fd\u6570\uff08\u4f8b\u5982\u7535\u9759\u529b)\uff0c\u7b2c\u4e8c\u90e8\u5206\u63cf\u8ff0\u4e86\u53ef\u80fd\u53d6\u51b3\u4e8e\u76f8\u5bf9\u8ddd\u79bb\u548c\u89d2\u5ea6\uff08\u4f8b\u5982\u7535\u78c1\u573a\u3001\u626d\u529b\uff09\u7684\u5916\u90e8\u529b\u573a\u7684\u5f71\u54cd\u3002\u56e0\u6b64\uff0c\\(U_{\\mathrm{ext}}\\left(x_i,x_j\\right)\\)\u7684\u68af\u5ea6\u53ef\u4ee5\u6cbf\u7740\u4efb\u610f\u65b9\u5411\u57283D\u7a7a\u95f4\u4e2d\u53d1\u6563\u3002\u5927\u591a\u6570\u51e0\u4f55GNN\u65b9\u6cd5\u5728\u5904\u7406\u8fd9\u79cd\u529b\u9884\u6d4b\u95ee\u9898\u65f6\uff0c\u90fd\u662f\u901a\u8fc7\u91c7\u7528\u4e0d\u53d8\u7279\u5f81\uff08\u4f8b\u5982\uff0c\u76f8\u5bf9\u8ddd\u79bb\uff09\u4f5c\u4e3a\u8f93\u5165\uff0c\u5e76\u8868\u793a\u6cbf\u5f84\u5411\u7684\u529b\uff0c\u5373\\(F_{i}=k\\left(x_{i}-x_{j}\\right)\\)\uff0c\u8fd9\u5728\u68af\u5ea6\\(\\nabla U_{\\mathrm{ext}}(x_i,x_j)\\)\u4e0d\u662f\u6cbf\u7740\u5f84\u5411\u65f6\u662f\u4e0d\u591f\u7684\u3002\u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898\uff0cclofNet\u63d0\u51fa\u4e86\u4e00\u4e2a\u6b63\u4ea4\u7684\u7b49\u53d8\u5c40\u90e8\u5750\u6807\u7cfb\u6846\u67b6\uff08\u4e00\u79cd\u6b63\u4ea4\u57fa\u7ec4\uff09\uff0c\u7528\u4e8e\u6784\u5efa\u548c\u8868\u793a\u51e0\u4f55\u6570\u636e\u66f4\u5b8c\u5907\u7684\u5c40\u90e8\u7ed3\u6784\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u53d6\\(x_{i}-x_{j}\\)\u4f5c\u4e3a\u7b2c\u4e00\u4e2a\u5206\u91cf\/\u65b9\u5411\\(\\mathrm{a}\\)\uff0c\u7136\u540e\\(x_i\\times x_j\\)\u4f5c\u4e3a\u7b2c\u4e8c\u65b9\u5411\\(\\mathrm{b}\\)\u3002\u6700\u540e\uff0c\u524d\u4e24\u4e2a\u65b9\u5411\u7684\u53c9\u79ef\u88ab\u4f5c\u4e3a\u7b2c\u4e09\u65b9\u5411\\(\\mathrm{c}\\)\u3002\u8fd9\u4e09\u4e2a\u5411\u91cf\\(\\mathrm{a}\\),\\(\\mathrm{b}\\)\u548c\\(\\mathrm{c}\\) \u5b9a\u4e49\u4e86\u4e00\u4e2a\u5c40\u90e8\u5750\u6807\u7cfb\uff0c\u7531\u4e8e\u53c9\u4e58\u7684\u6027\u8d28\uff0c\\(\\mathrm{b}\\)\u5782\u76f4\u4e8e\\(\\mathrm{a}\\)\uff0c\\(\\mathrm{c}\\)\u5782\u76f4\u4e8e\\(\\mathrm{a}\\)\u548c\\(\\mathrm{b}\\) \u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u8fd9\u4e2a\u5750\u6807\u7cfb\u5177\u6709\u6b63\u4ea4\u6027\u3002<\/p>\n\n\n\n
2\u3001clofNet\u6a21\u578b\u603b\u4f53\u67b6\u6784<\/h2>\n\n\n\n
clofNet\u6a21\u578b\u5904\u7406\u6570\u636e\u7684\u6d41\u7a0b\u53ef\u4ee5\u5206\u4e3a\u4e2d\u5fc3\u5316\uff08Centralization\uff09\u3001\u5c40\u90e8\u5b8c\u5907\u6846\u67b6\uff08Complete Frame\uff09\u3001\u6807\u91cf\u5316\uff08Scalarize\uff09\u3001\u6d88\u606f\u4f20\u9012\u6a21\u5757\uff08GMP\uff09\u3001\u5411\u91cf\u5316\uff08Vectorize\uff09\u548c\u8fed\u4ee3\u66f4\u65b0\u516d\u4e2a\u6b65\u9aa4\uff0c\u5982\u56fe2\u6240\u793a\u3002<\/p>\n\n\n
![\"\"](\"https:\/\/gnnclub-1311496010.cos.ap-beijing.myqcloud.com\/wp-content\/uploads\/2024\/12\/20241227225032814.png\")
\u9488\u5bf9\u8f93\u5165\u6570\u636e\uff0c\u6a21\u578b\u9996\u5148\u5c06\u6bcf\u4e2a\u70b9\u7684\u4f4d\u7f6e\u8fdb\u884c\u4e2d\u5fc3\u5316\u5904\u7406\u3002\u8fd9\u901a\u5e38\u610f\u5473\u7740\u4f1a\u4ece\u6bcf\u4e2a\u4f4d\u7f6e\u5411\u91cf\u4e2d\u51cf\u53bb\u6240\u6709\u4f4d\u7f6e\u5411\u91cf\u7684\u5e73\u5747\u503c\uff0c\u4ee5\u786e\u4fdd\u6a21\u578b\u5bf9\u5e73\u79fb\u4fdd\u6301\u4e0d\u53d8\u3002\u8fd9\u662f\u56e0\u4e3a\u901a\u8fc7\u51cf\u53bb\u6240\u6709\u70b9\u4f4d\u7f6e\u5411\u91cf\u7684\u5e73\u5747\u503c\uff0c\u5b9e\u9645\u4e0a\u662f\u5728\u91cd\u65b0\u5b9a\u4f4d\u6574\u4e2a\u7cfb\u7edf\u7684\u4e2d\u5fc3\uff08\u8d28\u5fc3\uff09\u5230\u539f\u70b9\uff08\u5750\u6807\u7cfb\u7684\u4e2d\u5fc3\uff09\u7684\u8ddd\u79bb\u3002\u8fd9\u6837\uff0c\u6a21\u578b\u5c31\u4e0d\u4f1a\u5c06\u4efb\u4f55\u7279\u5b9a\u7684\u4f4d\u7f6e\u4f5c\u4e3a\u53c2\u8003\u70b9\uff0c\u56e0\u6b64\u65e0\u8bba\u7cfb\u7edf\u5728\u7a7a\u95f4\u4e2d\u5982\u4f55\u5e73\u79fb\uff0c\u6a21\u578b\u7684\u8f93\u51fa\u90fd\u5c06\u4fdd\u6301\u4e0d\u53d8\u3002<\/p>\n\n\n\n
\u5c40\u90e8\u5b8c\u5907\u6846\u67b6\\(\\mathcal{F}_{ij}\\)\u5c31\u662f\u4e0a\u4e00\u5c0f\u8282\u89e3\u91ca\u7684\u5c40\u90e8\u6b63\u4ea4\u5750\u6807\u7cfb\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u4ee4\\(a_{ij}^{t}=\\frac{x_{i}(t)-x_{j}(t)}{\\begin{Vmatrix}x_{i}(t)-x_{j}(t)\\end{Vmatrix}}\\)\uff0c\u5219\u53ef\u4ee5\u5b9a\u4e49\\(b_{ij}^{t}=\\frac{x_{i}(t)\\times x_{j}(t)}{\\left|x_{i}(t)\\times x_{j}(t)\\right|}\\)\u548c\\(c_{ij}^t=a_{ij}^t\\times b_{ij}^t\\)\u3002\u8fd9\u4e09\u4e2a\u76f8\u4e92\u6b63\u4ea4\u7684\u57fa\u7ec4\u53ef\u4ee5\u6784\u5efa\u4e00\u4e2a\\(SO(3)\\)-\u7b49\u53d8\u6846\u67b6\\(\\mathcal{F}{ij}:\\mathrm{EquiFrame}\\left(x_{i},x_{j}\\right)=\\left(a_{ij}^{t},b_{ij}^{t},c_{ij}^{t}\\right)\\)\u3002\u5b9e\u9645\u64cd\u4f5c\u4e2d\uff0c\u6211\u4eec\u6dfb\u52a0\u4e00\u4e2a\u5c0f\u5e38\u6570\\(\\epsilon\\)\u5230\u5f52\u4e00\u5316\u56e0\u5b50\u4e2d\uff0c\u4ee5\u9632\\(x_{i}\\)\u548c\\(x_{j}\\)\u53d1\u751f\u91cd\u5408\u3002<\/p>\n\n\n\n
\u57fa\u4e8e\u4e0a\u8ff0\u7684\u5c40\u90e8\u5b8c\u5907\u6846\u67b6\\(\\mathcal{F}_{ij}\\)\uff0c\u53ef\u4ee5\u5c06\u4efb\u610f\u51e0\u4f55\u5411\u91cf\u8f6c\u6362\u4e3a\\(SO(3)\\)\u4e0d\u53d8\u7684\u6807\u91cf\u503c\u3002\u5177\u4f53\u65b9\u6cd5\u662f\u5c06\u51e0\u4f55\u5411\u91cf\u6295\u5f71\u5230\\(\\mathcal{F}_{ij}\\)\u7684\u57fa\u7ec4\u4e0a\uff0c\u901a\u8fc7\u5185\u79ef\u83b7\u5f97\u76f8\u5e94\u7684\u7cfb\u6570\uff0c\u8fd9\u4e9b\u7cfb\u6570\u662f\u4e0d\u53d8\u6807\u91cf\uff0c\u8868\u793a\u4e3a\\(t_{ij}=\\left(x_{i}\\cdot a_{ij}^{t},x_{i}\\cdot b_{ij}^{t},x_{i}\\cdot c_{ij}^{t}\\right)\\)\uff0c\u8fd9\u5c31\u662f\u51e0\u4f55\u5411\u91cf\u6807\u91cf\u5316\u7684\u6b65\u9aa4\u3002<\/p>\n\n\n\n
\u63a5\u4e0b\u6765\uff0c\\(t_{ij}\\)\u5c06\u4e0e\u8282\u70b9\u7279\u5f81\u5411\u91cf\\(h_{i},h_{j}\\)\u548c\u8fb9\u7279\u5f81\u5411\u91cf\\(e_{ij}\\)\u4e00\u8d77\u4f5c\u4e3a\u6d88\u606f\u4f20\u9012\u6a21\u5757\uff08GMP\uff09\u7684\u8f93\u5165\u4fe1\u606f\u3002GMP\u5206\u4e3a\u7279\u5f81\u7f16\u7801\u548c\u4fe1\u606f\u4f20\u9012\u4e24\u4e2a\u4fe1\u606f\u5904\u7406\u9636\u6bb5\uff0c\u7279\u5f81\u7f16\u7801\u9636\u6bb5\u7684\u5904\u7406\u975e\u5e38\u7b80\u5355\uff0c\u51e0\u4f55\u6807\u91cf\\(t_{ij}\\)\u3001\u8282\u70b9\u7279\u5f81\\(h_{i},h_{j}\\)\u548c\u8fb9\u7279\u5f81\\(e_{ij}\\)\u5206\u522b\u7ecf\u8fc7\u795e\u7ecf\u7f51\u7edc\u5c42\u7684\u5904\u7406\uff0c\u5b66\u4e60\u6620\u5c04\u5f97\u5230\u66f4\u65b0\u540e\u7684\u65b0\u7279\u5f81\u8868\u793a\u3002<\/p>\n\n\n\n
\u5728\u4fe1\u606f\u4f20\u9012\u9636\u6bb5\uff0cclofNet\u63d0\u51fa\u4e86\u6734\u7d20\u7684\u6d88\u606f\u4f20\u9012\u548c\u57fa\u4e8e\u6ce8\u610f\u529b\u7684\u6d88\u606f\u4f20\u9012\u4e24\u79cd\u4e0d\u540c\u7684\u673a\u5236\u3002\u6734\u7d20\u7684\u6d88\u606f\u4f20\u9012\u673a\u5236\u975e\u5e38\u7c7b\u4f3c\u4e0e\u6a21\u578bEGNN\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5c06\u51e0\u4f55\u6807\u91cf\\(t_{ij}\\)\u3001\u8282\u70b9\u7279\u5f81\\(h_{i},h_{j}\\)\u548c\u8fb9\u7279\u5f81\\(e_{ij}\\)\u540c\u65f6\u4f5c\u4e3a\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u5c42\\(\\phi_{_m}\\)\u7684\u8f93\u5165\uff0c\u8fdb\u884c\u8282\u70b9\u7279\u5f81\u3001\u8fb9\u7279\u5f81\u548c\u51e0\u4f55\u7279\u5f81\u4e4b\u95f4\u7684\u4fe1\u606f\u878d\u5408\uff0c\u8ba1\u7b97\u5f97\u5230\u65b0\u7684\u7279\u5f81\u5411\u91cf\\(m_{ij}\\)\uff0c\u8fd9\u4e2a\u7279\u5f81\u5411\u91cf\u4ee3\u8868\u76ee\u6807\u8282\u70b9\\(\\mathrm{i}\\)\u548c\u5176\u90bb\u5c45\\(\\mathrm{j}\\)\u4e4b\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u5173\u7cfb\u3002\u56e0\u6b64\uff0c\u76ee\u6807\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u7279\u6027\u66f4\u65b0\u65b9\u5f0f\u5c31\u662f\u5bf9\u6240\u6709\u7684\u90bb\u5c45\\(\\mathrm{j}\\)\u548c\u5176\u81ea\u8eab\u4fe1\u606f\u8fdb\u884c\u6c42\u548c\uff0c\u5e76\u7ecf\u8fc7\u53e6\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u5c42\\(\\phi_{h}\\)\u8ba1\u7b97\u5f97\u5230\uff0c\u516c\u5f0f\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n
\\(\\boldsymbol{m}_{ij}=\\phi_{m}\\left(s_{ij},\\boldsymbol{h}_{i},\\boldsymbol{h}_{j},\\boldsymbol{e}_{ij}\\right)
\\)<\/p>\n
\\(\\boldsymbol{h}_i^{l+1}=\\phi_h\\left(\\boldsymbol{h}_i^l,\\sum_{j\\in\\mathcal{N}(i)}\\boldsymbol{m}_{ij}\\right)\\)<\/p>\n\n\n
\u6b64\u5916\uff0c\u4e0eEGNN\u76f8\u4f3c\uff0cclofNet\u4e5f\u4f1a\u5bf9\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u4f4d\u7f6e\u4fe1\u606f\\(x_{i}\\)\u8fdb\u884c\u66f4\u65b0\uff1b\u4e0d\u540c\u7684\u662fEGNN\u7684\u66f4\u65b0\u65b9\u5f0f\u662f\u76f4\u63a5\u57fa\u4e8e\u7b1b\u5361\u5c14\u5750\u6807\u7684\uff0c\u800cclofNet\u7684\u66f4\u65b0\u65b9\u5f0f\u662f\u57fa\u4e8e\u5c40\u90e8\u5750\u6807\u7cfb \u7684\uff0c\u5176\u516c\u5f0f\u8868\u793a\u4e3a\uff1a<\/p>\n\n\n
\\(\\mathcal{F}_{ij}^{l+1}=\\mathrm{EquiFrame}\\left(x_i^l,x_j^l\\right)\\)<\/p>\n
\\(\\boldsymbol{x}_{i}^{l+1}=\\boldsymbol{x}_{i}^{l}+\\frac{1}{N}\\sum_{j\\in\\mathcal{N}(x_{i})}\\mathrm{Vectorize}\\left(\\boldsymbol{m}_{ij}^{l},\\mathcal{F}_{ij}^{l}\\right)\\)<\/p>\n\n\n
\u5176\u4e2d\u6709\u4e2a\u5173\u952e\u6027\u64cd\u4f5c\uff1a\u5411\u91cf\u5316\uff0c\u8fd9\u662f\u6807\u91cf\u5316\u7684\u9006\u64cd\u4f5c\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5bf9\u4e8e\u67d0\u4e2a\u4f4d\u7f6e\u51e0\u4f55\u5411\u91cf\\(\\mathrm{x}\\)\u5728\u7b49\u53d8\u6846\u67b6\u4e0b\u7684\u6807\u91cf\u5316\u8868\u793a\\({x^{a},x^{b},x^{c}}\\)\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u8fd9\u4e9b\u6807\u91cf\u5206\u91cf\u4e0e\u57fa\u5411\u91cf\u7684\u7ebf\u6027\u7ec4\u5408\u91cd\u6784\u539f\u59cb\u5411\u91cf\uff0c\u5373\\(x=x^{a}a+x^{b}b+x^{c}c\\)\u3002 \u8fd9\u5bf9\u4e8e\u4ece\u7b80\u5316\u7684\u6807\u91cf\u8868\u793a\u4e2d\u6062\u590d\u590d\u6742\u7684\u51e0\u4f55\u7ed3\u6784\u81f3\u5173\u91cd\u8981\uff0c\u8ba9GMP\u6a21\u5757\u7684\u8f93\u51fa\u80fd\u591f\u76f4\u63a5\u53cd\u6620\u51fa\u539f\u59cb\u7a7a\u95f4\u4e2d\u7684\u5411\u91cf\u6027\u8d28\uff0c\u5e76\u4fdd\u6301\u5728\\(SO(3)\\)\u7fa4\u4e0b\u7684\u4e0d\u53d8\u6027\u3002\u5728\u4e0a\u8ff0\u516c\u5f0f\u4e2d\uff0c\\(x_i^{l+1}\\)\u548c\\(h_{i}^{l+1}\\)\u4fbf\u662fGMP\u6a21\u5757\u8f93\u51fa\u7684\u51e0\u4f55\u5411\u91cf\u4fe1\u606f\u548c\u8282\u70b9\u6807\u91cf\u4fe1\u606f\u3002<\/p>\n\n\n\n
\u81f3\u4e8e\u6d88\u606f\u4f20\u9012\u7684\u53e6\u4e00\u79cd\u673a\u5236\uff1a\u57fa\u4e8e\u6ce8\u610f\u529b\u7684\u673a\u5236\u3002\u5176\u7075\u611f\u4e3b\u8981\u6e90\u4e8eTransformer\u7b97\u6cd5\u4e2d\u7684Self-attention\u673a\u5236\uff0c\u516c\u5f0f\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n\n\n
\\(\\boldsymbol{q}_{i}=\\phi_{q}\\left(h_{i}^{\\prime}\\right),\\boldsymbol{k}_{ij}=\\phi_{k}\\left(h_{i}^{\\prime},m_{ij}\\right),\\boldsymbol{v}_{ij}=\\phi_{\\nu}\\left(m_{ij}\\right)\\)<\/p>\n
\\(\\alpha_{ij}=\\frac{\\left\\langle q_i,k_{ij}\\right\\rangle}{\\sum_{j^{^{\\prime}}\\in\\mathcal{N}(i)}\\left\\langle q_i,k_{ij}\\right\\rangle}\\)<\/p>\n
\\(\\mathcal{M}_i=\\mathrm{LayerNorm}\\left(\\sum_{j\\in\\mathcal{N}(i)}\\alpha_{ij}\\boldsymbol{v}_{ij}\\right)\\)<\/p>\n
\\(h_i^{l+1}=\\phi_h\\left(h_i^l,\\mathcal{M}_i\\right)\\)<\/p>\n
\\(h_i^{l+1}=h_i^l+\\mathrm{LayerNorm}\\left(h_i^{l+1}\\right)\\)<\/p>\n\n\n
\u5176\u4e2d\uff1a<\/p>\n\n\n\n
\uff081\uff09\\(q_{i}\\)\u662f\u67e5\u8be2\u5411\u91cf\uff0c\u5b83\u7531\u4e00\u4e2a\u591a\u5c42\u611f\u77e5\u673a\\(\\phi_{q}\\)\u5bf9\u8282\u70b9\\(\\begin{array}{c}{i}\\end{array}\\)\u7684\u7279\u5f81\\(h_{i}^{l}\\)\u8fdb\u884c\u8f6c\u6362\u5f97\u5230\u3002<\/p>\n\n\n\n
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