{"id":2525,"date":"2024-12-28T12:06:12","date_gmt":"2024-12-28T04:06:12","guid":{"rendered":"https:\/\/www.gnn.club\/?p=2525"},"modified":"2025-01-07T16:54:38","modified_gmt":"2025-01-07T08:54:38","slug":"%e5%bc%a0%e9%87%8f%e5%9c%ba%e7%a5%9e%e7%bb%8f%e7%bd%91%e7%bb%9ctfn","status":"publish","type":"post","link":"http:\/\/gnn.club\/?p=2525","title":{"rendered":"\u5f20\u91cf\u573a\u795e\u7ecf\u7f51\u7edcTFN"},"content":{"rendered":"\n

\u5f20\u91cf\u573a\u7f51\u7edc\uff08Tensor Field Networks, TFNs\uff09\u6e90\u4e8e\u8bba\u6587Tensor Field Networks: Rotation- and translation-equivariant neural networks for 3D point clouds\uff0c\u662f\u4e00\u4e2a\u4e3a\u5206\u5b50\u548c\u7269\u8d28\u5efa\u6a21\u91cf\u8eab\u5b9a\u5236\u7684\u5de5\u5177\uff0c\u5176\u8bbe\u8ba1\u5145\u5206\u8003\u8651\u4e86\u5316\u5b66\u548c\u7269\u7406\u5b66\u7684\u539f\u7406\uff0c\u7279\u522b\u662f\u5bf9\u79f0\u6027\u548c\u4e0d\u53d8\u6027\u3002\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0cTFN\u63d0\u4f9b\u4e86\u4e00\u4e2a\u5f3a\u5927\u7684\u6846\u67b6\uff0c\u53ef\u4ee5\u7cbe\u786e\u5730\u6355\u6349\u5206\u5b50\u7684\u4e09\u7ef4\u7ed3\u6784\u548c\u5b83\u4eec\u5982\u4f55\u76f8\u4e92\u4f5c\u7528\u3002<\/p>\n\n\n\n

1\u3001TFN\u8f93\u5165\u6570\u636e\u7684Embedding<\/h2>\n\n\n\n

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\u5177\u4f53\u6765\u8bf4\uff0c\u5f20\u91cf\u573a\u7f51\u7edc\u7684\u6bcf\u4e00\u5c42\u7684\u8f93\u5165\u548c\u8f93\u51fa\u90fd\u662f\u4e00\u4e2a\u6709\u9650\u8282\u70b9\u96c6\\(\\mathrm{S}\\)\uff0c\u8fd9\u4e9b\u70b9\u4f4d\u4e8e3D\u7a7a\u95f4\\(\\mathbb{R}^3\\)\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e00\u4e2a\u4e0eSO(3)<\/em>\u7fa4\u7684\u8868\u793a\u76f8\u5173\u8054\u7684\u5411\u91cf\u3002\u6362\u53e5\u8bdd\u8bf4\uff0c\u7f51\u7edc\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u4e0d\u4ec5\u4ec5\u662f\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u5750\u6807\u70b9\uff0c\u5b83\u8fd8\u643a\u5e26\u4e86\u4e00\u5b9a\u7684\u9644\u52a0\u4fe1\u606f\uff0c\u8fd9\u4e9b\u4fe1\u606f\u4ee5\u5411\u91cf\u7684\u5f62\u5f0f\u4f53\u73b0\uff0c\u4e14\u8fd9\u4e2a\u5411\u91cf\u662f\u5982SO(3)<\/em>\u7fa4\u5143\u7d20\u90a3\u6837\u53ef\u4ee5\u53d8\u6362\u7684\u3002\u5728\u7269\u7406\u610f\u4e49\u4e0a\uff0c\u8fd9\u610f\u5473\u7740\u8fd9\u4e9b\u5411\u91cf\u80fd\u591f\u53cd\u6620\u70b9\u7684\u65cb\u8f6c\u6027\u8d28\u548c\u5bf9\u65cb\u8f6c\u7684\u54cd\u5e94\u65b9\u5f0f\uff0c\u786e\u4fdd\u7f51\u7edc\u5728\u5904\u7406\u65cb\u8f6c\u65f6\u80fd\u591f\u4fdd\u6301\u4e00\u81f4\u6027\u548c\u4e0d\u53d8\u6027\u3002\u8fd9\u4e9b\u5411\u91cf\u5b9e\u9645\u4e0a\u5c31\u662fWinger D\u77e9\u9635\uff08\u8be6\u89c1\u7b2c\u4e00\u7ae0\uff09\u4e0d\u540c\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\uff0c\u5bf9\u5e94\u4e8e\u65cb\u8f6c\u5bf9\u79f0\u6027\u7684\u4e0d\u540c\u7403\u8c10\u51fd\u6570\u9636\u6570\uff0c\u8fd9\u88ab\u79f0\u4f5c\\(\\mathrm{l}\\) -\u65cb\u8f6c\u9636\u6570\u4e0d\u53ef\u7ea6\u8868\u793a\u3002\u4f8b\u5982\uff0c\\(l=0\\)\u5bf9\u5e94\u4e8e\u6807\u91cf\uff08\u65e0\u65b9\u5411\u7684\u91cf\uff09\uff0c\\(l=1\\)\u5bf9\u5e94\u4e8e\u5411\u91cf\uff08\u6709\u65b9\u5411\u7684\u91cf\uff09\uff0c\u800c\u66f4\u9ad8\u7684\\(\\mathrm{l}\\) \u503c\u5bf9\u5e94\u4e8e\u66f4\u590d\u6742\u7684\u5bf9\u79f0\u6027\u3002\u5b83\u4eec\u53ef\u4ee5\u88ab\u7c7b\u6bd4\u4e8e\u6807\u51c6\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u5377\u79ef\u64cd\u4f5c\u7684\u5377\u79ef\u6838\u7684\u6570\u91cf\u3002\u5728CNN\u4e2d\uff0c\u6bcf\u4e2a\u5377\u79ef\u6838\u53ef\u4ee5\u6355\u6349\u8f93\u5165\u6570\u636e\u7684\u4e0d\u540c\u7279\u5f81\uff1b\u7c7b\u4f3c\u5730\uff0c\u5728\u5f20\u91cf\u573a\u7f51\u7edc\u4e2d\uff0c\u4e0d\u540c\u7684\\(\\mathrm{l}\\) -\u65cb\u8f6c\u9636\u6570\u4e0d\u53ef\u7ea6\u8868\u793a\u53ef\u4ee5\u6355\u6349\u6570\u636e\u7684\u4e0d\u540c\u65cb\u8f6c\u5bf9\u79f0\u7279\u5f81\u3002\u8fd9\u6837\uff0c\u5f53\u7f51\u7edc\u5904\u74063D\u6570\u636e\u65f6\uff0c\u5b83\u53ef\u4ee5\u540c\u65f6\u5904\u7406\u591a\u4e2a\u65cb\u8f6c\u5bf9\u79f0\u6027\u7ea7\u522b\u7684\u4fe1\u606f\uff0c\u4ece\u800c\u5141\u8bb8\u5b83\u66f4\u5168\u9762\u5730\u7406\u89e3\u6570\u636e\u7684\u65cb\u8f6c\u51e0\u4f55\u7ed3\u6784\u3002<\/p>\n\n\n\n

\u5728\u5f20\u91cf\u573a\u7f51\u7edc\u4e2d\uff0c\u6211\u4eec\u6784\u5efa\u4e00\u4e2a\u7279\u5f81\u5f20\u91cf\u5b57\u5178\\(V^{(L)}\\)\uff0c\u7528\u6765\u5b58\u50a8\u4e0d\u540c\u65cb\u8f6c\u9636\u6570\\(\\mathrm{l}\\)\u4e0b\u7684\u7279\u5f81\u5411\u91cf\u3002\u6bcf\u4e2a\u952e\\(\\mathrm{l}\\)\u5bf9\u5e94\u4e00\u4e2a\u503c\uff0c\u8fd9\u4e2a\u503c\u662f\u4e00\u4e2a\u591a\u7ef4\u6570\u7ec4\uff0c\u5f62\u5f0f\u4e3a\\([S,n_{_l},2l+1]\\)\uff0c\u5176\u4e2d\\(\\mathrm{S}\\)\u4ee3\u8868\u70b9\u7684\u6570\u91cf\uff0c\\(n_{l}\\)\u662f\u63d0\u53d6\u7279\u5b9a\u7c7b\u578b\u7279\u5f81\u7684\u201c\u5377\u79ef\u6838\u201d\u6570\u91cf\uff0c\u800c\\(\\text{2l+1}\\)\u5219\u662f\u53c2\u7167\u7403\u8c10\u51fd\u6570\u7684\u5b9a\u4e49\u63cf\u8ff0\u7684\u7ed9\u5b9a\\(\\mathrm{l}\\)\u4e0b\u7684\u7a7a\u95f4\u65b9\u5411\u7ef4\u5ea6\uff0c\u8bba\u6587\u7684\u540d\u5b57\u5f20\u91cf\u573a\uff08Tensor Fields\uff09\u6307\u4ee3\u7684\u5c31\u662f\u8fd9\u4e2a\u5f20\u91cf\u5b57\u5178\\(V^{(L)}\\)\u3002<\/p>\n\n\n\n

\u4e0b\u9762\u901a\u8fc7\u4e00\u4e2a\u7b80\u5316\u7684\u4f8b\u5b50\u6765\u7406\u89e3\u8fd9\u79cd\u7279\u5f81\u5f20\u91cf\u5b57\u5178\\(V^{(L)}\\)\u7684\u6982\u5ff5\u3002\u5047\u8bbe\u6211\u4eec\u6709\u4e00\u4e2a\u7cfb\u7edf\uff0c\u8fd9\u4e2a\u7cfb\u7edf\u91cc\u6709\u4e24\u4e2a\u661f\u7403\uff0c\u6211\u4eec\u60f3\u8981\u8ddf\u8e2a\u6bcf\u4e2a\u661f\u7403\u7684\u4e00\u4e9b\u7279\u6027\uff1a\u8d28\u91cf\u3001\u4f4d\u7f6e\u548c\u901f\u5ea6\u3002\u6b64\u65f6\u53ef\u4ee5\u5c06\u8fd9\u4e9b\u7279\u6027\u7f16\u7801\u4e3a\u7279\u5f81\u5f20\u91cf\uff0c\u5e76\u4f7f\u7528\u5f20\u91cf\u573a\u7f51\u7edc\u6765\u5904\u7406\u3002<\/p>\n\n\n\n

\\(l=0\\)\uff1a\u5bf9\u4e8e\u5f20\u91cf\u573a\u4e2d\u7684\u96f6\u9636\uff08\u6807\u91cf\u573a\uff09\uff0c\u6b64\u65f6\u53ea\u5173\u5fc3\u6bcf\u4e2a\u661f\u7403\u7684\u8d28\u91cf\u3002\u56e0\u6b64\uff0c\u5bf9\u4e8e\u6bcf\u4e2a\u661f\u7403\uff08\u6216\u70b9\uff09\uff0c\u53ef\u4ee5\u7528\u4e00\u4e2a\u6570\u503c\uff08\u6807\u91cf\uff09\u6765\u8868\u793a\u5176\u8d28\u91cf\u3002\u4ee4\u661f\u74031\u7684\u8d28\u91cf\u662f\\(m_{1}\\)\uff0c\u661f\u74032\u7684\u8d28\u91cf\u662f\\(m_{2}\\)\u3002\u5b57\u5178\u7684\u5f62\u5f0f\u53ef\u4ee5\u662f\uff1a<\/p>\n\n\n

\\(\\begin{aligned}&\\boldsymbol{V}^{(0)}=\\{\\\\&1{:}[m_1]\\text{ \/\/\u661f\u74031\u7684\u8d28\u91cf}\\\\&2{:}[m_2]\\text{ \/\/\u661f\u74032\u7684\u8d28\u91cf}\\\\&\\}\\end{aligned}\\)<\/p>\n\n\n

\\(l=1\\)\uff1a\u5bf9\u4e8e\u4e00\u9636\u5f20\u91cf\u573a\uff08\u4e5f\u5c31\u662f\u5411\u91cf\u573a\uff09\uff0c\u901a\u5e38\u6d89\u53ca\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u5411\u91cf\uff0c\u56e0\u6b64\u53ef\u4ee5\u7528\u6765\u8868\u793a\u8bf8\u5982\u4f4d\u7f6e\u3001\u901f\u5ea6\u3001\u52a0\u901f\u5ea6\u7b49\u7269\u7406\u91cf\u3002\u8fd9\u4e9b\u5411\u91cf\u7684\u6bcf\u4e2a\u5206\u91cf\u90fd\u5bf9\u5e94\u7740\u7a7a\u95f4\u7684\u4e00\u4e2a\u7ef4\u5ea6\uff08\u901a\u5e38\u662fx\u3001y\u3001z\u5750\u6807\u8f74\uff09\u3002\u5047\u8bbe\u5f53\u524d\u60f3\u8981\u8868\u793a\u6bcf\u4e2a\u661f\u7403\u7684\u4f4d\u7f6e\uff0c\u901f\u5ea6\u548c\u52a0\u901f\u5ea6\u3002\u4ee4\u661f\u74031\u7684\u4f4d\u7f6e\u662f\\((x_1,y_1,z_1)\\)\uff0c\u901f\u5ea6\u662f\\((vx_1,vy_1,vz_1)\\)\uff0c\u52a0\u901f\u5ea6\u662f\\((ax_1,ay_1,az_1)\\)\u3002\u661f\u74032\u7684\u4f4d\u7f6e\u662f\\((x_2,y_2,z_2)\\)\uff0c\u901f\u5ea6\u662f\\((vx_2,vy_2,vz_2)\\)\uff0c\u52a0\u901f\u5ea6\u662f\\((ax_2,ay_2,az_2)\\)\u3002\u5b57\u5178\u7684\u5f62\u5f0f\u53ef\u4ee5\u662f\uff1a<\/p>\n\n\n

$V^{(1)}=\\{$<\/p>\n

$1: \\{$ \/\/\u661f\u74031<\/p>\n

position:$[ x_1, y_1, z_1]$ \/\/\u4f4d\u7f6e\u5411\u91cf
velocity$:[\\boldsymbol{\\nu x}_1,\\boldsymbol{\\nu y}_1,\\boldsymbol{\\nu z}_1]$ $\\|$\u901f\u5ea6\u5411\u91cf<\/p>\n

acceleration$:[ax_1,ay_1,az_1]$ \/\/\u52a0\u901f\u5ea6\u5411\u91cf$\\}$,<\/p>\n

$2: \\{$ \/\/\u661f\u74032<\/p>\n

position:$[ \\boldsymbol{x}_2, \\boldsymbol{y}_2, \\boldsymbol{z}_2]$ $\\|$\u4f4d\u7f6e\u5411\u91cf<\/p>\n

velocity:$[\\boldsymbol{\\nu x}_2,\\boldsymbol{\\nu y}_2,\\boldsymbol{\\nu z}_2]$ \/\/\u901f\u5ea6\u5411\u91cf<\/p>\n

acceleration:$[ax_2,ay_2,az_2]\/\/$\u52a0\u901f\u5ea6\u5411\u91cf<\/p>\n

$\\}\\}$<\/p>\n\n\n

\u5b9e\u9645\u4e0a\uff0c\u5728\u673a\u5668\u5b66\u4e60\u548c\u7279\u522b\u662f\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u4e2d\uff0c\u5bf9\u4e8e\u521d\u6b65\u5b9a\u4e49\u7684\u7279\u5f81\u5411\u91cf\u7684\u96c6\u5408\uff08\u5b57\u5178\\(\\mathrm{V}\\)\uff09\u662f\u57fa\u4e8e\u7403\u9762\u8c10\u51fd\u6570\u4ece\u6570\u636e\u4e2d\u76f4\u63a5\u63d0\u53d6\u7684\u3002\u8fd9\u4e9b\u7279\u5f81\u5411\u91cf\u4e0d\u662f\u7b80\u5355\u5730\u968f\u673a\u521d\u59cb\u5316\uff0c\u800c\u662f\u901a\u8fc7\u8ba1\u7b97\u6bcf\u4e2a\u70b9\u7684\u7403\u9762\u8c10\u51fd\u6570\uff0c\u4ece\u800c\u6355\u83b7\u4e0eSO(3)<\/em>\u7fa4\u76f8\u5173\u7684\u65cb\u8f6c\u7b49\u53d8\u4fe1\u606f\u3002\u8fd9\u6837\u751f\u6210\u7684\u7279\u5f81\u5411\u91cf\u66f4\u80fd\u7cbe\u786e\u5730\u53cd\u6620\u5206\u5b50\u6216\u51e0\u4f55\u5f62\u4f53\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u7684\u7269\u7406\u548c\u5316\u5b66\u6027\u8d28\u3002<\/p>\n\n\n\n

2\u3001\u70b9\u5377\u79ef<\/h2>\n\n\n\n

\u70b9\u5377\u79ef\uff08Point Convolution\uff09\u662f\u4e00\u79cd\u4e13\u95e8\u4e3a\u5904\u7406\u51e0\u4f55\u6570\u636e\u8bbe\u8ba1\u7684\u5377\u79ef\u64cd\u4f5c\u3002\u5728\u4f20\u7edf\u7684\u5377\u79ef\u795e\u7ecf\u7f51\u7edc\uff08CNN\uff09\u4e2d\uff0c\u5377\u79ef\u662f\u5728\u89c4\u5219\u7684\u7f51\u683c\u4e0a\u6267\u884c\u7684\uff08\u4f8b\u5982\u56fe\u50cf\u7684\u50cf\u7d20\u7f51\u683c\uff09\uff0c\u800c\u70b9\u5377\u79ef\u5219\u9700\u8981\u5728\u4e0d\u89c4\u5219\u7684\u70b9\u96c6\u4e0a\u6267\u884c\uff0c\u8fd9\u4e9b\u70b9\u96c6\u6ca1\u6709\u56fa\u5b9a\u7684\u62d3\u6251\u7ed3\u6784\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u5728\u6807\u51c6\u7684\u5377\u79ef\u7f51\u7edc\u4e2d\uff0c\u6743\u91cd\u5171\u4eab\u610f\u5473\u7740\u5377\u79ef\u6838\u5728\u6574\u4e2a\u8f93\u5165\u7279\u5f81\u56fe\u4e0a\u6ed1\u52a8\uff0c\u5e76\u5728\u6bcf\u4e2a\u5c40\u90e8\u533a\u57df\u5e94\u7528\u76f8\u540c\u7684\u6743\u91cd\u3002\u5728\u70b9\u5377\u79ef\u4e2d\uff0c\u8fd9\u4e00\u6982\u5ff5\u9700\u8981\u88ab\u91cd\u65b0\u89e3\u91ca\uff0c\u56e0\u4e3a\u6ca1\u6709\u56fa\u5b9a\u7684\u5c40\u90e8\u533a\u57df\u53ef\u4ee5\u7b80\u5355\u5730\u201c\u6ed1\u52a8\u201d\u3002\u6240\u4ee5\uff0c\u70b9\u5377\u79ef\u901a\u8fc7\u5b9a\u4e49\u6bcf\u4e2a\u70b9\u7684\u5c40\u90e8\u90bb\u57df\u6765\u64cd\u4f5c\u3002\u5c40\u90e8\u90bb\u57df\u53ef\u4ee5\u901a\u8fc7\u627e\u5230\u6bcf\u4e2a\u70b9\u5728\u7a7a\u95f4\u4e2d\u6700\u8fd1\u7684\\(\\mathrm{k}\\)\u4e2a\u90bb\u70b9\u6765\u786e\u5b9a\uff0c\u6216\u8005\u901a\u8fc7\u5728\u4e00\u4e2a\u56fa\u5b9a\u534a\u5f84\u5185\u641c\u7d22\u90bb\u70b9\u3002\u5bf9\u4e8e\u6bcf\u4e2a\u70b9\uff0c\u5176\u5c40\u90e8\u90bb\u57df\u4e2d\u7684\u6bcf\u4e2a\u8282\u70b9\u90fd\u6709\u4e00\u4e2a\u7279\u5f81\u5411\u91cf\uff0c\u4f8b\u5982\u7b2c1\u8282\u6240\u8bb2\u7684\u4f8b\u5b50\u3002\u70b9\u5377\u79ef\u64cd\u4f5c\u6d89\u53ca\u5230\u5c06\u6bcf\u4e2a\u90bb\u70b9\u7684\u7279\u5f81\u5411\u91cf\u4e0e\u4e00\u4e2a\u6743\u91cd\u77e9\u9635\u76f8\u4e58\uff0c\u8fd9\u4e2a\u6743\u91cd\u77e9\u9635\u5bf9\u6240\u6709\u70b9\u662f\u5171\u4eab\u7684\uff0c\u4ee5\u4fbf\u63d0\u53d6\u5c40\u90e8\u7279\u5f81\u3002\u6700\u540e\uff0c\u5c06\u90bb\u5c45\u70b9\u7684\u53d8\u6362\u540e\u7279\u5f81\u805a\u5408\uff08\u4f8b\u5982\uff0c\u901a\u8fc7\u52a0\u548c\u6216\u6700\u5927\u6c60\u5316\uff09\u6765\u751f\u6210\u4e00\u4e2a\u65b0\u7684\u7279\u5f81\u63cf\u8ff0\u7b26\uff0c\u4ee3\u8868\u4e2d\u5fc3\u70b9\u7684\u5c40\u90e8\u7ed3\u6784\u3002<\/p>\n\n\n\n

\u4e0e\u4f20\u7edf\u7684\u5377\u79ef\u4e0d\u540c\uff0c\u70b9\u5377\u79ef\u9664\u4e86\u80fd\u591f\u5904\u7406\u65e0\u5e8f\u548c\u4e0d\u89c4\u5219\u7684\u6570\u636e\u7ed3\u6784\u4ee5\u5916\uff0c\u8fd8\u9700\u8981\u5728\u5904\u7406\u8fc7\u7a0b\u4e2d\u4fdd\u8bc1\u51e0\u4f55\u56fe\u6570\u636e\u7684\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e0b\u7684\u5bf9\u79f0\u6027\u3002\u8fd9\u91cc\u7684\u5173\u952e\u662f\u4f7f\u7528Wigner-D\u77e9\u9635\uff0cWigner-D\u77e9\u9635\u63d0\u4f9b\u4e86\u4e00\u4e2a\u65cb\u8f6c\u7fa4SO(3)<\/em>\u7684\u4e0d\u53ef\u7ea6\u8868\u793a\uff0c\u8fd9\u53ef\u4ee5\u7528\u6765\u6784\u5efa\u65cb\u8f6c\u7b49\u53d8\u7684\u7279\u5f81\u3002\u6570\u5b66\u4e0a\uff0c\u5982\u679c\\(Y^{(l)}\\)\u662f\u4e00\u4e2a\u5728SO(3)<\/em>\u4e0b\u7684\u7b49\u53d8\u51fd\u6570\uff0c\u5219\u5bf9\u4e8e\u6240\u6709\\(g\\in SO(3)\\)\u548c\u8f93\u5165\u5411\u91cf\\(\\mathrm{r}\\)\uff0c\u6709\uff1a<\/p>\n\n\n

\\(Y^{(l)}(R(g)\\boldsymbol{r})=\\sum_{m^{\\prime}}D_{mm^{\\prime}}^{(l)}(g)\\boldsymbol{Y}_{m^{\\prime}}^{(l)}(\\boldsymbol{r})\\)<\/p>\n\n\n

\u5176\u4e2d\\(D^{(l)}(g)\\)\u662fWigner-D\u77e9\u9635\uff0c\\(\\mathrm{m}\\)\u548c\\(\\mathrm{m~'}\\)\u662f\u7d22\u5f15\u8fd9\u4e9b\u7279\u5f81\u5411\u91cf\u7684\u91cf\u5b50\u6570\u3002\u4e3a\u4e86\u8bbe\u8ba1\u65cb\u8f6c\u7b49\u53d8\u7684\u70b9\u5377\u79ef\uff0c\u57fa\u4e8e\u4e0a\u5f0f\u5b9a\u4e49\u7684\\(Y^{(l)}\\)\uff0c\u53ef\u4ee5\u5b9a\u4e49\u5377\u79ef\u6838\u7684\u5f62\u5f0f\u4e3a\uff1a<\/p>\n\n\n

\\(F_{crm}^{\\left(l_f,l_i\\right)}(r)=R_c^{l_f,l_i}(r)Y_m^{\\left(l_f\\right)}(\\hat{r})\\)<\/p>\n\n\n

\u5176\u4e2d\uff1a<\/p>\n\n\n\n

\uff081\uff09\\(Y_m^{(l_f)}(\\hat{r})\\)\u662f\u5728\u5355\u4f4d\u5411\u91cf\\(\\hat{r}\\)\u4e0a\u8ba1\u7b97\u7684\u7403\u9762\u8c10\u51fd\u6570\u7684\\(\\mathrm{m}\\)\u9636\u5206\u91cf\u3002<\/p>\n\n\n\n

\uff082\uff09\\(\\hat{r}\\)\u662f\u5355\u4f4d\u957f\u5ea6\u7684\\(\\mathbf{r}\\)\u5411\u91cf\uff0c\u800c\\(\\mathrm{r}\\)\u662f\\(\\mathbf{r}\\)\u7684\u957f\u5ea6\u3002<\/p>\n\n\n\n

\uff083\uff09\\(R_c^{l_f,l_i}(r)\\)\u662f\u4e00\u4e2a\u5173\u4e8e\\(\\mathrm{r}\\)\uff08\u4e5f\u5c31\u662f\u70b9\u4e0e\u5377\u79ef\u4e2d\u5fc3\u4e4b\u95f4\u7684\u8ddd\u79bb\uff09\u7684\u5f84\u5411\u51fd\u6570\u3002<\/p>\n\n\n\n

\uff084\uff09\\(l_{i}\\)\u8868\u793a\u8f93\u5165\u7279\u5f81\u7684\u65cb\u8f6c\u9636\u6570\uff0c\\(l_{f}\\)\u8868\u793a\u70b9\u5377\u79ef\u6838\u7684\u65cb\u8f6c\u9636\u6570\u3002<\/p>\n\n\n\n

\uff085\uff09\u4e0b\u6807\\(\\mathrm{c}\\)\u901a\u5e38\u8868\u793a\u7279\u5f81\u6216\u70b9\u5377\u79ef\u6838\u7684\u901a\u9053\u3002<\/p>\n\n\n\n

\u56e0\u6b64\uff0c\\(R_c^{l_f,l_i}(r)\\)\u5f84\u5411\u51fd\u6570\u8d1f\u8d23\u5bf9\u6bcf\u4e2a\u534a\u5f84\\(\\mathrm{r}\\)\u7684\u70b9\u63d0\u4f9b\u4e00\u4e2a\u6743\u91cd\uff0c\u8fd9\u4e2a\u6743\u91cd\u4f1a\u5f71\u54cd\u5377\u79ef\u6838\u5982\u4f55\u5904\u7406\u8ddd\u79bb\u4e2d\u5fc3\u70b9\u4e0d\u540c\u8fdc\u8fd1\u7684\u5176\u4ed6\u70b9\u7684\u4fe1\u53f7\u3002\u5728\u5b9e\u8df5\u4e2d\uff0c\\(R_c^{l_f,l_i}(r)\\)\u53ef\u4ee5\u662f\u67d0\u79cd\u56fa\u5b9a\u7684\u8ddd\u79bb\u51fd\u6570\uff0c\u5982cos\u51fd\u6570\u3001-log\u51fd\u6570\u7b49\uff1b\u4e5f\u53ef\u4ee5\u662f\u53ef\u5b66\u4e60\u7684\u53c2\u6570\u5316\u51fd\u6570\uff0c\u5982\u9ad8\u65af\u5f84\u5411\u57fa\u51fd\u6570\uff0c\u53c2\u6570\uff08\u5982\u5bbd\u5ea6\u3001\u4e2d\u5fc3\u7b49\uff09\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u88ab\u4f18\u5316\u3002\u53e6\u4e00\u79cd\u5e38\u89c1\u7684\u505a\u6cd5\u662f\u5c06\\(r,l_i,l_f\\)\u548c\\(\\mathrm{c}\\)\u4f5c\u4e3a\u7279\u5f81\u8f93\u5165\u5230\u4e00\u4e2a\u5c0f\u7684\u795e\u7ecf\u7f51\u7edc\uff08\u4f8b\u5982\u591a\u5c42\u611f\u77e5\u673aMLP\uff09\uff0c\u8be5\u7f51\u7edc\u5b66\u4e60\u5728\u7ed9\u5b9a\u8fd9\u4e9b\u8f93\u5165\u7684\u60c5\u51b5\u4e0b\u8f93\u51fa\u6b63\u786e\u7684\u8ddd\u79bb\u6743\u91cd\u3002<\/p>\n\n\n\n

\u5728\u8fdb\u884c\u70b9\u5377\u79ef\u65f6\uff0c\u4f20\u7edf\u7684\u5377\u79ef\u64cd\u4f5c\u4e0d\u80fd\u76f4\u63a5\u5e94\u7528\uff0c\u56e0\u4e3a\u51e0\u4f55\u7a7a\u95f4\u7684\u70b9\u7f3a\u4e4f\u50cf\u56fe\u50cf\u50cf\u7d20\u90a3\u6837\u7684\u89c4\u5219\u7ed3\u6784\u3002\u56e0\u6b64\uff0c\u9700\u8981\u4e00\u79cd\u65b9\u6cd5\u6765\u5b9a\u4e49\u70b9\u4e91\u4e2d\u70b9\u7684\u201c\u5c40\u90e8\u533a\u57df\u201d\u4ee5\u53ca\u5982\u4f55\u5728\u8fd9\u4e9b\u5c40\u90e8\u533a\u57df\u5185\u805a\u5408\u4fe1\u606f\u3002\u8fd9\u5c31\u662f\\(R_c^{l_f,l_i}(r)\\)\u548c\u7403\u9762\u8c10\u51fd\u6570\\(Y_{m}^{(l_{f})}(\\hat{r})\\)\u53d1\u6325\u4f5c\u7528\u7684\u5730\u65b9\uff0c\u5177\u4f53\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n\n\n\n

\uff081\uff09\u9996\u5148\u9009\u62e9\u4e00\u4e2a\u4e2d\u5fc3\u70b9\u3002<\/p>\n\n\n\n

\uff082\uff09\u5bf9\u4e8e\u4e2d\u5fc3\u70b9\u7684\u6bcf\u4e2a\u90bb\u5c45\uff0c\u8ba1\u7b97\u4ece\u4e2d\u5fc3\u70b9\u5230\u90bb\u5c45\u70b9\u7684\u5411\u91cf\\(\\mathbf{r}\\)\u3002<\/p>\n\n\n\n

\uff083\uff09\u5c06\u8fd9\u4e2a\u5411\u91cf\u89c4\u8303\u5316\u5230\u5355\u4f4d\u5411\u91cf\\(\\hat{r}\\)\uff0c\u5e76\u8ba1\u7b97\u5b83\u7684\u957f\u5ea6\\(\\mathrm{r}\\)\u3002<\/p>\n\n\n\n

\uff084\uff09\u5e94\u7528\u7403\u9762\u8c10\u51fd\u6570\\(Y_m^{(l_f)}(\\hat{r})\\)\u6765\u6355\u6349\u4e2d\u5fc3\u70b9\u548c\u5b83\u90bb\u5c45\u4e4b\u95f4\u7684\u89d2\u5ea6\u5173\u7cfb\u3002<\/p>\n\n\n\n

\uff085\uff09\u4f7f\u7528\\(R_c^{l_f,l_i}(r)\\)\u51fd\u6570\u7f16\u7801\u8ddd\u79bb\\(\\mathrm{r}\\)\u7684\u4fe1\u606f\uff0c\u8fd9\u6837\u8fd1\u7684\u548c\u8fdc\u7684\u70b9\u53ef\u4ee5\u6709\u4e0d\u540c\u7684\u5f71\u54cd\u529b\u3002<\/p>\n\n\n\n

\uff086\uff09\u5b9a\u4e49\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\u8fd9\u4e2a\u5377\u79ef\u6838\u7efc\u5408\u8003\u8651\u7684\u8ddd\u79bb\u4fe1\u606f\u548c\u89d2\u5ea6\u5173\u7cfb\uff0c\u4f7f\u7528\u5b83\u805a\u5408\u90bb\u5c45\u7684\u4fe1\u606f\uff0c\u5f62\u6210\u8be5\u4e2d\u5fc3\u70b9\u7684\u65b0\u7279\u5f81\u5411\u91cf\u3002<\/p>\n\n\n\n

\u5b9e\u9645\u4e0a\uff0c\u70b9\u5377\u79ef\u5c31\u662f\u5728SchNet\u6a21\u578b\u4e2d\u63d0\u51fa\u7684\u8fde\u7eed\u6027\u6ee4\u6ce2\u5377\u79ef\u7684\u6ee4\u6ce2\u5668\u751f\u6210\u51fd\u6570\u57fa\u7840\u4e0a\uff0c\u4e58\u4e86\u4e00\u4e2a\u88abWinger D\u77e9\u9635\u548c\u7403\u8c10\u51fd\u6570\u8ba1\u7b97\u5f97\u5230\u7684\u89d2\u5ea6\u5206\u5e03\uff0c\u5176\u4ed6\u7684\u6ca1\u6709\u533a\u522b\u3002\u8fd9\u6837\u7684\u8bbe\u8ba1\u989d\u5916\u8003\u8651\u4e86\u89d2\u5ea6\u5206\u5e03\uff0c\u5141\u8bb8\u6ee4\u6ce2\u5668\u6355\u6349\u5c40\u90e8\u90bb\u57df\u5185\u7684\u65b9\u5411\u4f9d\u8d56\u7279\u6027\uff0c\u4ece\u800c\u8fbe\u5230\u65cb\u8f6c\u7b49\u53d8\u6027\u3002<\/p>\n\n\n\n

3\u3001Clebsch-Gordan coefficients<\/h2>\n\n\n\n

\u6b64\u5916\uff0c\u5728\u4e0a\u8ff0\u70b9\u5377\u79ef\u8ba1\u7b97\u6d41\u7a0b\u7684\u6700\u540e\u4e00\u6b65\uff08\u7b2c\u516d\u6b65\uff09\u5b9a\u4e49\u5b8c\u5377\u79ef\u6838\u540e\uff0c\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u7279\u5f81\u4e0e\u5176\u90bb\u5c45\u8282\u70b9\\(\\mathrm{j}\\)\u7684\u7279\u5f81\u8fdb\u884c\u5377\u79ef\u64cd\u4f5c\u66f4\u65b0\u5f97\u5230\u65b0\u7684\u70b9\\(\\mathrm{i}\\)\u7279\u5f81\u65f6\uff0c\u9700\u8981\u4f7f\u7528CG\u7cfb\u6570\u3002\u56e0\u4e3a\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u7279\u5f81\u4e0e\u5176\u90bb\u5c45\u8282\u70b9\\(\\mathrm{j}\\)\u7684\u7279\u5f81\u662f\u5728\u591a\u79cd\u65cb\u8f6c\u9636\\(\\mathrm{l}\\)\u4e0b\u5b9a\u4e49\u7684\uff0c\u4e24\u4e2a\u5177\u6709\u4e0d\u540c\u89d2\u91cf\u5b50\u6570\\(\\mathrm{l}\\)\u548c\\(\\mathrm{m}\\)\u7684\u7279\u5f81\u60f3\u8981\u8fdb\u884c\u4fe1\u606f\u4ea4\u4e92\u66f4\u65b0\u5f97\u5230\u65b0\u7279\u5f81\u65f6\uff0c\u4e0d\u80fd\u7b80\u5355\u5730\u5c06\u5b83\u4eec\u8fdb\u884c\u76f8\u52a0\u6216\u76f8\u4e58\uff0c\u5426\u5219\u4f1a\u4f7f\u5f97\u65b0\u7279\u5f81\u4e0d\u80fd\u4fdd\u7559\u7cfb\u7edf\u7684\u65cb\u8f6c\u7b49\u53d8\u6027\u3002\u6b64\u65f6\u9700\u8981\u4e00\u79cd\u65b9\u6cd5\u6765\u786e\u4fdd\u5408\u5e76\u540e\u7684\u65b0\u7279\u5f81\u4ecd\u7136\u7b26\u5408\u65cb\u8f6c\u7b49\u53d8\u6027\uff0cCG\u7cfb\u6570\u5c31\u662f\u7528\u6765\u505a\u8fd9\u4ef6\u4e8b\u7684\u3002CG\u7cfb\u6570\u5728\u91cf\u5b50\u529b\u5b66\u4e2d\u7528\u4e8e\u5408\u5e76\u4e24\u4e2a\u5e26\u6709\u89d2\u52a8\u91cf\u7684\u72b6\u6001\uff0c\u751f\u6210\u4e00\u4e2a\u65b0\u7684\u72b6\u6001\uff0c\u8fd9\u4e2a\u72b6\u6001\u5177\u6709\u660e\u786e\u5b9a\u4e49\u7684\u603b\u89d2\u52a8\u91cf\uff0c\u8fd9\u79cd\u89d2\u52a8\u91cf\u5b88\u6052\u662f\u5b9e\u73b0\u65cb\u8f6c\u7b49\u53d8\u6027\u7684\u57fa\u7840\u3002\u5728\u5f20\u91cf\u573a\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u7c7b\u4f3c\u7684\u539f\u7406\u88ab\u5e94\u7528\u4e8e\u7279\u5f81\u878d\u5408\uff0c\u4f7f\u7528CG\u7cfb\u6570\u610f\u5473\u7740\u53ef\u4ee5\u751f\u6210\u5177\u6709\u826f\u597d\u65cb\u8f6c\u7279\u6027\u7684\u590d\u5408\u7279\u5f81\uff0c\u8fd9\u4e9b\u7279\u5f81\u53ef\u4ee5\u7528\u4e8e\u8bad\u7ec3\u6a21\u578b\uff0c\u4f7f\u5176\u80fd\u591f\u9884\u6d4b\u65cb\u8f6c\u4e0d\u53d8\u7684\u5206\u5b50\u6027\u8d28\u3002\u7ed3\u5408CG\u7cfb\u6570\u540e\uff0c\u5728\u4e00\u4e2a\u66f4\u5168\u9762\u7684\u70b9\u5377\u79ef\u516c\u5f0f\u8868\u8ff0\u5982\u4e0b\uff1a<\/p>\n\n\n

\\(\\mathcal{L}_{icm_{out}}^{(l_{out})}\\left(\\boldsymbol{r}_i,\\boldsymbol{V}_{icm_{in}}^{(l_m)}\\right)=\\sum_{m_f,m_{in}}C_{l_{out},m_{out}}^{(l_{in},m_{in}),(l_f,m_f)}\\sum_{j\\in S}F_{cm_f}^{(l_f,l_{in})}\\left(\\boldsymbol{r}_{ij}\\right)\\boldsymbol{V}_{jcm_{in}}^{(l_{in})}\\)<\/p>\n\n\n

\u5176\u4e2d\uff1a<\/p>\n\n\n\n

\uff081\uff09\\(\\mathcal{L}_{icm_{out}}^{(l_{out})}\\left(r_i,V_{icm_{in}}^{(l_{in})}\\right)\\)\u662f\u7ecf\u8fc7\u70b9\u5377\u79ef\u64cd\u4f5c\u540e\u5f97\u5230\u7684\u65b0\u7279\u5f81\u5411\u91cf\uff0c\u5bf9\u5e94\u4e8e\u8f93\u5165\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u4e00\u4e2a\u7279\u5b9a\u65cb\u8f6c\u9636\u6570\\(l_{out}\\)\u3002<\/p>\n\n\n\n

\uff082\uff09\\(r_{ij}=r_i-r_j\\)\u8868\u793a\u4ece\u70b9\\(\\mathrm{i}\\)\u5230\u70b9\\(\\mathrm{j}\\)\u7684\u4f4d\u7f6e\u5411\u91cf\u3002<\/p>\n\n\n\n

\uff083\uff09\\(\\sum_{j\\in S}F_{cm_f}^{\\left(l_f,l_{in}\\right)}\\left(r_{ij}\\right)\\)\u4ee3\u8868\u4e86\u8fde\u7eed\u5377\u79ef\u6838\u751f\u6210\u51fd\u6570\uff0c\u5b83\u4f9d\u8d56\u4e8e\u8ddd\u79bb\u5411\u91cf\\(r_{ij}\\)\u3002\u4e3a\u6bcf\u4e2a\u901a\u9053\\(\\mathrm{c}\\)\u6839\u636e\u8ddd\u79bb\u751f\u6210\u5377\u79ef\u6838\u7684\u76f8\u5173\u6743\u91cd\u3002<\/p>\n\n\n\n

\uff084\uff09\\(V_{jcm_{in}}^{(l_{in})}\\)\u662f\u70b9\\(\\mathrm{j}\\)\u7684\u65cb\u8f6c\u9636\u6570\\(l_{in}\\)\u7684\u8f93\u5165\u7279\u5f81\uff0c\u8be5\u7279\u5f81\u662f\u7403\u8c10\u51fd\u6570\u7f16\u7801\u7684\u3002<\/p>\n\n\n\n

\uff085\uff09\\(C_{l_{out},m_{out}}^{(l_{in},m_{in}),(l_{f},m_{f})}\\)\u662fClebsch-Gordan\u7cfb\u6570\uff0c\u7528\u4e8e\u5904\u7406\u8f93\u5165\u7279\u5f81\u4e0e\u5377\u79ef\u6838\u7279\u5f81\u7684\u65cb\u8f6c\u9636\u6570\u5408\u5e76\uff0c\u8fd9\u4e2a\u5408\u5e76\u8fc7\u7a0b\u9700\u8981\u9075\u5b88\u7269\u7406\u89c4\u5219\u4ee5\u786e\u4fdd\u7ed3\u679c\u7279\u5f81\u7684\u65cb\u8f6c\u7b49\u53d8\u6027\uff0c\u4e09\u4e2a\u9650\u5236\u5982\u4e0b\u6240\u793a\u3002<\/p>\n\n\n\n

1.\u603b\u78c1\u91cf\u5b50\u6570\u5b88\u6052\uff1a\\(M=m_1+m_2\\)<\/p>\n\n\n\n

\u5728\u91cf\u5b50\u7cfb\u7edf\u4e2d\uff0c\u603b\u78c1\u91cf\u5b50\u6570\u5b88\u6052\u610f\u5473\u7740\u5728\u89d2\u52a8\u91cf\u5408\u5e76\u8fc7\u7a0b\u4e2d\uff0c\u5b50\u7cfb\u7edf\u7684\u78c1\u91cf\u5b50\u6570\u4e4b\u548c\u5fc5\u987b\u7b49\u4e8e\u5408\u5e76\u540e\u7cfb\u7edf\u7684\u603b\u78c1\u91cf\u5b50\u6570\u3002\u5728\u795e\u7ecf\u7f51\u7edc\u7684\u80cc\u666f\u4e0b\uff0c\u8fd9\u53ef\u4ee5\u88ab\u770b\u4f5c\u662f\u7279\u5f81\u7ef4\u5ea6\u6216\u7279\u5f81\u4fe1\u606f\u5728\u5904\u7406\u8fc7\u7a0b\u4e2d\u7684\u4e00\u79cd\u201c\u4fe1\u606f\u5b88\u6052\u201d\u3002\u5f53\u7f51\u7edc\u5c42\u901a\u8fc7CG\u7cfb\u6570\u5408\u5e76\u8f93\u5165\u7279\u5f81\u65f6\uff0c\u7cfb\u6570\u786e\u4fdd\u4fe1\u606f\u7684\u603b\u548c\u5728\u65cb\u8f6c\u53d8\u6362\u540e\u4ecd\u7136\u5f97\u5230\u4fdd\u6301\uff0c\u8fd9\u4f7f\u5f97\u7f51\u7edc\u7684\u8f93\u51fa\u5bf9\u4e8e\u8f93\u5165\u7684\u65cb\u8f6c\u4fdd\u6301\u4e0d\u53d8\u6216\u9002\u5e94\u3002<\/p>\n\n\n\n

2.\u4e09\u89d2\u4e0d\u7b49\u5f0f\uff1a\\(\\left|l_1-l_2\\right|\\leq L\\leq l_1+l_2\\)<\/p>\n\n\n\n

\u4e09\u89d2\u4e0d\u7b49\u5f0f\u5728\u7269\u7406\u4e0a\u4fdd\u8bc1\u4e86\u89d2\u52a8\u91cf\u7684\u53ef\u80fd\u503c\u5fc5\u987b\u6ee1\u8db3\u8fd9\u4e00\u8303\u56f4\u3002\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\uff0c\u901a\u8fc7 CG\u7cfb\u6570\u5e94\u7528\u8fd9\u4e00\u539f\u5219\uff0c\u5e2e\u52a9\u6a21\u578b\u5728\u878d\u5408\u4e0d\u540c\u5c42\u6216\u4e0d\u540c\u90e8\u5206\u7684\u7279\u5f81\u65f6\uff0c\u7ef4\u6301\u4e00\u4e2a\u6709\u6548\u7684\u7279\u5f81\u201c\u89c4\u6a21\u201d\u6216\u201c\u5c42\u7ea7\u201d\uff0c\u786e\u4fdd\u8f93\u51fa\u7279\u5f81\u4e0d\u4f1a\u8d85\u51fa\u5408\u9002\u7684\u5904\u7406\u8303\u56f4\u3002\u8fd9\u5bf9\u4e8e\u7ef4\u6301\u6a21\u578b\u5bf9\u65cb\u8f6c\u7684\u9002\u5e94\u6027\u81f3\u5173\u91cd\u8981\uff0c\u56e0\u4e3a\u5b83\u4fdd\u8bc1\u4e86\u7279\u5f81\u5408\u5e76\u7684\u6b63\u786e\u6027\u548c\u9002\u5e94\u6027\u3002<\/p>\n\n\n\n

3.\u5b87\u79f0\u5b88\u6052\uff1a \\(l_1+l_2+L\\)\u901a\u5e38\u4e3a\u5076\u6570<\/p>\n\n\n\n

\u5728\u7269\u7406\u4e2d\uff0c\u7279\u522b\u662f\u5728\u91cf\u5b50\u529b\u5b66\u7684\u89d2\u52a8\u91cf\u7406\u8bba\u4e2d\uff0c\u5b87\u79f0\u5b88\u6052\u4e0e\u5b87\u79f0\uff08parity\uff09\u7684\u6982\u5ff5\u5bc6\u5207\u76f8\u5173\u3002\u5b87\u79f0\u63cf\u8ff0\u4e86\u7269\u7406\u7cfb\u7edf\u5728\u7a7a\u95f4\u5750\u6807\u53cd\u6f14\uff08\u5982\u70b9\\((x,y,z)\\)\u53d8\u6210\\((-x,-y,-z)\\)\u65f6\u7684\u884c\u4e3a\u3002\u5bf9\u4e8e\u8f68\u9053\u89d2\u52a8\u91cf\uff0c\u5b87\u79f0\u53ef\u4ee5\u901a\u8fc7\u8f68\u9053\u89d2\u52a8\u91cf\u7684\u91cf\u5b50\u6570\\(\\mathrm{l}\\)\u6765\u63cf\u8ff0\uff0c\u5176\u5b87\u79f0\u4e3a\\(\\left(-1\\right)^l\\)\u3002\u8fd9\u610f\u5473\u7740\uff0c\u5982\u679c\u8f68\u9053\u89d2\u52a8\u91cf\u7684\u91cf\u5b50\u6570\\(\\mathrm{l}\\)\u662f\u5076\u6570\uff0c\u5b87\u79f0\u4e3a\u6b63\uff08\u4e0d\u53d8\uff09\uff0c\u5982\u679c\\(\\mathrm{l}\\)\u662f\u5947\u6570\uff0c\u5219\u5b87\u79f0\u4e3a\u8d1f\uff08\u53cd\u8f6c\uff09\u3002<\/p>\n\n\n\n

\u5728\u7ec4\u5408\u4e24\u4e2a\u89d2\u52a8\u91cf\u7cfb\u7edf\u65f6\uff08\u6bd4\u5982\u4e24\u4e2a\u8f68\u9053\u89d2\u52a8\u91cf\uff0c\u91cf\u5b50\u6570\u5206\u522b\u4e3a\\(l_{1}\\)\u548c\\(l_{2}\\)\uff0c\u7ed3\u679c\u7684\u603b\u89d2\u52a8\u91cf\u91cf\u5b50\u6570\u4e3a\\(\\text{L}\\)\uff09\uff0c\u5b87\u79f0\u7684\u4e58\u79ef\u4e3a\\((-1)^{l_1}\\times(-1)^{l_2}\\times(-1)^L=(-1)^{l_1+l_2+L}\\)\u3002\u56e0\u6b64\uff0c\u5f53\\(l_{1}+l_{2}+L\\)\u662f\u5076\u6570\u65f6\uff0c\u8fd9\u4e2a\u4e58\u79ef\u662f1\uff08\u5373\u603b\u7cfb\u7edf\u5b87\u79f0\u4e0d\u53d8\uff09\uff0c\u800c\u5f53\\(l_{1}+l_{2}+L\\)\u662f\u5947\u6570\u65f6\uff0c\u5b87\u79f0\u4e3a-1 \uff08\u603b\u7cfb\u7edf\u5b87\u79f0\u53cd\u8f6c\uff09\u3002<\/p>\n\n\n\n

\u5982\u679c\u7cfb\u7edf\u7684\u7269\u7406\u5c5e\u6027\u5728\u7a7a\u95f4\u5750\u6807\u53cd\u6f14\u540e\u4fdd\u6301\u4e0d\u53d8\uff0c\u5373\u7cfb\u7edf\u7684\u5b87\u79f0\u662f\u6b63\u7684\uff0c\u5219\u7cfb\u7edf\u88ab\u79f0\u4e3a\u5b87\u79f0\u4e0d\u53d8\u7684\u3002\u4f8b\u5982\uff0c\u5982\u679c\u4e00\u4e2a\u7cfb\u7edf\u6216\u72b6\u6001\u5728\u5750\u6807\u53cd\u8f6c\u540e\u770b\u8d77\u6765\u4e0e\u53cd\u8f6c\u524d\u76f8\u540c\uff0c\u5219\u8fd9\u4e2a\u7cfb\u7edf\u6216\u72b6\u6001\u7684\u5b87\u79f0\u662f\u4e0d\u53d8\u7684\u3002\u53e6\u4e00\u65b9\u9762\uff0c\u5bf9\u79f0\u6027\u901a\u5e38\u6307\u7684\u662f\u5728\u8fdb\u884c\u67d0\u4e9b\u53d8\u6362\uff08\u5982\u5e73\u79fb\u3001\u65cb\u8f6c\u3001\u53cd\u5c04\uff09\u540e\u7cfb\u7edf\u7684\u7269\u7406\u6cd5\u5219\u4fdd\u6301\u4e0d\u53d8\u7684\u6027\u8d28\u3002\u56e0\u6b64\uff0c\u5b87\u79f0\u662f\u5bf9\u79f0\u64cd\u4f5c\u7684\u4e00\u79cd\u5f62\u5f0f\uff0c\u5728\u795e\u7ecf\u7f51\u7edc\u4e2d\u4f7f\u7528CG\u7cfb\u6570\u65f6\uff0c\u8fd9\u79cd\u5b88\u6052\u539f\u5219\u6709\u52a9\u4e8e\u4fdd\u8bc1\u5728\u65cb\u8f6c\u5bf9\u79f0\u6027\u7684\u60c5\u51b5\u4e0b\uff0c\u7f51\u7edc\u80fd\u6b63\u786e\u5904\u7406\u5bf9\u79f0\u4e0e\u975e\u5bf9\u79f0\u7279\u5f81\u7684\u53d8\u6362\u3002<\/p>\n\n\n\n

\u4e0a\u8ff0\u8fd9\u4e9b\u5b88\u6052\u539f\u5219\u4e3a\u795e\u7ecf\u7f51\u7edc\u63d0\u4f9b\u4e86\u4e00\u79cd\u5f3a\u5927\u7684\u6570\u5b66\u5de5\u5177\uff0c\u7528\u4ee5\u4fdd\u8bc1\u5728\u5904\u7406\u5177\u6709\u65cb\u8f6c\u53d8\u6362\u7684\u8f93\u5165\u6570\u636e\u65f6\uff0c\u7f51\u7edc\u80fd\u591f\u9002\u5f53\u5730\u7ef4\u6301\u8f93\u5165\u7279\u5f81\u7684\u65cb\u8f6c\u7279\u6027\u3002\u5982\u679c\u4e0d\u6ee1\u8db3\u4e0a\u8ff0\u4e09\u4e2a\u5b88\u6052\u89c4\u5219\uff0c\u5219CG\u7cfb\u6570\u7684\u503c\u4e3a0\u3002<\/p>\n\n\n\n

\u5b9e\u9645\u4e0a\uff0c\u4e0a\u8ff0\u516c\u5f0f\u53ef\u4ee5\u770b\u4f5c\u4e00\u4e2a\u53cc\u7ebf\u6027\u5c42\u7684\u8ba1\u7b97\u3002\u53cc\u7ebf\u6027\u5c42\uff08Bilinear Layer\uff09\u662f\u4e00\u79cd\u901a\u8fc7\u4e24\u4e2a\u8f93\u5165\u5411\u91cf\u7684\u5f20\u91cf\u79ef\u751f\u6210\u8f93\u51fa\u7684\u5c42\u3002\u5176\u4e00\u822c\u5f62\u5f0f\u4e3a\uff1a\\(z=x^{\\mathrm{T}}Wy\\)\uff0c\u5176\u4e2d\\(\\mathbf{x}\\)\u548c\\(\\mathbf{y}\\)\u662f\u8f93\u5165\u5411\u91cf\uff0c\\(\\mathrm{W}\\)\u662f\u6743\u91cd\u77e9\u9635\uff0c\u53ef\u4ee5\u5c06\u516c\u5f0f\u5206\u89e3\u6210\u51e0\u4e2a\u90e8\u5206\uff1a<\/p>\n\n\n\n

\uff081\uff09\u8f93\u5165\u7279\u5f81\\(V_{jcm_\\mathrm{in}}^{(l_\\mathrm{in})}\\)\uff1a\u8868\u793a\u8f93\u5165\u7684\u7279\u5f81\u6620\u5c04\uff0c\u5e26\u6709\u4e0d\u53ef\u7ea6\u8868\u793a\\(l_{\\mathrm{in}}\\)\u548c\u901a\u9053\u7d22\u5f15\\(C_{in}\\)\u3002<\/p>\n\n\n\n

\uff082\uff09\u5377\u79ef\u6838\\(F_{cm_f}^{\\left(l_f,l_{\\mathrm{in}}\\right)}(r_{ij})\\)\uff1a\u8868\u793a\u57fa\u4e8e\u76f8\u5bf9\u4f4d\u7f6e\u5411\u91cf\\(r_{ij}\\)\u7684\u5377\u79ef\u6838\u3002<\/p>\n\n\n\n

\uff083\uff09Clebsch-Gordan\u7cfb\u6570\\(C_{(l_{\\mathrm{out}},m_{\\mathrm{out}})}^{(l_{\\mathrm{in}},m_{\\mathrm{in}}),(l_{f},m_{f})}\\)\uff1a\u7528\u4e8e\u5c06\u8f93\u5165\u7279\u5f81\u548c\u5377\u79ef\u6838\u7684\u89d2\u52a8\u91cf\u6001\u7ed3\u5408\u8d77\u6765\u3002<\/p>\n\n\n\n

\u4ece\u53cc\u7ebf\u6027\u5c42\u7684\u89d2\u5ea6\u6765\u770b\uff0c\u8f93\u5165\u7279\u5f81\\(\\mathrm{V}\\)\u548c\u5377\u79ef\u6838\\(\\text{F}\\)\u53ef\u4ee5\u5206\u522b\u770b\u4f5c\u53cc\u7ebf\u6027\u5c42\u7684\u4e24\u4e2a\u8f93\u5165\uff0c\u800cClebsch-Gordan\u7cfb\u6570\u5219\u7c7b\u4f3c\u4e8e\u6743\u91cd\u5f20\u91cf\\(\\boldsymbol{W}\\)\uff1a<\/p>\n\n\n\n

\uff081\uff09\u8f93\u5165\u5411\u91cf\uff1a\\(x=V_{jc_{\\mathrm{in}}}^{(l_{\\mathrm{in}})}\\)<\/p>\n\n\n\n

\uff082\uff09\u8f93\u5165\u5411\u91cf\uff1a\\(\\boldsymbol{y}=F_{c_{mf}}^{(l_{f},l_{\\mathrm{in}})}(\\boldsymbol{r}_{ij})\\)<\/p>\n\n\n\n

\uff083\uff09\u6743\u91cd\u5f20\u91cf\uff1a\\(\\boldsymbol{W}=C_{(l_{\\mathrm{out}},m_{\\mathrm{out}})}^{(l_{\\mathrm{in}},m_{\\mathrm{in}})(l_{f},m_{f})}\\)<\/p>\n\n\n\n

\u53cc\u7ebf\u6027\u5c42\u7684\u8ba1\u7b97\u65b9\u5f0f\u53ef\u4ee5\u770b\u4f5c\u662f\u5177\u6709\u4e00\u4e2a\u9690\u85cf\u5c42\u7684\u591a\u5c42\u611f\u77e5\u5668\uff0c\u901a\u8fc7\u4e24\u4e2a\u8f93\u5165\u5411\u91cf\u548c\u4e00\u4e2a\u6743\u91cd\u5f20\u91cf\u8fdb\u884c\u52a0\u6743\u6c42\u548c\uff0c\u751f\u6210\u8f93\u51fa\u3002Clebsch-Gordan \u7cfb\u6570\u7684\u4f5c\u7528\u662f\u786e\u4fdd\u4e86\u5f20\u91cf\u79ef\u7684\u6b63\u786e\u6027\u548c\u6709\u6548\u6027\uff0c\u786e\u5b9a\u54ea\u4e9b\u8f93\u5165\u7279\u5f81\u7ec4\u5408\u662f\u7269\u7406\u4e0a\u5141\u8bb8\u7684\uff0c\u4ee5\u53ca\u8fd9\u4e9b\u7ec4\u5408\u5982\u4f55\u751f\u6210\u7b26\u5408\u8f93\u51fa\u8868\u793a\u7684\u7279\u5f81\uff0c\u5373\u6839\u636e\u7269\u7406\u89c4\u5219\u51b3\u5b9a\u9690\u85cf\u5c42\u7684\u6743\u91cd\u77e9\u9635\u54ea\u4e9b\u5e94\u8be5\u7f6e0\u3002<\/p>\n\n\n\n

CG\u7cfb\u6570\u7684\u8ba1\u7b97\u65b9\u5f0f\u4f9d\u8d56\u4e8e\u56fa\u5b9a\u4e14\u590d\u6742\u7684\u6570\u5b66\u516c\u5f0f\uff0c\u5176\u7684\u503c\u53d6\u51b3\u4e8e\u4e24\u4e2a\u539f\u59cb\u89d2\u52a8\u91cf\u7684\u91cf\u5b50\u6570\\(l_{1},m_{1}\\)\u548c\\(l_{2},m_{2}\\)\uff0c\u4ee5\u53ca\u7ed3\u679c\u89d2\u52a8\u91cf\u7684\u91cf\u5b50\u6570\\(L,M\\)\uff0c\u5982\u4e0b\u6240\u793a\uff1a<\/p>\n\n\n

\\(\\begin{aligned}&C\\left(l_1,m_1;l_2,m_2;L,M\\right)=\\\\&\\delta_{m_1+m_2,M}\\sqrt[]{\\frac{(2L+1)(L+l_1-l_2)!(L-l_1+l_2)!(l_1+l_2-L)!}{\\left(l_1+l_2+L+1\\right)!}}\\sqrt{\\frac{(L+M)!(L-M)!}{\\left(l_1+m_1\\right)!\\left(l_1-m_1\\right)!\\left(l_2+m_2\\right)!\\left(l_2-m_2\\right)!}}\\times f\\end{aligned}\\)<\/p>\n\n\n

\u5176\u4e2d\uff0c\\(\\mathrm{f}\\)\u662f\u4e00\u4e2a\u6d89\u53caWigner 3j\u7b26\u53f7\u7684\u8868\u8fbe\u5f0f\uff0c\u5176\u5177\u4f53\u5f62\u5f0f\u4e3a\uff1a<\/p>\n\n\n

\\(f=(-1)^{l_1-l_2+M}\\begin{pmatrix}l_1&l_2&L\\\\m_1&m_2&-M\\end{pmatrix}\\)<\/p>\n\n

\\(\\left.\\left(\\begin{array}{ccc}{l_{1}}&{l_{2}}&{L}\\\\{m_{1}}&{m_{2}}&{-M}\\end{array}\\right.\\right)\\)\u662fWigner 3j\u7b26\u53f7\uff0c\u5b83\u53cd\u6620\u4e86\u89d2\u52a8\u91cf\u5408\u6210\u7684\u51e0\u4f55\u548c\u5bf9\u79f0\u6027\u5c5e\u6027\uff0c\u5e76\u76f4\u63a5\u6d89\u53ca\u5230\u89d2\u52a8\u91cf\u7684\u9009\u62e9\u89c4\u5219\u548c\u5408\u6210\u3002\u5728\u5b9e\u9645\u5e94\u7528\u4e2d\uff0c\u8fd9\u4e9b\u7cfb\u6570\u901a\u5e38\u4f7f\u7528\u6570\u5b66\u8f6f\u4ef6\u5982 Mathematica\u3001Python \u4e2d\u7684 SymPy \u5e93\u7b49\u6765\u8ba1\u7b97\uff0c\u4ee5\u8868\u683c\u7684\u5f62\u5f0f\u5b58\u50a8\uff0c\u4ee5\u4fbf\u5feb\u901f\u67e5\u627e\u548c\u4f7f\u7528\uff0c\u65e0\u9700\u6bcf\u6b21\u4f7f\u7528\u65f6\u91cd\u65b0\u8ba1\u7b97\uff0c\u5982\u88681\u6240\u793a\u3002<\/p>\n\n

\n
\"\"
\u88681 \u7cfb\u6570\u8ba1\u7b97<\/figcaption><\/figure><\/div>\n\n\n

\u5728\u795e\u7ecf\u7f51\u7edc\u6216\u5176\u4ed6\u8ba1\u7b97\u5e94\u7528\u4e2d\uff0c\u901a\u5e38\u4f1a\u9884\u5148\u52a0\u8f7d\u4e00\u4e2a\u5305\u542b\u6240\u6709\u6240\u9700\\(l_{1},l_{2},l\\)\u7ec4\u5408\u7684CG\u7cfb\u6570\u7684\u6570\u636e\u5e93\u6216\u6587\u4ef6\u3002\u8fd9\u6837\u505a\u53ef\u4ee5\u907f\u514d\u8ba1\u7b97\u4e0a\u7684\u5ef6\u8fdf\uff0c\u5e76\u63d0\u9ad8\u6548\u7387\u3002\u5728\u8f6f\u4ef6\u4e2d\uff0c\u8fd9\u4e9b\u7cfb\u6570\u53ef\u80fd\u5df2\u7ecf\u4ee5\u4f18\u5316\u7684\u6570\u636e\u7ed3\u6784\u5b58\u50a8\uff0c\u4ee5\u4fbf\u5feb\u901f\u8bbf\u95ee\u548c\u4f7f\u7528\u3002<\/p>\n\n\n\n

4\u3001\u81ea\u76f8\u4e92\u4f5c\u7528\u5c42<\/h2>\n\n\n\n

TFN\u901a\u8fc7\u4e0a\u4e00\u5c0f\u8282\u4ecb\u7ecd\u7684\u70b9\u5377\u79ef\u5c42\u6765\u805a\u5408\u67d0\u8282\u70b9\u7684\u90bb\u5c45\u4fe1\u606f\uff0c\u66f4\u65b0\u5f97\u5230\u65b0\u7684\u8282\u70b9\u8868\u793a\u3002\u6b64\u5916\uff0cTFN\u8fd8\u901a\u8fc7\u5b50\u76f8\u4e92\u4f5c\u7528\u5c42\u6765\u4ec5\u66f4\u65b0\u67d0\u8282\u70b9\u81ea\u8eab\u7684\u7279\u5f81\u5411\u91cf\uff08\u4e0d\u8003\u8651\u90bb\u5c45\u4fe1\u606f\uff09\uff0c\u81ea\u76f8\u4e92\u4f5c\u7528\u5c42\u53ef\u4ee5\u770b\u4f5c\u7c7b\u4f3c\u4e8e\u8ba1\u7b97\u673a\u89c6\u89c9\u4e2d\u76841\u00d71\u76842D\u5377\u79ef\u64cd\u4f5c\uff0c\u5176\u516c\u5f0f\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n\n\n

\\(V_{\\mathrm{acm}}^{(l+1)}=\\sum_{c^{\\prime}}W_{\\mathrm{cc}}^{(l)}V_{\\mathrm{ac}^{\\prime}\\mathrm{m}}^{(l)}\\)<\/p>\n\n\n

\u8be5\u516c\u5f0f\u53ef\u4ee5\u89e3\u91ca\u4e3a\uff1a\\(V_{\\mathrm{ac^{\\prime}m}}^{(l)}\\)\u662f\u5c42\\(\\mathrm{l}\\)\u4e2d\u8282\u70b9\\(\\mathrm{a}\\)\u7ecf\u7403\u8c10\u51fd\u6570\u7f16\u7801\u7684\u8f93\u5165\u7279\u5f81\u5411\u91cf\uff0c\u5176\u4e2d\\(\\mathrm{m}\\)\u8868\u793a\u7279\u5f81\u5411\u91cf\u5728\u5f20\u91cf\u573a\u4e2d\u7684\u4e00\u4e2a\u7279\u5b9a\u5206\u91cf\uff0c\\(c^{\\prime}\\)\u8868\u793a\u8f93\u5165\u901a\u9053\u3002\\(W_{cc^{\\prime}}^{(c)}\\)\u662f\u6743\u91cd\u77e9\u9635\uff0c\u5b83\u5b9a\u4e49\u4e86\u4ece\u8f93\u5165\u901a\u9053\\(c^{\\prime}\\)\u5230\u8f93\u51fa\u901a\u9053\\(\\mathrm{c}\\)\u7684\u7ebf\u6027\u53d8\u6362\u65b9\u5f0f\u3002\u603b\u548c\\(\\sum_{c^{\\prime}}\\)\u8868\u793a\u5bf9\u6240\u6709\u8f93\u5165\u901a\u9053\u7684\u7d2f\u52a0\uff0c\u8fd9\u5141\u8bb8\u6a21\u578b\u4ece\u4e0d\u540c\u7684\u8f93\u5165\u901a\u9053\u5b66\u4e60\u5230\u4e0d\u540c\u7684\u7279\u5f81\uff0c\u5e76\u5c06\u5b83\u4eec\u5408\u5e76\u6210\u65b0\u7684\u8282\u70b9\u8868\u793a\u3002\u8be5\u81ea\u76f8\u4e92\u4f5c\u7528\u5c42\u7684\u8f93\u51fa\u53ef\u4ee5\u8868\u793a\u4e3a\\(V_{\\mathrm{acm}}^{(l+1)}\\)\uff0c\u5373\u8282\u70b9\\(\\mathrm{a}\\)\u5728\u5c42\\(\\text{l+1}\\)\u7684\u7279\u5f81\u5411\u91cf\uff0c\u5176\u6bcf\u4e2a\u901a\u9053\\(\\mathrm{c}\\)\u7684\u503c\u7531\u4e0a\u8ff0\u7ebf\u6027\u7ec4\u5408\u8ba1\u7b97\u5f97\u5230\u3002\u8fd9\u6837\u7684\u64cd\u4f5c\u4f7f\u5f97\u6bcf\u4e2a\u8282\u70b9\u80fd\u591f\u72ec\u7acb\u66f4\u65b0\u5176\u7279\u5f81\u8868\u793a\uff0c\u800c\u4e0d\u53d7\u90bb\u5c45\u8282\u70b9\u7684\u76f4\u63a5\u5f71\u54cd\uff0c\u8fd9\u5bf9\u4e8e\u4fdd\u6301\u7279\u5f81\u7684\u5c40\u90e8\u6027\u975e\u5e38\u91cd\u8981\u3002\u5728\u8fdb\u4e00\u6b65\u7684\u7f51\u7edc\u5c42\u4e2d\uff0c\u8fd9\u4e9b\u66f4\u65b0\u8fc7\u7684\u7279\u5f81\u5411\u91cf\\(V_{\\mathrm{acm}}^{(l+1)}\\)\u53ef\u4ee5\u88ab\u7528\u4f5c\u8f93\u5165\u5230\u540e\u7eed\u7684SO(3<\/em>)\u5377\u79ef\u5c42\u6216\u5176\u4ed6\u7c7b\u578b\u7684\u5c42\uff0c\u4ee5\u5b9e\u73b0\u66f4\u590d\u6742\u7684\u7279\u5f81\u63d0\u53d6\u548c\u53d8\u6362\u3002\u8fd9\u79cd\u8bbe\u8ba1\u4f7fTFN\u80fd\u591f\u9002\u5e94\u5404\u79cd\u5404\u6837\u7684\u6570\u636e\u7ed3\u6784\u548c\u4efb\u52a1\uff0c\u63d0\u4f9b\u7075\u6d3b\u800c\u5f3a\u5927\u7684\u6a21\u578b\u67b6\u6784\u4ee5\u5904\u7406\u5177\u6709\u590d\u6742\u51e0\u4f55\u548c\u62d3\u6251\u7ed3\u6784\u7684\u6570\u636e\u3002<\/p>\n\n\n\n

5\u3001\u6fc0\u6d3b\u51fd\u6570<\/h2>\n\n\n\n

\u4e0e\u4f20\u7edf\u795e\u7ecf\u7f51\u7edc\u4e00\u6837\uff0cTFNs\u4e2d\u7684\u5377\u79ef\u5c42\u540e\u901a\u5e38\u4f1a\u8ddf\u968f\u6fc0\u6d3b\u51fd\u6570\u3002\u8fd9\u4e9b\u975e\u7ebf\u6027\u51fd\u6570\u662f\u5fc5\u9700\u7684\uff0c\u56e0\u4e3a\u5b83\u4eec\u5141\u8bb8\u7f51\u7edc\u5b66\u4e60\u548c\u8868\u793a\u590d\u6742\u7684\u975e\u7ebf\u6027\u5173\u7cfb\u3002\u5728\u5904\u7406\u5206\u5b50\u6570\u636e\u65f6\uff0c\u5408\u9002\u7684\u6fc0\u6d3b\u51fd\u6570\u53ef\u4ee5\u5e2e\u52a9\u6a21\u578b\u8bc6\u522b\u5206\u5b50\u5185\u90e8\u7684\u5316\u5b66\u53cd\u5e94\u8def\u5f84\u548c\u80fd\u91cf\u53d8\u5316\u7b49\u590d\u6742\u73b0\u8c61\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5728\u5904\u7406\u65cb\u8f6c\u9636\u6570\u4e3a0\u548c\u5927\u4e8e0\u4e24\u79cd\u60c5\u51b5\u65f6\uff0c\u6fc0\u6d3b\u51fd\u6570\u7684\u5b9a\u4e49\u662f\u4e0d\u540c\u7684\u3002\u5177\u4f53\u6765\u8bf4\u5bf9\u4e8e\u65cb\u8f6c\u9636\u6570\u4e3a0\u7684\u7279\u5f81\uff0c\u901a\u5e38\u53ef\u4ee5\u5e94\u7528\u6807\u51c6\u7684\u975e\u7ebf\u6027\u6fc0\u6d3b\u51fd\u6570\uff0c\u5982ReLU\u3001Sigmoid\u6216Tanh\uff0c\u8fd9\u4e9b\u6fc0\u6d3b\u51fd\u6570\u80fd\u591f\u5904\u7406\u6807\u91cf\u8f93\u5165\uff0c\u9002\u7528\u4e8e\u4e0d\u53d7\u65cb\u8f6c\u53d8\u6362\u5f71\u54cd\u7684\u7279\u5f81\u3002<\/p>\n\n\n\n

\u7136\u800c\uff0c\u5bf9\u4e8e\u65cb\u8f6c\u9636\u6570\u5927\u4e8e0\u7684\u60c5\u51b5\uff0c\u5373\u90a3\u4e9b\u5728\u65cb\u8f6c\u4e0b\u53d8\u6362\u7684\u7279\u5f81\uff0c\u6807\u51c6\u7684\u6fc0\u6d3b\u51fd\u6570\u4e0d\u80fd\u76f4\u63a5\u5e94\u7528\uff0c\u56e0\u4e3a\u8fd9\u4e9b\u7279\u5f81\u5177\u6709\u65b9\u5411\u6027\uff0c\u9700\u8981\u4fdd\u6301\u5176\u5728\u7a7a\u95f4\u65cb\u8f6c\u4e0b\u7684\u4e00\u81f4\u6027\u3002\u4e3a\u4e86\u4fdd\u6301\u65cb\u8f6c\u7b49\u53d8\u6027\uff0c\u6fc0\u6d3b\u51fd\u6570\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\u53ea\u5e94\u7528\u4e8e\u7279\u5f81\u5411\u91cf\u7684\u6a21\u957f\uff0c\u56e0\u4e3a\u6a21\u957f\u662f\u65cb\u8f6c\u4e0d\u53d8\u7684\u3002\u5728\u65cb\u8f6c\u9636\u6570\u5927\u4e8e0\u7684\u60c5\u51b5\u4e0b\uff0c\u7528\u975e\u7ebf\u6027\u51fd\u6570\u5904\u7406\u540e\u7684\u6a21\u957f\u6765\u7f29\u653e\u539f\u59cb\u7279\u5f81\u5411\u91cf\u7684\u6bcf\u4e2a\u5206\u91cf\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u53ef\u4ee5\u60f3\u8c61\u6210\u5bf9\u539f\u59cb\u5411\u91cf\u7684\u957f\u5ea6\u8fdb\u884c\u7f29\u653e\uff0c\u4f46\u662f\u4fdd\u6301\u5176\u6307\u5411\u4e0d\u53d8\u3002\u901a\u8fc7\u8fd9\u6837\u7684\u64cd\u4f5c\uff0c\u7279\u5f81\u5411\u91cf\u5728\u7ecf\u8fc7\u975e\u7ebf\u6027\u53d8\u6362\u540e\uff0c\u4ecd\u7136\u4fdd\u6301\u5bf9\u65cb\u8f6c\u7684\u654f\u611f\u6027\u3002\u8fd9\u6837\uff0c\u5f53\u6574\u4e2a\u7cfb\u7edf\u65cb\u8f6c\u65f6\uff0c\u53d8\u6362\u540e\u7684\u7279\u5f81\u5411\u91cf\u4ecd\u7136\u4ee5\u4e00\u79cd\u4e00\u81f4\u7684\u65b9\u5f0f\u8868\u8fbe\u540c\u4e00\u4e2a\u7269\u7406\u73b0\u8c61\u6216\u7ed3\u6784\u7279\u5f81\u3002<\/p>\n\n\n\n

6\u3001\u9876\u5c42\u7ed3\u6784<\/h2>\n\n\n\n

\u5728TFNs\u4e2d\uff0c\u9876\u5c42\u8bbe\u8ba1\u5305\u542b\u6c60\u5316\u5c42\u548c\u5168\u8fde\u63a5\u5c42\u3002\u6c60\u5316\u5c42\u7684\u804c\u80fd\u662f\u6574\u5408\u6574\u4e2a\u5206\u5b50\u7684\u4fe1\u606f\u3002\u901a\u8fc7\u5bf9\u6240\u6709\u539f\u5b50\u7684\u9ad8\u7ef4\u8868\u793a\u8fdb\u884c\u6c47\u603b\uff0c\u7f51\u7edc\u53ef\u4ee5\u83b7\u5f97\u4e00\u4e2a\u5168\u5c40\u63cf\u8ff0\uff0c\u8fd9\u5bf9\u4e8e\u9884\u6d4b\u5206\u5b50\u7ea7\u522b\u7684\u6027\u8d28\uff08\u5982\u5206\u5b50\u7684\u603b\u80fd\u91cf\u6216\u5076\u6781\u77e9\uff09\u662f\u5fc5\u8981\u7684\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u53ef\u80fd\u5305\u62ec\u7b80\u5355\u7684\u64cd\u4f5c\uff0c\u6bd4\u5982\u8ba1\u7b97\u6240\u6709\u539f\u5b50\u7279\u5f81\u7684\u5e73\u5747\u503c\uff0c\u6216\u8005\u66f4\u590d\u6742\u7684\u52a0\u6743\u548c\uff0c\u8fd9\u4e9b\u52a0\u6743\u53ef\u80fd\u4f9d\u8d56\u4e8e\u539f\u5b50\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\u3002<\/p>\n\n\n\n

\u5728\u7ecf\u8fc7\u4e00\u7cfb\u5217\u5377\u79ef\u548c\u6c60\u5316\u64cd\u4f5c\u4e4b\u540e\uff0cTFN\u53ef\u4ee5\u5305\u62ec\u4e00\u6216\u591a\u4e2a\u5168\u8fde\u63a5\u5c42\u3002\u8fd9\u4e9b\u5c42\u53ef\u4ee5\u8fdb\u4e00\u6b65\u7ec4\u5408\u548c\u8f6c\u6362\u5168\u5c40\u63cf\u8ff0\uff0c\u4ee5\u4ea7\u751f\u6700\u7ec8\u7684\u8f93\u51fa\uff0c\u5982\u5206\u5b50\u7684\u6027\u8d28\u9884\u6d4b\u3002\u5168\u8fde\u63a5\u5c42\u80fd\u591f\u5b66\u4e60\u4e0d\u540c\u539f\u5b50\u7279\u5f81\u4e4b\u95f4\u590d\u6742\u7684\u76f8\u4e92\u4f5c\u7528\uff0c\u5e76\u5c06\u5176\u6620\u5c04\u5230\u6240\u9700\u7684\u8f93\u51fa\u7a7a\u95f4\u3002<\/p>\n\n\n\n

\u6700\u7ec8\uff0cTFN\u7684\u8f93\u51fa\u5c42\u5c06\u7f51\u7edc\u7684\u9ad8\u7ef4\u8868\u793a\u6620\u5c04\u5230\u6240\u5173\u5fc3\u7684\u9884\u6d4b\u4e0a\u3002\u5728\u5206\u5b50\u52a8\u529b\u5b66\u6a21\u62df\u4e2d\uff0c\u8fd9\u53ef\u80fd\u662f\u80fd\u91cf\u7ea7\u522b\u3001\u529b\u7684\u5927\u5c0f\u548c\u65b9\u5411\u3001\u7535\u8377\u5206\u5e03\u7b49\u3002\u8fd9\u4e2a\u5c42\u901a\u5e38\u4f1a\u7ecf\u8fc7\u8bad\u7ec3\u6765\u6700\u5c0f\u5316\u9884\u6d4b\u548c\u5b9e\u9645\u503c\u4e4b\u95f4\u7684\u5dee\u8ddd\uff0c\u4f7f\u7528\u7684\u662f\u56de\u5f52\u5206\u6790\u4e2d\u5e38\u89c1\u7684\u635f\u5931\u51fd\u6570\uff0c\u5982\u5747\u65b9\u8bef\u5dee\u3002<\/p>\n\n\n\n

\u6574\u4e2aTFN\u67b6\u6784\u901a\u8fc7\u76d1\u7763\u5b66\u4e60\u6765\u8bad\u7ec3\u3002\u7ed9\u5b9a\u4e00\u7ec4\u5206\u5b50\u7ed3\u6784\u548c\u76f8\u5e94\u7684\u76ee\u6807\u5c5e\u6027\uff08\u5982\u5b9e\u9a8c\u6d4b\u91cf\u7684\u80fd\u91cf\uff09\uff0c\u7f51\u7edc\u901a\u8fc7\u4f18\u5316\u7b97\u6cd5\uff08\u5982\u68af\u5ea6\u4e0b\u964d\uff09\u8c03\u6574\u5176\u6743\u91cd\uff0c\u4ee5\u6700\u5c0f\u5316\u9884\u6d4b\u8f93\u51fa\u548c\u76ee\u6807\u8f93\u51fa\u4e4b\u95f4\u7684\u5dee\u5f02\u3002\u5728\u8fd9\u4e2a\u8fc7\u7a0b\u4e2d\uff0c\u7f51\u7edc\u5b66\u4e60\u5230\u4e86\u5982\u4f55\u7cbe\u786e\u6355\u6349\u5206\u5b50\u5185\u90e8\u7684\u590d\u6742\u76f8\u4e92\u4f5c\u7528\uff0c\u4ee5\u53ca\u8fd9\u4e9b\u76f8\u4e92\u4f5c\u7528\u5982\u4f55\u5f71\u54cd\u5206\u5b50\u7684\u5b8f\u89c2\u6027\u8d28\u3002<\/p>\n\n\n\n

7\u3001TFN\u7684\u6d88\u606f\u4f20\u9012\u673a\u5236<\/h2>\n\n\n\n

TFN\u7684\u6d88\u606f\u4f20\u9012\u8fc7\u7a0b\u53ef\u4ee5\u5206\u4e3a\u4ee5\u4e0b\u4e09\u4e2a\u6b65\u9aa4\uff1a<\/p>\n\n\n\n

1\u3001\u6d88\u606f\u4f20\u9012\u4e0e\u66f4\u65b0\uff1a<\/p>\n\n\n\n

\u7b2c\u4e00\u6b65\u662f\u8282\u70b9\u4e4b\u95f4\u7684\u4fe1\u606f\u4f20\u9012\u548c\u66f4\u65b0\u3002\u5177\u4f53\u7684\u6570\u5b66\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n\n\n

\\(\\boldsymbol{M}_{ij}^{(L)}=Y^{(L)}\\left(\\frac{\\boldsymbol{x}_{ij}}{\\|\\boldsymbol{x}_{ij}\\|}\\right)\\odot\\boldsymbol{W}_{\\mathrm{CG}}\\boldsymbol{V}_j^{(L)}\\)<\/p>\n\n\n

\u5176\u4e2d\uff1a<\/p>\n\n\n\n

\uff081\uff09 \\(x_{ij}=x_i-x_j\\)\u8868\u793a\u8282\u70b9\\(\\mathrm{i}\\)\u548c\u8282\u70b9\\(\\mathrm{j}\\)\u4e4b\u95f4\u7684\u76f8\u5bf9\u4f4d\u7f6e\u5411\u91cf\uff1b<\/p>\n\n\n\n

\uff082\uff09\\(\\boldsymbol{W}\\)\u662f\u4e00\u4e2a\u53ef\u5b66\u4e60\u7684\u6743\u91cd\u77e9\u9635\uff0c\u5e76\u9075\u5b88\u4e8eCG\u7cfb\u6570\u7684\u9650\u5236\uff1b<\/p>\n\n\n\n

\uff083\uff09\\(Y(|x_{ij}|)\\)\u662f\u4e00\u4e2a\u4f9d\u8d56\u4e8e\u76f8\u5bf9\u4f4d\u7f6e\u5411\u91cf\\(\\mathcal{X}_{ij}\\)\u7684\u7403\u8c10\u57fa\u51fd\u6570\uff1b<\/p>\n\n\n\n

\uff084\uff09\\(\\odot\\)\u8868\u793a\u5143\u7d20\u9010\u4f4d\u76f8\u4e58\uff1b<\/p>\n\n\n\n

\uff085\uff09\\(V_j^{(L)}\\)\u662f\u8282\u70b9\\(\\mathrm{j}\\)\u5728\u7b2c\\(\\mathrm{L}\\)\u5c42\u7684\u7279\u5f81\u5411\u91cf\u3002<\/p>\n\n\n\n

\u5728\u4e0a\u8ff0\u516c\u5f0f\u4e2d\uff0c\\(\\boldsymbol{M}_{ij}^{(L)}\\)\u8868\u793a\u4ece\u8282\u70b9\\(\\mathrm{j}\\)\u4f20\u9012\u5230\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u6d88\u606f\uff0c\u8fd9\u4e2a\u6d88\u606f\u901a\u8fc7\u8282\u70b9\\(\\mathrm{j}\\)\u7684\u7279\u5f81\\(V_j^{(L)}\\)\u548c\u4e00\u4e2a\u57fa\u4e8e\u76f8\u5bf9\u4f4d\u7f6e\u5411\u91cf\u7684\u7403\u8c10\u51fd\u6570\\(\\text{Y}\\)\u8fdb\u884c\u52a0\u6743\u5e73\u5747\u540e\u5f97\u5230\u3002<\/p>\n\n\n\n

\u63a5\u4e0b\u6765\uff0c\u8282\u70b9\\(\\mathrm{i}\\)\u5728\u7b2c\\(\\mathrm{L}\\)\u5c42\u7684\u66f4\u65b0\u7279\u5f81\u5411\u91cf\\(U_i^{(L)}\\)\u901a\u8fc7\u4ee5\u4e0b\u516c\u5f0f\u8ba1\u7b97\uff1a<\/p>\n\n\n

\\(U_i^{(L)}=V_i^{(L)}+\\sum_{j\\in\\mathcal{N}(i)}M_{ij}^{(L)}\\)<\/p>\n\n\n

\u5176\u4e2d\uff1a<\/p>\n\n\n\n

\uff081\uff09 \\(U_i^{(L)}\\)\u8868\u793a\u8282\u70b9\\(\\mathrm{i}\\)\u5728\u7b2c\\(\\mathrm{L}\\)\u5c42\u7684\u66f4\u65b0\u7279\u5f81\u5411\u91cf\uff1b<\/p>\n\n\n\n

\uff082\uff09 \\(V_{i}^{(L)}\\)\u8868\u793a\u8282\u70b9\\(\\mathrm{i}\\)\u5728\u7b2c\\(\\mathrm{L}\\)\u5c42\u7684\u521d\u59cb\u7279\u5f81\u5411\u91cf\uff1b<\/p>\n\n\n\n

\uff083\uff09\\(\\mathcal{N}(i)\\)\u662f\u8282\u70b9\\(\\mathrm{i}\\)\u7684\u90bb\u5c45\u8282\u70b9\u96c6\u5408\u3002<\/p>\n\n\n\n

\u901a\u8fc7\u4e0a\u8ff0\u516c\u5f0f\uff0c\u53ef\u4ee5\u5c06\u6765\u81ea\u90bb\u5c45\u8282\u70b9\u7684\u6d88\u606f\\(\\boldsymbol{M}_{ij}^{(L)}\\)\u805a\u5408\u5230\u5f53\u524d\u8282\u70b9\u7684\u7279\u5f81\\(\\tilde{V}_i^{(L)}\\)\u4e0a\uff0c\u5b9e\u73b0\u7279\u5f81\u7684\u66f4\u65b0\u548c\u4fe1\u606f\u7684\u4f20\u64ad\u3002<\/p>\n\n\n\n

2\u3001\u81ea\u6211\u4ea4\u4e92\uff1a<\/p>\n\n\n\n

\u7b2c\u4e8c\u6b65\u662f\u5bf9\u805a\u5408\u540e\u7684\u8282\u70b9\u7279\u5f81\u8fdb\u884c\u81ea\u6211\u4ea4\u4e92\uff0c\u516c\u5f0f\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n\n\n

\\(\\tilde{V}_i^{(L)}=\\{U_i^{(L)}W^{(L)}\\}_{\\ell\\in L}\\)<\/p>\n\n\n

\u5176\u4e2d\uff0c\\(W^{(L)}\\)\u662f\u4e00\u4e2a\u53ef\u5b66\u4e60\u7684\u6743\u91cd\u77e9\u9635\u3002\u81ea\u6211\u4ea4\u4e92\u662f\u5bf9\u7279\u5f81\u5411\u91cf\u8fdb\u884c\u7684\u7ebf\u6027\u53d8\u6362\uff0c\u5728\u8fd9\u4e2a\u6b65\u9aa4\u4e2d\uff0c\u6bcf\u4e2a\u8282\u70b9\u7684\u7279\u5f81\u901a\u8fc7\u81ea\u6211\u4ea4\u4e92\u7684\u6743\u91cd\u77e9\u9635\\(W^{(L)}\\)\u8fdb\u884c\u53d8\u6362\uff0c\u4ece\u800c\u6355\u6349\u5230\u7279\u5f81\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\u3002<\/p>\n\n\n\n

3\u3001\u975e\u7ebf\u6027\u5316(\u6fc0\u6d3b\u51fd\u6570)\uff1a<\/p>\n\n\n\n

\u6d88\u606f\u4f20\u9012\u8fc7\u7a0b\u7684\u7b2c\u4e09\u6b65\u662f\u5bf9\u8282\u70b9\u7279\u5f81\u8fdb\u884c\u975e\u7ebf\u6027\u5316\u5904\u7406\u3002\u8fd9\u4e2a\u6b65\u9aa4\u7684\u76ee\u7684\u662f\u5f15\u5165\u975e\u7ebf\u6027\u56e0\u7d20\uff0c\u4f7f\u5f97\u6a21\u578b\u80fd\u591f\u6355\u6349\u5230\u66f4\u52a0\u590d\u6742\u7684\u7279\u5f81\u548c\u5173\u7cfb\u3002\u5177\u4f53\u7684\u6570\u5b66\u8868\u793a\u5982\u4e0b\uff1a<\/p>\n\n\n

\\(\\tilde{\\boldsymbol{V}}_i^{(L)}=\\left\\{\\tilde{\\boldsymbol{V}}_i^{(\\ell)}\\sigma\\left(\\left\\|\\tilde{\\boldsymbol{V}}_i^{(\\ell)}\\right\\|_2+b_i^{(\\ell)}\\right)\\right\\}_{\\ell\\in L}\\)<\/p>\n\n\n

\u5176\u4e2d\uff1a<\/p>\n\n\n\n

\uff081\uff09\u6fc0\u6d3b\u51fd\u6570\\(\\sigma\\)\uff1a\u6fc0\u6d3b\u51fd\u6570\u662f\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u4e2d\u5f15\u5165\u975e\u7ebf\u6027\u7684\u91cd\u8981\u7ec4\u4ef6\u3002\u5e38\u89c1\u7684\u6fc0\u6d3b\u51fd\u6570\u6709ReLU\u548cSigmoid\u7b49\u3002ReLU\u5c06\u8d1f\u503c\u6620\u5c04\u4e3a0\uff0c\u4fdd\u7559\u6b63\u503c\uff0c\u800cSigmoid\u5219\u5c06\u503c\u6620\u5c04\u52300\u548c1\u4e4b\u95f4\u3002\u901a\u8fc7\u5f15\u5165\u975e\u7ebf\u6027\u6fc0\u6d3b\u51fd\u6570\uff0c\u6a21\u578b\u80fd\u591f\u66f4\u597d\u5730\u62df\u5408\u590d\u6742\u7684\u975e\u7ebf\u6027\u5173\u7cfb\u3002<\/p>\n\n\n\n

\uff082\uff09\u7279\u5f81\u5411\u91cf\u7684\u6a21\u957f\uff1a\u8ba1\u7b97\u8282\u70b9\\(\\text{i}\\)\u5728\u7b2c\\(\\ell\\)\u5c42\u7279\u5f81\u5411\u91cf\u7684L2\u8303\u6570\uff08\u6b27\u51e0\u91cc\u5f97\u8303\u6570\uff09\u5373\u4e3a\u8be5\u7279\u5f81\u5411\u91cf\u7684\u6a21\u957f\u3002\u901a\u8fc7\u8ba1\u7b97\u6a21\u957f\uff0c\u53ef\u4ee5\u83b7\u5f97\u7279\u5f81\u5411\u91cf\u7684\u5927\u5c0f\u4fe1\u606f\uff0c\u8fd9\u5bf9\u4e8e\u540e\u7eed\u7684\u7f29\u653e\u548c\u5f52\u4e00\u5316\u64cd\u4f5c\u81f3\u5173\u91cd\u8981\u3002<\/p>\n\n\n\n

\uff083\uff09\u504f\u7f6e\u9879\\(b_{i}^{(\\ell)}\\)\uff1a\u504f\u7f6e\u9879\u662f\u4e00\u4e2a\u53ef\u5b66\u4e60\u7684\u53c2\u6570\uff0c\u7528\u4e8e\u8c03\u6574\u7279\u5f81\u5411\u91cf\u7684\u6a21\u957f\uff0c\u4f7f\u5f97\u6a21\u578b\u80fd\u591f\u66f4\u7075\u6d3b\u5730\u9002\u5e94\u4e0d\u540c\u7684\u7279\u5f81\u5206\u5e03\u3002<\/p>\n\n\n\n

\uff084\uff09\u975e\u7ebf\u6027\u5316\u7684\u610f\u4e49\uff1a\u901a\u8fc7\u975e\u7ebf\u6027\u5316\u5904\u7406\uff0c\u53ef\u4ee5\u5f15\u5165\u590d\u6742\u7684\u7279\u5f81\u5173\u7cfb\uff0c\u63d0\u5347\u6a21\u578b\u7684\u8868\u8fbe\u80fd\u529b\u3002\u975e\u7ebf\u6027\u5316\u540e\u7684\u7ed3\u679c\u53ef\u4ee5\u89c6\u4e3a\u5bf9\u8282\u70b9\\(\\begin{array}{c}{i}\\end{array}\\)\u7684\u7279\u5f81\u8fdb\u884c\u7f29\u653e\uff0c\u4f7f\u5176\u66f4\u9002\u5e94\u6a21\u578b\u7684\u5b66\u4e60\u9700\u6c42\u3002\u5177\u4f53\u7684\u8ba1\u7b97\u7ed3\u679c\u8868\u793a\u4e3a\uff1a\\(\\sigma\\left(\\left|\\tilde{V}_i^{(\\ell)}\\right|_2+b_i^{(\\ell)}\\right)\\)\u3002\u8fd9\u4e00\u7ed3\u679c\u53ef\u4ee5\u770b\u4f5c\u662f\u4e00\u4e2a\u7f29\u653e\u56e0\u5b50\uff0c\u5bf9\u8282\u70b9\\(\\begin{array}{c}{i}\\end{array}\\)\u7684\u7279\u5f81\\(\\tilde{V}_i^{(\\ell)}\\)\u8fdb\u884c\u7f29\u653e\uff0c\u4ece\u800c\u4f7f\u5f97\u7279\u5f81\u66f4\u5177\u8fa8\u8bc6\u5ea6\u548c\u9002\u5e94\u6027\u3002<\/p>\n\n\n\n

\u901a\u8fc7\u4ee5\u4e0a\u6b65\u9aa4\uff0cTFN\u80fd\u591f\u5728\u7279\u5f81\u4f20\u9012\u548c\u66f4\u65b0\u8fc7\u7a0b\u4e2d\u6709\u6548\u5730\u6355\u6349\u5230\u8282\u70b9\u4e4b\u95f4\u7684\u590d\u6742\u5173\u7cfb\uff0c\u5e76\u901a\u8fc7\u975e\u7ebf\u6027\u5316\u5904\u7406\u589e\u5f3a\u6a21\u578b\u7684\u8868\u8fbe\u80fd\u529b\u548c\u62df\u5408\u6548\u679c\u3002\u8fd9\u4e00\u6b65\u9aa4\u5728\u6574\u4e2a\u6d88\u606f\u4f20\u9012\u8fc7\u7a0b\u4e2d\u8d77\u5230\u4e86\u81f3\u5173\u91cd\u8981\u7684\u4f5c\u7528\uff0c\u786e\u4fdd\u4e86\u6a21\u578b\u7684\u7075\u6d3b\u6027\u548c\u9c81\u68d2\u6027\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u5f20\u91cf\u573a\u7f51\u7edc\uff08Tensor Field Networks, TFNs\uff09\u6e90\u4e8e\u8bba\u6587Tensor Field Netw […]<\/p>\n","protected":false},"author":1,"featured_media":2590,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[27,23],"tags":[],"class_list":["post-2525","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\/2525","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=2525"}],"version-history":[{"count":20,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2525\/revisions"}],"predecessor-version":[{"id":2546,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/posts\/2525\/revisions\/2546"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=\/wp\/v2\/media\/2590"}],"wp:attachment":[{"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2525"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2525"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/gnn.club\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2525"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}