{"id":2310,"date":"2024-10-18T18:54:45","date_gmt":"2024-10-18T10:54:45","guid":{"rendered":"https:\/\/www.gnn.club\/?p=2310"},"modified":"2024-10-22T10:49:28","modified_gmt":"2024-10-22T02:49:28","slug":"dimenet","status":"publish","type":"post","link":"http:\/\/www.gnn.club\/?p=2310","title":{"rendered":"DimeNet"},"content":{"rendered":"\n

DimeNet\uff08Directional Message Passing Neural Network\uff09\u662f\u4e00\u79cd\u4e13\u95e8\u7528\u4e8e\u5206\u5b50\u56fe\u7ed3\u6784\u7684\u795e\u7ecf\u7f51\u7edc\uff0c\u5728\u9884\u6d4b\u5206\u5b50\u6027\u8d28\u548c\u6a21\u62df\u5206\u5b50\u95f4\u76f8\u4e92\u4f5c\u7528\u65b9\u9762\u8868\u73b0\u51fa\u8272\u3002DimeNet \u7684\u72ec\u7279\u4e4b\u5904\u5728\u4e8e\u5b83\u5f15\u5165\u4e86\u65b9\u5411\u6027\u6d88\u606f\u4f20\u9012\u673a\u5236\uff0c\u4f7f\u5176\u80fd\u591f\u66f4\u597d\u5730\u6355\u6349\u5206\u5b50\u4e2d\u7684\u51e0\u4f55\u548c\u89d2\u5ea6\u4fe1\u606f\u3002DimeNet\u7684\u5173\u952e\u7279\u70b9\u4e4b\u4e00\u662f\u65b9\u5411\u6027\u6d88\u606f\u4f20\u9012\uff0c\u4f20\u7edf\u7684\u6d88\u606f\u4f20\u9012\u795e\u7ecf\u7f51\u7edc\uff08Message Passing Neural Network, MPNN\uff09\u4e3b\u8981\u8003\u8651\u8282\u70b9\u548c\u8fb9\u4e4b\u95f4\u7684\u6d88\u606f\u4f20\u9012\u3002SchNet\u8fdb\u4e00\u6b65\u7684\u5bf9\u8282\u70b9\u4e4b\u524d\u7684\u76f8\u5bf9\u8ddd\u79bb\u8fdb\u884c\u5efa\u6a21\u3002\u800cDimeNet\u5219\u66f4\u8fdb\u4e00\u6b65\u5f15\u5165\u4e86\u65b9\u5411\u6027\u6d88\u606f\u4f20\u9012\u7684\u6982\u5ff5\uff0c\u8fd9\u610f\u5473\u7740\u5b83\u4e0d\u4ec5\u8003\u8651\u8282\u70b9\u548c\u8fb9\u7684\u4fe1\u606f\uff0c\u8fd8\u5c06\u76f8\u90bb\u8fb9\u4e4b\u95f4\u7684\u89d2\u5ea6\u4fe1\u606f\u7eb3\u5165\u8ba1\u7b97\u3002<\/p>\n\n\n\n

1.1\u5f15\u5165\u89d2\u5ea6\u4fe1\u606f<\/h3>\n\n\n\n

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\n
\"\"
\u56fe1 \u5206\u5b50\u793a\u610f\u56fe<\/figcaption><\/figure><\/div>\n\n\n

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1.2\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u5f84\u5411\u6a21\u578b<\/h3>\n\n\n\n

\u5728\u6784\u5efa\u5206\u5b50\u7684\u57fa\u7840\u8868\u793a\u65f6\uff0cDimeNet\u91c7\u7528\u4e86\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u5f84\u5411\u6a21\u578b\u8fd9\u4e24\u7c7b\u6570\u5b66\u51fd\u6570\u3002\u7403\u8c10\u51fd\u6570\u7528\u4e8e\u6355\u6349\u5206\u5b50\u5f62\u72b6\u53ca\u539f\u5b50\u5728\u4e0d\u540c\u65b9\u5411\u4e0a\u7684\u5206\u5e03\uff0c\u8fd9\u5bf9\u4e8e\u7406\u89e3\u5206\u5b50\u7684\u4e09\u7ef4\u7ed3\u6784\u81f3\u5173\u91cd\u8981\u3002\u8d1d\u585e\u5c14\u5f84\u5411\u6a21\u578b\u5219\u5728\u5904\u7406\u95ee\u9898\u65f6\u66f4\u52a0\u5173\u6ce8\u4e8e\u539f\u5b50\u95f4\u7684\u5b9e\u9645\u8ddd\u79bb\uff0c\u8fd9\u4e9b\u51fd\u6570\u7279\u522b\u9002\u5408\u4e8e\u63cf\u8ff0\u539f\u5b50\u95f4\u8ddd\u79bb\u76f8\u5173\u7684\u6027\u8d28\u3002<\/p>\n\n\n\n

\u5728DimeNet\u6a21\u578b\u4e2d\uff0c\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u5f84\u5411\u6a21\u578b\u7684\u5e94\u7528\u662f\u4e00\u79cd\u5173\u952e\u7684\u6570\u636e\u9884\u5904\u7406\u6b65\u9aa4\uff0c\u65e8\u5728\u4ece\u5206\u5b50\u7684\u4e09\u7ef4\u7ed3\u6784\u4e2d\u63d0\u53d6\u7a7a\u95f4\u7279\u5f81\uff0c\u5c24\u5176\u662f\u539f\u5b50\u95f4\u7684\u8ddd\u79bb\u548c\u89d2\u5ea6\u4fe1\u606f\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u5728\u5904\u7406\u539f\u5b50\u95ee\u9898\u65f6\u7684\u8f93\u5165\u4e00\u822c\u662f\u539f\u5b50\u5728\u7b1b\u5361\u5c14\u5750\u6807\u7cfb(x,y,z)<\/em>\u4e0b\u7684\u4e09\u7ef4\u5750\u6807\u3002\u901a\u8fc7\u8ba1\u7b97\u4efb\u610f\u4e24\u4e2a\u539f\u5b50\u4e4b\u95f4\u7684\u6b27\u51e0\u91cc\u5f97\u8ddd\u79bb\uff0c\u5e76\u628a\u8fd9\u4e2a\u8ddd\u79bb\u4f5c\u4e3a\u8d1d\u585e\u5c14\u5f84\u5411\u6a21\u578b\u7684\u8f93\u5165\uff0c\u8d1d\u585e\u5c14\u5f84\u5411\u6a21\u578b\u80fd\u591f\u6355\u6349\u5230\u539f\u5b50\u95f4\u8ddd\u79bb\u4e0a\u7684\u6ce2\u52a8\u7279\u5f81\u3002\u7c7b\u4f3c\u7684\uff0c\u53ef\u4ee5\u5c06\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0b\u539f\u5b50\u7684\u4f4d\u7f6e\u8868\u793a\u8f6c\u6362\u4e3a\u7403\u5750\u6807\u7cfb\u4e2d\u7684\u89d2\u5ea6\u8868\u793a\uff0c\u8fd9\u4e9b\u89d2\u5ea6\u4fe1\u606f\u518d\u4f5c\u4e3a\u7403\u8c10\u51fd\u6570\u7684\u8f93\u5165\uff0c\u7403\u8c10\u51fd\u6570\u80fd\u591f\u6355\u6349\u5230\u539f\u5b50\u95f4\u65b9\u5411\u4e0a\u7684\u89d2\u5ea6\u7279\u5f81\u3002\u57fa\u4e8e\u4e0a\u8ff0\u8ddd\u79bb\u548c\u89d2\u5ea6\uff0c\u6211\u4eec\u4f7f\u7528\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u51fd\u6570\u516c\u5f0f\u4e2d\u7684\u7cfb\u6570\u7ec4\u5408\u4f5c\u4e3a\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u7684\u8f93\u5165\u7279\u5f81\u3002\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u51fd\u6570\u7684\u6570\u5b66\u7279\u5f81\u6b63\u4ea4\u6027\u53ef\u4ee5\u786e\u4fdd\u4ece\u8fd9\u4e9b\u8f6c\u6362\u5f97\u5230\u7684\u7279\u5f81\u5728\u4fe1\u606f\u4e0a\u662f\u72ec\u7acb\u4e14\u4e92\u8865\u7684\uff0c\u51cf\u5c11\u4e86\u7279\u5f81\u95f4\u7684\u5197\u4f59\u3002<\/p>\n\n\n\n

\u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u65b9\u7a0b\u516c\u5f0f\u4e2d\u7684\u7cfb\u6570\u662f\u56fa\u5b9a\u7684\u6570\u5b66\u51fd\u6570\uff0c\u4e0d\u9700\u8981\u88ab\u8bad\u7ec3\u3002\u53ea\u8981\u6709\u70b9\u7684\u4e09\u7ef4\u5750\u6807\u4f5c\u4e3a\u8f93\u5165\uff0c\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u65b9\u7a0b\u62c6\u89e3\u7684\u7ed3\u679c\u5c31\u662f\u56fa\u5b9a\u7684\u3002\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u65b9\u7a0b\u7684\u4f7f\u7528\u66f4\u50cf\u662f\u4e00\u79cd\u57fa\u4e8e\u539f\u5b50\u5750\u6807\u7684\u7279\u5f81\u5de5\u7a0b\u624b\u6bb5\uff0c\u7528\u4e8e\u5c06\u539f\u59cb\u7684\u51e0\u4f55\u6570\u636e\u8f6c\u6362\u6210\u4e00\u79cd\u5bf9\u6a21\u578b\u66f4\u6709\u7528\u7684\u5f62\u5f0f\u3002\u8fd9\u79cd\u7279\u5f81\u8f6c\u6362\u65b9\u6cd5\u4e3aDimeNet\u63d0\u4f9b\u4e86\u4e00\u79cd\u201c\u5f3a\u5927\u7684\u8bed\u8a00\u201d\uff0c\u4f7f\u5176\u80fd\u591f\u7cbe\u786e\u63cf\u8ff0\u5e76\u5229\u7528\u539f\u5b50\u95f4\u7684\u8ddd\u79bb\u548c\u89d2\u5ea6\u76f8\u4e92\u4f5c\u7528\u4fe1\u606f\u3002\u6b63\u4ea4\u6027\u4fdd\u8bc1\u4e86\u8f6c\u6362\u5f97\u5230\u7684\u7279\u5f81\u96c6\u4e2d\u6bcf\u4e2a\u7279\u5f81\u90fd\u643a\u5e26\u6709\u7528\u4e14\u72ec\u7279\u7684\u4fe1\u606f\uff0c\u6709\u52a9\u4e8e\u6a21\u578b\u4ece\u6570\u636e\u4e2d\u5b66\u4e60\u5230\u66f4\u591a\u72ec\u7acb\u7684\u7279\u5f81\uff0c\u4ece\u800c\u63d0\u9ad8\u9884\u6d4b\u5206\u5b50\u6027\u8d28\u7684\u51c6\u786e\u6027\u3002<\/p>\n\n\n\n

\u6700\u540e\uff0c\u8fd8\u6709\u4e00\u4e2a\u5173\u952e\u95ee\u9898\u662f\u6a21\u578b\u5982\u4f55\u4e60\u5f97\u8fd9\u4e9b\u7cfb\u6570\u7279\u5f81\u7684\u542b\u4e49\uff1f\u6bd5\u7adf\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u51fd\u6570\u5e76\u6ca1\u6709\u88ab\u5185\u7f6e\u5728\u6a21\u578b\u7684\u7ed3\u6784\u4e2d\u4f5c\u4e3a\u6a21\u578b\u7684\u4e00\u90e8\u5206\u3002\u5b9e\u9645\u4e0a\uff0cDimeNet\u5e76\u4e0d\u9700\u8981\u201c\u77e5\u9053\u201d\u8fd9\u4e9b\u7cfb\u6570\u4ee3\u8868\u4e86\u4ec0\u4e48\u3002\u6a21\u578b\u6709\u80fd\u529b\u901a\u8fc7\u8bad\u7ec3\u8fc7\u7a0b\u5b66\u4e60\u8fd9\u4e9b\u8f6c\u6362\u540e\u7684\u7cfb\u6570\u7279\u5f81\u4e0e\u76ee\u6807\u5c5e\u6027\uff08\u5982\u5206\u5b50\u7684\u80fd\u91cf\u6216\u5176\u4ed6\u5316\u5b66\u6027\u8d28\uff09\u4e4b\u95f4\u7684\u5173\u7cfb\u3002\u901a\u8fc7\u5927\u91cf\u7684\u8bad\u7ec3\u6570\u636e\uff0cDimeNet\u80fd\u591f\u8bc6\u522b\u54ea\u4e9b\u6a21\u5f0f\uff08\u5728\u8fd9\u4e9b\u7cfb\u6570\u8868\u793a\u7684\u7279\u5f81\u4e2d\uff09\u4e0e\u9884\u6d4b\u7684\u5316\u5b66\u6027\u8d28\u6709\u5173\u8054\u3002\u8fd9\u79cd\u5b66\u4e60\u8fc7\u7a0b\u662f\u6570\u636e\u9a71\u52a8\u7684\uff0c\u4e0d\u9700\u8981\u6a21\u578b\u5bf9\u7cfb\u6570\u7684\u7269\u7406\u610f\u4e49\u6709\u5148\u9a8c\u7684\u7406\u89e3\u3002\u4f7f\u7528\u7403\u8c10\u51fd\u6570\u548c\u8d1d\u585e\u5c14\u51fd\u6570\u4f5c\u4e3a\u8f93\u5165\u9884\u5904\u7406\u6b65\u9aa4\u662f\u56e0\u4e3a\u5b83\u4eec\u63d0\u4f9b\u4e86\u4e00\u79cd\u6709\u6548\u7684\u65b9\u5f0f\u6765\u8868\u8fbe\u5206\u5b50\u7ed3\u6784\u4e2d\u7684\u4e09\u7ef4\u7a7a\u95f4\u4fe1\u606f\u3002\u8fd9\u4e9b\u4fe1\u606f\u662f\u4f20\u7edf\u56fe\u795e\u7ecf\u7f51\u7edc\uff08\u4f8b\u5982GCN\u3001GraphSAGE\u3001GAT\u7b49\uff09\u96be\u4ee5\u76f4\u63a5\u5904\u7406\u7684\uff0c\u56e0\u4e3a\u5b83\u4eec\u901a\u5e38\u8bbe\u8ba1\u7528\u4e8e\u5904\u7406\u56fe\u7ed3\u6784\u6570\u636e\uff0c\u800c\u6ca1\u6709\u8003\u8651\u5230\u539f\u5b50\u95f4\u590d\u6742\u7684\u7a7a\u95f4\u548c\u65b9\u5411\u6027\u5173\u7cfb\u3002<\/p>\n\n\n\n

DimeNet\u4e2d\u4f7f\u7528\u7684\u8d1d\u585e\u5c14\u51fd\u6570\u8be6\u89c1\u516c\u5f0f1\uff1a<\/p>\n\n\n

\\(\\tilde{e}_{\\mathrm{RBF},n}(d)=\\sqrt{\\frac2c}\\frac{\\sin(\\frac{n\\pi}cd)}d\\)<\/p>\n\n\n

\u516c\u5f0f1\u5c55\u793a\u7684\u662f\u5f84\u5411\u8d1d\u8d5b\u5c14\u57fa\u51fd\u6570\\(\\tilde{e}_{\\mathrm{RBF},n}(d)\\)\uff0c\u8fd9\u662f\u4e00\u79cd\u53ea\u4e0e\u539f\u5b50\u95f4\u8ddd\u79bbd<\/em>\u6709\u5173\u7684\u51fd\u6570\u3002\u5176\u4e2dn<\/em>\u662f\u7d22\u5f15\uff0c\u8868\u793a\u4e0d\u540c\u7684\u57fa\u51fd\u6570\uff1bc<\/em>\u662f\u622a\u65ad\u8ddd\u79bb\uff0c\u8868\u793a\u53ea\u6709\u5728\u6b64\u8303\u56f4\u5185\u7684\u539f\u5b50\u95f4\u76f8\u4e92\u4f5c\u7528\u624d\u88ab\u8003\u8651\u3002\u8fd9\u4e2a\u51fd\u6570\u662f\u901a\u8fc7\u5c06\u6b63\u5f26\u51fd\u6570\u8c03\u6574\u548c\u7f29\u653e\u5230[0,c<\/em>]\u533a\u95f4\u5185\u5f97\u5230\u7684\u3002\u8fd9\u4e2a\u533a\u95f4\u662f\u56e0\u4e3a\u5728\u5206\u5b50\u6a21\u578b\u4e2d\uff0c\u901a\u5e38\u53ea\u5173\u5fc3\u539f\u5b50\u95f4\u5728\u4e00\u5b9a\u8ddd\u79bb\u5185\u7684\u76f8\u4e92\u4f5c\u7528\u3002\u6b63\u5f26\u51fd\u6570\u81ea\u7136\u5730\u4ea7\u751f\u4e86\u4e00\u4e2a\u5728\u4e24\u7aef\u4e3a\u96f6\u4e14\u5728\u4e2d\u95f4\u632f\u8361\u7684\u5f62\u72b6\uff0c\u8fd9\u4e2a\u7279\u6027\u5bf9\u4e8e\u63cf\u8ff0\u539f\u5b50\u95f4\u8ddd\u79bb\u4e0a\u7684\u6ce2\u52a8\u7279\u5f81\u662f\u6709\u7528\u7684\u3002\u516c\u5f0f\u4e2d\u7684\\(\\sqrt{\\frac2c}\\frac{\\sin(\\frac{n\\pi}cd)}d\\) \u662f\u5177\u4f53\u7684\u51fd\u6570\u5f62\u5f0f\uff0c\u5176\u4e2d\\(\\sqrt{\\frac2c}\\)\u662f\u4e3a\u4e86\u786e\u4fdd\u51fd\u6570\u5728\u533a\u95f4[0,c<\/em>]\u4e0a\u662f\u5f52\u4e00\u5316\u7684\uff0c\u4e5f\u5c31\u662f\u8bf4\u8fd9\u4e2a\u51fd\u6570\u7684\u503c\u4e0d\u4f1a\u968f\u7740\u533a\u95f4\u7684\u5927\u5c0f\u800c\u53d8\u6362\u3002<\/p>\n\n\n

\n
\"\"
\u56fe2 \u5f84\u5411\u8d1d\u8d5b\u5c14\u57fa\u51fd\u6570\u66f2\u7ebf<\/figcaption><\/figure><\/div>\n\n\n

\u5f84\u5411\u8d1d\u8d5b\u5c14\u57fa\u51fd\u6570\u66f2\u7ebf\u5982\u56fe2\u6240\u793a\uff0c\u5176\u4e2d\u4e0d\u540c\u989c\u8272\u7684\u7ebf\u4ee3\u8868\u4e0d\u540c\u7684\u5f84\u5411\u8d1d\u585e\u5c14\u57fa\u51fd\u6570\u3002\u6bcf\u6761\u7ebf\u5bf9\u5e94\u4e8e\u516c\u5f0f1\u4e2d\u4e0d\u540c\u7684n<\/em>\u503c\uff0c\u8fd9\u4e9b\u503c\u4ee3\u8868\u4e86\u8d1d\u585e\u5c14\u51fd\u6570\u7684\u4e0d\u540c\u9636\u3002 n<\/em>=1\u7684\u57fa\u51fd\u6570\u5728\u6574\u4e2a\u533a\u95f4\u5185\u53ea\u6709\u4e00\u4e2a\u660e\u663e\u7684\u5cf0\u503c\uff08d<\/em>=0 \u65f6\uff09\uff0c\u5e76\u4e14\u5728d<\/em>=c <\/em>\u5904\u4e0b\u964d\u81f3\u6700\u5c0f\u503c0\u3002\u8fd9\u4e2a\u6027\u8d28\u4e0e\u7269\u7406\u76f4\u89c9\u4e00\u81f4\uff1a\u539f\u5b50\u95f4\u76f8\u4e92\u4f5c\u7528\u5728\u63a5\u8fd1\u65f6\u6700\u5f3a\uff0c\u5728\u8fdc\u79bb\u65f6\u51cf\u5f31\uff0c\u5e76\u5728\u4e00\u5b9a\u8ddd\u79bb\u540e\u6d88\u5931\u3002\u968f\u7740 d<\/em>\/c<\/em>\u7684\u589e\u5927\uff0c\u9636\u6570n<\/em>\u5927\u4e8e1\u7684\u57fa\u51fd\u6570\u4f1a\u6709\u66f4\u591a\u7684\u6ce2\u5cf0\u548c\u6ce2\u8c37\u3002\u8fd9\u53ef\u4ee5\u8868\u793a\u5728\u539f\u5b50\u95f4\u8ddd\u79bb\u4e0a\u7684\u53d8\u5316\u66f4\u52a0\u590d\u6742\u7684\u6a21\u5f0f\u3002\u5728\u67d0\u4e9b\u60c5\u51b5\u4e0b\uff0c\u968f\u7740\u8ddd\u79bb\u7684\u589e\u52a0\uff0c\u539f\u5b50\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u53ef\u80fd\u4f1a\u7ecf\u5386\u4e00\u7cfb\u5217\u7684\u589e\u5f3a\u548c\u51cf\u5f31\uff0c\u8fd9\u53ef\u80fd\u4e0e\u5206\u5b50\u5185\u90e8\u539f\u5b50\u7684\u7a7a\u95f4\u6392\u5217\u6709\u5173\u3002\u6b63\u662f\u8fd9\u79cd\u590d\u6742\u6027\u4f7f\u5f97\u57fa\u4e8e\u8d1d\u585e\u5c14\u51fd\u6570\u7684\u57fa\u51fd\u6570\u6210\u4e3a\u6355\u6349\u548c\u63cf\u8ff0\u539f\u5b50\u95f4\u76f8\u4e92\u4f5c\u7528\u7279\u5f81\u7684\u6709\u529b\u5de5\u5177\u3002<\/p>\n\n\n\n

\u8d1d\u585e\u5c14\u51fd\u6570\u548c\u7403\u8c10\u51fd\u6570\u7684\u7ed3\u5408\u65b9\u5f0f\u7684\u6570\u5b66\u516c\u5f0f\u8be6\u89c1\u516c\u5f0f2\uff0c\u79f0\u4e3a\u89d2\u5411\u57fa\u51fd\u6570\uff1a<\/p>\n\n\n

\\(\\tilde{a}_{\\mathrm{SBF},ln}(d,\\alpha)=\\sqrt{\\frac2{c^3j_{l+1}^2(z_{ln})}}j_l(\\frac{z_{ln}}cd)Y_l^0(\\alpha)\\)<\/p>\n\n\n

\u516c\u5f0f2\u8868\u793a\u4e86\u5982\u4f55\u7ed3\u5408\u8d1d\u8d5b\u5c14\u51fd\u6570\\(j_{l}\\)\u548c\u7403\u8c10\u51fd\u6570\\(Y_{l}^0\\)\u6765\u6784\u9020\u4e00\u4e2a\u7403\u9762\u5085\u91cc\u53f6-\u8d1d\u8d5b\u5c14\u57fa\\(\\tilde{a}_{\\mathrm{sBF},ln}\\)\uff0c\u8fd9\u4e2a\u57fa\u51fd\u6570\u7528\u4e8e\u5206\u5b50\u5185\u90e8\u539f\u5b50\u95f4\u5173\u7cfb\u7684\u5efa\u6a21\u3002\u5176\u4e2d\uff0c \\(j_1(\\frac{z_{ln}}cd)\\)\u662f\u8d1d\u8d5b\u5c14\u51fd\u6570\u7684l<\/em>\u9636\u8868\u793a\u3002\\(Y_l^0\\left(\\alpha\\right)\\)\u662fl<\/em>\u9636\u7684\u7403\u8c10\u51fd\u6570\uff0c\u8fd9\u91cc\u8003\u8651\u7684\u662fm=0\u7684\u7279\u6b8a\u60c5\u51b5\uff0c\u53ea\u4f9d\u8d56\u4e8e\u89d2\u5ea6\u03b1\u3002\u5177\u4f53\u6765\u8bf4\uff0c\u7403\u8c10\u51fd\u6570\u7684\u4e00\u822c\u5f62\u5f0f\u4e3a\uff1a<\/p>\n\n\n

\\(Y_l^m(\\theta,\\phi)=\\sqrt{\\frac{(2l+1)}{4\\pi}\\frac{(l-m)!}{(l+m)!}}P_l^m(\\cos\\theta)e^{im\\phi}\\)<\/p>\n\n\n

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

\uff081\uff09l<\/em>\u662f\u9636\u6570\uff08\u6216\u89d2\u91cf\u5b50\u6570\uff09\uff0c\u8868\u793a\u7403\u8c10\u51fd\u6570\u7684\u6b21\u6570\u3002<\/p>\n\n\n\n

\uff082)m<\/em>\u662f\u78c1\u91cf\u5b50\u6570\uff0c\u8303\u56f4\u662f\\(-l\\leq m\\leq l\\)\u3002<\/p>\n\n\n\n

\uff083\uff09\\(P_l^m\\)\u662f\u7f14\u5408\u52d2\u8ba9\u5fb7\u591a\u9879\u5f0f\u3002<\/p>\n\n\n\n

\uff084\uff09\\(\\theta \\)\u662f\u6781\u89d2\uff08\u6216\u7eac\u5ea6\uff09\uff0c\u8303\u56f4\u662f\\([0,\\pi]\\)\u3002<\/p>\n\n\n\n

\uff085\uff09\\(\\phi \\)\u662f\u65b9\u4f4d\u89d2\uff08\u6216\u7ecf\u5ea6\uff09\uff0c\u8303\u56f4\u662f\\([0,2\\pi]\\)\u3002<\/p>\n\n\n\n

\u7279\u6b8a\u60c5\u51b5\uff1a\u5f53m<\/em>=0\u65f6\uff0c\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u7403\u8c10\u51fd\u6570\u53ea\u4f9d\u8d56\u4e8e\u89d2\u5ea6\\(\\theta \\)\uff1a<\/p>\n\n\n

\\(Y_l^0(\\theta)=\\sqrt{\\frac{(2l+1)}{4\\pi}}P_l(\\cos\\theta)\\)<\/p>\n\n\n

\u5176\u4e2d\\(P_l(\\cos\\theta)\\)\u662f\u52d2\u8ba9\u5fb7\u591a\u9879\u5f0f\u3002<\/p>\n\n\n\n

\u800c\u516c\u5f0f2\u4e2d\u7684\\(\\sqrt{\\frac{2}{c^{3}j_{l+1}^{2}(z_{ln})}}\\)\u662f\u5f52\u4e00\u5316\u7cfb\u6570\uff0c\u7528\u4e8e\u786e\u4fdd\u4e0d\u540cl<\/em>\u548cn<\/em>\u5bf9\u5e94\u7684\u57fa\u51fd\u6570\u5728\u7ed9\u5b9a\u8303\u56f4\u5185\u662f\u6b63\u4ea4\u7684\uff0c\u5e76\u5177\u6709\u5408\u9002\u7684\u89c4\u6a21\u3002<\/p>\n\n\n\n

\u4e3a\u4e86\u5e2e\u52a9\u7406\u89e3\u516c\u5f0f\uff0c\u5bf9\u8d1d\u585e\u5c14\u51fd\u6570\u548c\u7403\u8c10\u51fd\u6570\u6784\u6210\u7684\u7403\u9762\u5085\u91cc\u53f6-\u8d1d\u585e\u5c14\u57fa\u8fdb\u884c\u4e86\u53ef\u89c6\u5316\uff0c\u5982\u56fe3\u6240\u793a\uff0c\u5176\u4e2d\u6bcf\u4e2a\u5706\u5708\u4ee3\u8868\u4e86\u4e0d\u540c\u7684l<\/em>\u548cn<\/em>\u503c\u4e0b\u57fa\u51fd\u6570\u7684\u6a21\u5f0f\u3002<\/p>\n\n\n

\n
\"\"
\u56fe3 \u7403\u9762\u5085\u91cc\u53f6-\u8d1d\u585e\u5c14\u57fa\u7684\u53ef\u89c6\u5316<\/figcaption><\/figure><\/div>\n\n\n

\u968f\u7740l<\/em>\u503c\u7684\u589e\u52a0\uff0c\u53ef\u4ee5\u53d1\u73b0\u770b\u5230\u7684\u662f\u5728\u89d2\u5ea6\u4e0a\u7684\u53d8\u5316\u8d8a\u6765\u8d8a\u590d\u6742\uff1b\u968f\u7740n<\/em>\u503c\u7684\u589e\u52a0\uff0c\u53ef\u4ee5\u53d1\u73b0\u5728\u5f84\u5411\u4e0a\u632f\u8361\u7684\u6b21\u6570\u589e\u52a0\u3002\u56fe3\u4e2d\u7684\u8fd9\u4e9b\u5706\u5f62\u7684\u5b50\u56fe\u662f\u4e8c\u7ef4\u8868\u793a\u4e2d\u7684\u57fa\u7840\u6a21\u5f0f\uff0c\u53ef\u4ee5\u7528\u6765\u8868\u793a\u539f\u5b50\u95f4\u5173\u7cfb\u7684\u591a\u79cd\u53ef\u80fd\u6027\u3002\u901a\u8fc7\u7ec4\u5408\u8fd9\u4e9b\u57fa\u7840\u6a21\u5f0f\uff0c\u6211\u4eec\u53ef\u4ee5\u7cbe\u786e\u5730\u63cf\u8ff0\u539f\u5b50\u4e4b\u95f4\u7684\u7a7a\u95f4\u5173\u7cfb\uff0c\u5e76\u5c06\u8fd9\u4e9b\u4fe1\u606f\u8f93\u5165\u5230\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u4e2d\u8fdb\u884c\u8fdb\u4e00\u6b65\u7684\u5904\u7406\u548c\u5b66\u4e60\u3002\u8fd9\u4e9b\u57fa\u51fd\u6570\u4e3a\u5206\u5b50\u7684\u51e0\u4f55\u7ed3\u6784\u63d0\u4f9b\u4e86\u4e00\u79cd\u6570\u5b66\u63cf\u8ff0\uff0c\u5b83\u65e2\u8003\u8651\u5230\u4e86\u539f\u5b50\u95f4\u7684\u8ddd\u79bb\uff0c\u4e5f\u8003\u8651\u5230\u4e86\u5b83\u4eec\u4e4b\u95f4\u7684\u89d2\u5ea6\uff0c\u56e0\u6b64\u80fd\u591f\u5728\u4e0d\u540c\u7684\u5c3a\u5ea6\u548c\u65b9\u5411\u4e0a\u5206\u6790\u5206\u5b50\u3002<\/p>\n\n\n\n

\u603b\u7684\u6765\u8bf4\uff0c\u516c\u5f0f1\u548c\u516c\u5f0f2\u63d0\u4f9b\u4e86\u5173\u4e8e\u89d2\u5ea6\u548c\u8ddd\u79bb\u7684\u4e0d\u540c\u4fe1\u606f\uff0c\u5e76\u4e14\u88ab\u8bbe\u8ba1\u7528\u4e8e\u51e0\u4f55\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u4e2d\u7684\u4e0d\u540c\u76ee\u7684\u3002\u516c\u5f0f2\u4f7f\u7528\u8d1d\u585e\u5c14\u51fd\u6570\u548c\u7403\u8c10\u51fd\u6570\u7684\u7ec4\u5408\u6765\u540c\u65f6\u6355\u83b7\u8ddd\u79bb\u548c\u89d2\u5ea6\u4e0a\u7684\u4fe1\u606f\uff0c\u8fd9\u79cd\u8868\u793a\u5728\u8003\u8651\u539f\u5b50\u95f4\u7684\u76f8\u4e92\u4f5c\u7528\u65f6\u975e\u5e38\u6709\u7528\uff0c\u56e0\u4e3a\u8fd9\u4e9b\u76f8\u4e92\u4f5c\u7528\u901a\u5e38\u53d6\u51b3\u4e8e\u5b83\u4eec\u4e4b\u95f4\u7684\u8ddd\u79bb\u548c\u5b83\u4eec\u76f8\u5bf9\u4e8e\u5f7c\u6b64\u7684\u65b9\u5411\u3002\u8fd9\u4e2a\u590d\u5408\u57fa\u7840\u80fd\u591f\u63d0\u4f9b\u4e00\u4e2a\u7efc\u5408\u89c6\u56fe\uff0c\u8868\u5f81\u539f\u5b50\u5982\u4f55\u5728\u7a7a\u95f4\u4e2d\u76f8\u5bf9\u4e8e\u5176\u4ed6\u539f\u5b50\u6392\u5217\u3002\u516c\u5f0f1\u5219\u662f\u7eaf\u7cb9\u7684\u5f84\u5411\u57fa\u51fd\u6570\uff0c\u5b83\u53ea\u5173\u6ce8\u539f\u5b50\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\u800c\u4e0d\u8003\u8651\u89d2\u5ea6\u3002\u8fd9\u79cd\u8868\u793a\u63d0\u4f9b\u4e86\u5206\u5b50\u5927\u5c0f\u548c\u5f62\u72b6\u7684\u66f4\u5b8f\u89c2\u7684\u89c6\u89d2\uff0c\u5bf9\u4e8e\u90a3\u4e9b\u53ea\u4f9d\u8d56\u4e8e\u8ddd\u79bb\u7684\u5c5e\u6027\uff08\u4f8b\u5982\u67d0\u4e9b\u7c7b\u578b\u7684\u529b\uff09\u7279\u522b\u6709\u7528\u3002<\/p>\n\n\n\n

\u5728\u540e\u7eed\u7684DimeNet\u6a21\u578b\u7ed3\u6784\u4e2d\uff0c\u7f51\u7edc\u5c42\u53ef\u4ee5\u5229\u7528\u8fd9\u4e9b\u4e0d\u540c\u7684\u4fe1\u606f\u6765\u5b66\u4e60\u9884\u6d4b\u5206\u5b50\u7684\u6027\u8d28\uff0c\u7efc\u5408\u4f7f\u7528\u8fd9\u4e9b\u57fa\u51fd\u6570\u53ef\u4ee5\u8ba9\u6a21\u578b\u66f4\u5168\u9762\u5730\u7406\u89e3\u5206\u5b50\u5185\u90e8\u7684\u590d\u6742\u76f8\u4e92\u4f5c\u7528\uff0c\u63d0\u9ad8\u9884\u6d4b\u7684\u51c6\u786e\u6027\u548c\u9c81\u68d2\u6027\uff0c\u4ece\u800c\u5728\u5404\u81ea\u9886\u57df\u63d0\u4f9b\u6700\u4f73\u6027\u80fd\u3002<\/p>\n\n\n\n

1.3 DimeNet\u7684\u6d88\u606f\u4f20\u9012\u673a\u5236<\/h3>\n\n\n\n

\u5728 DimeNet \u4e4b\u524d\uff0c\u51e0\u4f55\u56fe\u795e\u7ecf\u7f51\u7edc\u7684\u7f51\u7edc\u7ed3\u6784\u6700\u591a\u53ea\u5229\u7528\u4e86\u51e0\u4f55\u4fe1\u606f\u4e2d\u7684\u8ddd\u79bb\u4fe1\u606f\uff0c\u4f8b\u5982SchNet\u6a21\u578b\uff0c\u800c DimeNet \u662f\u7b2c\u4e00\u4e2a\u5c06\u89d2\u5ea6\u4fe1\u606f\u5f15\u5165\u6d88\u606f\u4f20\u9012\u9636\u6bb5\u7684\uff0c\u5176\u6d88\u606f\u4f20\u9012\u673a\u5236\u5982\u56fe4\u6240\u793a\u3002<\/p>\n\n\n

\n
\"\"
\u56fe4 DimeNet\u6d88\u606f\u4f20\u9012\u673a\u5236<\/figcaption><\/figure><\/div>\n\n\n

\uff081\uff09\u8282\u70b9\uff1a\u56fe4\u4e2d\u7684\u5706\u5708\u4ee3\u8868\u539f\u5b50\uff0c\u6807\u8bb0\u4e3ai<\/em>\u548cj<\/em>\uff0c\u8fd9\u4e9b\u539f\u5b50\u5728\u5206\u5b50\u56fe\u4e2d\u901a\u8fc7\u8fb9\u76f8\u8fde\u3002<\/p>\n\n\n\n

\uff082\uff09\u6d88\u606f\uff1a\u65b9\u5411\u6027\u5d4c\u5165\\(m_{ji}\\)\u88ab\u89c6\u4e3a\u4ece\u539f\u5b50j<\/em>\u53d1\u9001\u7ed9\u539f\u5b50i<\/em>\u7684\u4e00\u6761\u6d88\u606f\u3002<\/p>\n\n\n\n

\uff083\uff09\u8fb9\uff1a\u7bad\u5934\u8868\u793a\u6d88\u606f\u7684\u65b9\u5411\uff0c\u4ece\u4e00\u4e2a\u539f\u5b50\u5230\u53e6\u4e00\u4e2a\u539f\u5b50\u3002\u4f8b\u5982\uff0c\\(m_{ji}\\)\u8868\u793a\u4ecej<\/em>\u5230i<\/em>\u7684\u6d88\u606f\u3002<\/p>\n\n\n\n

\uff084\uff09\u90bb\u5c45\uff1a\\(N_{j}\\)\u8868\u793a\u539f\u5b50j<\/em>\u7684\u90bb\u5c45\u96c6\u5408\u3002\u5728\u56fe\u4e2d\uff0c\u90bb\u5c45\u88ab\u6807\u8bb0\u4e3a\\(k_1,k_2\\text{\u548c}k_3\\)\u3002\u6bcf\u4e2a\u90bb\u5c45\u90fd\u5411\u539f\u5b50j<\/em>\u53d1\u9001\u5b83\u4eec\u7684\u6d88\u606f\\(m_{kj}\\)\u3002<\/p>\n\n\n\n

\uff085\uff09\u805a\u5408\uff1a\u539f\u5b50i<\/em>\u901a\u8fc7\u805a\u5408\u6765\u81eaj<\/em>\u7684\u6240\u6709\u90bb\u5c45k<\/em>\u7684\u6d88\u606f\u6765\u66f4\u65b0\\(m_{ji}\\)\u7684\u6d88\u606f\u3002<\/p>\n\n\n\n

\u5728\u56fe4-17\u4e2d\uff0c\u6211\u4eec\u7684\u76ee\u6807\u662f\u6839\u636e\u8282\u70b9i<\/em>\u7684\u90bb\u5c45\u4fe1\u606f\u6765\u66f4\u65b0\u8282\u70b9i<\/em>\u7684\u7279\u5f81\u3002\u901a\u4fd7\u7684\u89e3\u91ca\uff0c\u53ef\u4ee5\u8bbe\u60f3\u8282\u70b9i<\/em>\u662f\u4e00\u4e2a\u56fd\u738b\uff0c\u4ed6\u5c06\u5411\u5b50\u6c11\u6536\u7a0e\u3002\u9996\u5148\uff0c\u56fd\u738b\u5411\u5927\u81e3\uff08\u6700\u8fd1\u7684\u4e00\u5708\u90bb\u5c45\uff0c\u4f8b\u5982\u8282\u70b9j<\/em>\uff09\u6536\u7a0e\u3002\u5927\u81e3\u4eec\u6536\u5230\u547d\u4ee4\u540e\uff0c\u5411\u8001\u767e\u59d3\u6536\u7a0e\uff08k<\/em>\uff09\uff0c\u5927\u81e3\u4eec\u6c47\u96c6\u4e4b\u540e\uff0c\u518d\u5411\u56fd\u738b\u7f34\u7a0e\u3002\u5f53\u7136\u8fd9\u662f\u4e00\u4e2a\u7b80\u5316\u7684\u6a21\u578b\u3002\u4e0d\u8fc7\uff0c\u53ef\u4ee5\u5f88\u6e05\u6670\u7684\u770b\u5230\uff0cDimeNet\u7684\u6d88\u606f\u4f20\u9012\u4e0d\u4ec5\u6d89\u53ca\u90bb\u5c45\u4fe1\u606f\uff0c\u8fd8\u6d89\u53ca\u90bb\u5c45\u7684\u90bb\u5c45\u3002\u8fd9\u79cd\u6d89\u53ca\u4e24\u8df3\uff082-hop\uff09\u7684\u6d88\u606f\u4f20\u9012\u673a\u5236\u4f7f\u5f97 DimeNet \u80fd\u591f\u83b7\u53d6\u66f4\u591a\u7684\u5c40\u90e8\u4fe1\u606f\uff0c\u53e6\u4e00\u65b9\u9762\uff0c\u5176\u7b97\u529b\u6d88\u8017\u4e5f\u5927\u5e45\u589e\u52a0\u3002<\/p>\n\n\n\n

DimeNet \u662f\u7b2c\u4e00\u4e2a\u5c06\u89d2\u5ea6\u4fe1\u606f\u5f15\u5165\u6d88\u606f\u4f20\u9012\u9636\u6bb5\u7684\uff0c\u5b83\u4e0d\u4ec5\u5305\u62ec\u4e86\u4e24\u4e2a\u8282\u70b9\uff08\u6bd4\u5982\u8bf4\u539f\u5b50i<\/em>\u548cj<\/em>\uff09\u4e4b\u95f4\u7684\u76f4\u7ebf\u8ddd\u79bb\uff0c\u8fd8\u5305\u62ec\u4e86\u4e00\u4e2a\u7b2c\u4e09\u4e2a\u8282\u70b9k<\/em>\uff0c\u5f62\u6210\u4e86\u4e00\u4e2a\u89d2\u5ea6\u3002\u8fd9\u4e2a\u89d2\u5ea6\u662f\u901a\u8fc7\u8ba1\u7b97\u8282\u70b9 i<\/em>\u548c\u8282\u70b9j<\/em> \u4e4b\u95f4\u7684\u5411\u91cf\u4ee5\u53ca\u8282\u70b9i<\/em> \u548c\u8282\u70b9k<\/em> \u4e4b\u95f4\u7684\u5411\u91cf\u7684\u5939\u89d2\u5f97\u5230\u7684\u3002\u901a\u8fc7\u8fd9\u79cd\u65b9\u5f0f\uff0c\u5f53\u8282\u70b9\u66f4\u65b0\u5176\u8868\u793a\u65f6\uff0c\u5b83\u5df2\u7ecf\u96c6\u6210\u4e86\u5176\u5468\u56f4\u73af\u5883\u7684\u8be6\u7ec6\u51e0\u4f55\u7ed3\u6784\u4fe1\u606f\uff0c\u516c\u5f0f\u8868\u8ff0\u5982\u4e0b\uff1a<\/p>\n\n\n

\\(m_{ji}^{(l+1)}=f_{\\mathrm{update}}(m_{ji}^{(l)},\\sum_{k\\in\\mathcal{N}_{j}\\in\\{i\\}}f_{\\mathrm{int}}(m_{kj}^{(l)},e_{\\mathrm{RBF}}^{(ji)},a_{\\mathrm{SBF}}^{(kj,ji)}))\\)<\/p>\n\n\n

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

\uff081\uff09\\(m_{ji}^{(l+1)}\\)\u662f\u66f4\u65b0\u540e\u7684\u6d88\u606f\uff0c\u4f20\u9012\u4ece\u539f\u5b50j<\/em>\u5230 i<\/em>\u7684\u4fe1\u606f\u3002<\/p>\n\n\n\n

\uff082\uff09\\(f_{\\mathrm{update}}\\)\u662f\u6d88\u606f\u66f4\u65b0\u51fd\u6570\u3002<\/p>\n\n\n\n

\uff083\uff09\\(m_{ji}^{(l)}\\)\u662f\u5f53\u524d\u5c42l<\/em>\u7684\u6d88\u606f\u5d4c\u5165\u3002<\/p>\n\n\n\n

\uff084\uff09\\(\\text{E}\\)\u8868\u793a\u5bf9j<\/em> \u7684\u6240\u6709\u90bb\u5c45k<\/em>\uff08\u9664\u4e86i<\/em>\uff09\u8fdb\u884c\u6c42\u548c\u3002<\/p>\n\n\n\n

\uff085\uff09\\(f_{\\mathrm{int}}\\)\u662f\u5185\u90e8\u6d88\u606f\u6574\u5408\u51fd\u6570\u3002<\/p>\n\n\n\n

\uff086\uff09\\(m_{kj}^{(l)}\\)\u662f\u4ece\u539f\u5b50k<\/em>\u5230j<\/em> \u7684\u5f53\u524d\u6d88\u606f\u5d4c\u5165\u3002<\/p>\n\n\n\n

\uff087\uff09\\(e_{\\mathrm{RBF}}^{(ji)}\\)\u662f\u57fa\u4e8ej<\/em>\u548ci<\/em>\u4e4b\u95f4\u7684\u8ddd\u79bb\u7684\u5f84\u5411\u57fa\u51fd\u6570\u3002<\/p>\n\n\n\n

\uff088\uff09\\(a_{\\mathrm{SBF}}^{(kj,ji)}\\)\u662f\u89d2\u5411\u57fa\u51fd\u6570\uff0c\u5b83\u5305\u542b\u4e86\u7531\u539f\u5b50k<\/em>\uff0cj<\/em> \u548ci<\/em>\u5f62\u6210\u7684\u89d2\u5ea6\u4fe1\u606f\u3002<\/p>\n\n\n\n

\u5176\u4e2d\uff0c\u5f84\u5411\u57fa\u51fd\u6570\u548c\u89d2\u5411\u57fa\u51fd\u6570\u57281.2\u8282\u5747\u6709\u4ecb\u7ecd\u3002\\(f_{\\mathrm{update}}\\)\u7684\u5b9e\u73b0\u5f88\u7b80\u5355\uff0c\u901a\u8fc7\u4e00\u4e2a\u795e\u7ecf\u7f51\u7edc\u5c42\u8fdb\u884c\u6620\u5c04\u5b66\u4e60\u5373\u53ef\uff0c\\(f_{\\mathrm{int}}\\)\u7684\u5b9e\u73b0\u8be6\u89c1\u516c\u5f0f6\u30017\u30018\u3002<\/p>\n\n\n

\\(f_{\\mathrm{int}}(m,d_{ji},d_{kj},\\alpha_{(kj,ji)})=\\sum_n[\\sigma(\\boldsymbol{W}m+b)]_nf_{\\mathrm{filter}1,n}(d_{ji})f_{\\mathrm{filter}2,n}(d_{kj},\\alpha_{(kj,ji)})\\)<\/p>\n

\\(f_{\\text{filter}1,n}(d){=}(W_{\\mathrm{RBF}}e_{\\mathrm{RBF}}(d))_n\\)<\/p>\n

\\(f_{\\text{filter}2,n}(d,\\alpha){=}(W_{\\mathrm{SBF}}a_{\\mathrm{SBF}}(d,\\alpha))^TW_n\\)<\/p>\n\n\n

\uff081\uff09\u4ea4\u4e92\u51fd\u6570\\(f_{\\mathrm{int}}\\)<\/h4>\n\n\n\n

\u516c\u5f0f6\u63cf\u8ff0\u4e86DimeNet\u6a21\u578b\u4e2d\u7528\u4e8e\u5efa\u6a21\u539f\u5b50\u95f4\u4ea4\u4e92\u7684\u51fd\u6570\\(f_{\\mathrm{int}}\\)\u3002\u8fd9\u4e2a\u51fd\u6570\u7684\u4f5c\u7528\u662f\u6355\u6349\u539f\u5b50i<\/em>\u548cj<\/em>\uff0c\u4ee5\u53ca\u4e00\u4e2a\u7b2c\u4e09\u4e2a\u539f\u5b50k<\/em>\u4e4b\u95f4\u7684\u4ea4\u4e92\u3002\u5bf9\u4e8e\u6bcf\u5bf9\u539f\u5b50\u95f4\u7684\u4ea4\u4e92\uff0c\u8be5\u51fd\u6570\u8ba1\u7b97\u4e00\u4e2a\u8868\u793a\u4ea4\u4e92\u5f3a\u5ea6\u7684\u503c\uff0c\u8fd9\u4e2a\u503c\u901a\u8fc7\u5c06\u8ddd\u79bb\u548c\u89d2\u5ea6\u4fe1\u606f\u4e0e\u5b66\u4e60\u5f97\u5230\u7684\u6ee4\u6ce2\u5668\\(f_{\\text{filter}1,n}\\)\u548c\\(f_{\\text{filter}2,n}\\)\u7ed3\u5408\u6765\u5b9e\u73b0\u3002m<\/em>\u4ee3\u8868\u7684\u662f\u4e0e\u539f\u5b50i<\/em>\u76f8\u5173\u7684\u6d88\u606f\u6216\u7279\u5f81\uff0c \\(d_{ji}\\)\u548c\\(d_{kj}\\)\u4ee3\u8868\u539f\u5b50i<\/em>\u4e0ej<\/em>\u548c\u539f\u5b50k<\/em>\u4e0ej<\/em>\u4e4b\u95f4\u7684\u8ddd\u79bb\uff0c\\(\\alpha_{(kj,ji)}\\)\u662f\u7531\u8fd9\u4e09\u4e2a\u539f\u5b50\u5f62\u6210\u7684\u5939\u89d2\u3002<\/p>\n\n\n\n

\u51fd\u6570\u5185\u90e8\u7684\u6c42\u548c\u8868\u793a\u6a21\u578b\u5c06\u6240\u6709\u8fd9\u4e9b\u4e0d\u540c\u7684\u4ea4\u4e92\u52a0\u6743\u7d2f\u79ef\u8d77\u6765\uff0c\u5176\u4e2d\\(\\sigma\\ )\u662f\u6fc0\u6d3b\u51fd\u6570\uff0cW<\/em><\/strong>\u548cb<\/em>\u662f\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u5b66\u4e60\u7684\u53c2\u6570\uff0c\u8fd9\u5141\u8bb8\u6a21\u578b\u5728\u5904\u7406\u6bcf\u4e2a\u539f\u5b50\u5bf9\u65f6\u5bf9\u5176\u4ea4\u4e92\u5f3a\u5ea6\u8fdb\u884c\u7f16\u7801\u3002<\/p>\n\n\n\n

\uff082\uff09\u7b2c\u4e00\u4e2a\u6ee4\u6ce2\u5668\\(f_{\\text{filter}1,n}\\)<\/h4>\n\n\n\n

\u8be5\u6ee4\u6ce2\u5668\u4e13\u6ce8\u4e8e\u5904\u7406\u539f\u5b50\u95f4\u7684\u8ddd\u79bb\u4fe1\u606f\u3002\\(W_{\\mathrm{RBF}}\\)\u662f\u6743\u91cd\u77e9\u9635\uff0c\\(e_{\\mathrm{RBF}}(d)\\)\u662f\u5f84\u5411\u57fa\u51fd\u6570\u7684\u8868\u793a\uff0c\u5b83\u80fd\u5c06\u539f\u5b50\u95f4\u7684\u8ddd\u79bb\u8f6c\u6362\u6210\u4e00\u7ec4\u80fd\u53cd\u6620\u8ddd\u79bb\u53d8\u5316\u7684\u7279\u5f81\u3002\u516c\u5f0f7\u8868\u793a\uff1a\u901a\u8fc7\u5c06\u8ddd\u79bbd<\/em>\u7ecf\u8fc7\\(e_{\\mathrm{RBF}}\\)\u6620\u5c04\u548c\u4e0e\\(W_{\\mathrm{RBF}}\\)\u76f8\u4e58\uff0c\u6211\u4eec\u5f97\u5230\u4e86\u53cd\u6620\u8be5\u8ddd\u79bb\u4fe1\u606f\u7684\u7279\u5f81\u3002<\/p>\n\n\n\n

\uff083)\u7b2c\u4e00\u4e2a\u6ee4\u6ce2\u5668\\(f_{\\text{filter}2,n}\\)<\/h4>\n\n\n\n

\u4e0e\u516c\u5f0f7\u4e0d\u540c\u7684\u662f\uff0c\u516c\u5f0f8\u8003\u8651\u7684\u4e0d\u4ec5\u4ec5\u662f\u8ddd\u79bb\uff0c\u8fd8\u6709\u89d2\u5ea6\u4fe1\u606f\u3002\\(W_{\\mathrm{SBF}}\\)\u548c\\(W_{\\mathrm{n}}\\)\u4e5f\u662f\u53ef\u5b66\u4e60\u7684\u6743\u91cd\u77e9\u9635\u3002\\(a_{\\mathrm{SBF}}(d,\\alpha)\\)\u662f\u4e8c\u7ef4\u7403\u5f62Fourier-Bessel\u8868\u793a\uff0c\u5b83\u5c06\u8ddd\u79bbd<\/em>\u548c\u89d2\u5ea6\\(\\alpha)\\)\u7ed3\u5408\u8d77\u6765\u8f6c\u6362\u4e3a\u4e00\u7ec4\u7279\u5f81\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f\u53ef\u4ee5\u6355\u6349\u539f\u5b50\u95f4\u66f4\u590d\u6742\u7684\u7a7a\u95f4\u5173\u7cfb\uff0c\u4f8b\u5982\u539f\u5b50\u5982\u4f55\u56f4\u7ed5\u4e2d\u5fc3\u539f\u5b50\u6392\u5217\u3002<\/p>\n\n\n\n

\u516c\u5f0f8\u8bf4\u660e\u4e86\u5982\u4f55\u7ed3\u5408\u8fd9\u4e9b\u7279\u5f81\u548c\u6743\u91cd\u6765\u8868\u793a\u539f\u5b50\u95f4\u7684\u4ea4\u4e92\u6a21\u5f0f\uff0c\u5f53\u539f\u5b50\u7684\u4f4d\u7f6e\u6216\u5b83\u4eec\u4e4b\u95f4\u7684\u89d2\u5ea6\u53d1\u751f\u53d8\u5316\u65f6\uff0c\u4ea4\u4e92\u7279\u5f81\u4e5f\u4f1a\u53d1\u751f\u76f8\u5e94\u53d8\u5316\u3002\u8fd9\u662fDimeNet\u6a21\u578b\u80fd\u591f\u7406\u89e3\u5206\u5b50\u51e0\u4f55\u7ed3\u6784\u5e76\u51c6\u786e\u9884\u6d4b\u5176\u6027\u8d28\u7684\u5173\u952e\u3002<\/p>\n\n\n\n

\u901a\u8fc7\u4e0a\u8ff0\u4e24\u4e2a\u6ee4\u6ce2\u5668\uff0cDimeNet\u4e0d\u4ec5\u6355\u6349\u4e86\u5206\u5b50\u5185\u5404\u539f\u5b50\u95f4\u7684\u8ddd\u79bb\u4fe1\u606f\uff0c\u8fd8\u6355\u6349\u4e86\u539f\u5b50\u4e4b\u95f4\u590d\u6742\u7684\u89d2\u5ea6\u4fe1\u606f\uff0c\u8fd9\u6837\u7684\u5904\u7406\u65b9\u5f0f\u662f\u6a21\u4eff\u5206\u5b50\u5185\u90e8\u5982\u4f55\u5b9e\u9645\u53d1\u751f\u76f8\u4e92\u4f5c\u7528\u7684\u3002DimeNet\u901a\u8fc7\u8fd9\u4e9b\u7cbe\u5fc3\u8bbe\u8ba1\u7684\u6ee4\u6ce2\u5668\u80fd\u591f\u6355\u6349\u590d\u6742\u7684\u7a7a\u95f4\u4f9d\u8d56\u6027\u3002<\/p>\n\n\n\n

1.4 \u6a21\u578b\u7ed3\u6784<\/h3>\n\n\n\n

DimeNet\u6a21\u578b\u7684\u6982\u89c8\u5982\u56fe5\u6240\u793a\uff0c\u5b83\u5c55\u793a\u4e86\u5982\u4f55\u5904\u7406\u539f\u5b50\u95f4\u7684\u8ddd\u79bb\u548c\u89d2\u5ea6\u4fe1\u606f\u4ee5\u751f\u6210\u5206\u5b50\u7684\u6700\u7ec8\u5d4c\u5165\u8868\u793at<\/em><\/strong>\uff0c\u8be6\u7ec6\u7684\u6b65\u9aa4\u5982\u4e0b\uff1a<\/p>\n\n\n

\n
\"\"
\u56fe5 DimeNet\u6a21\u578b\u7ed3\u6784<\/figcaption><\/figure><\/div>\n\n\n

1.\u8ddd\u79bb\u548c\u89d2\u5ea6\u8ba1\u7b97\uff1a<\/p>\n\n\n\n

\uff081\uff09\u5bf9\u4e8e\u4efb\u610f\u4e00\u5bf9\u539f\u5b50i<\/em>\u548cj<\/em>\uff0c\u8ba1\u7b97\u5b83\u4eec\u4e4b\u95f4\u7684\u6b27\u6c0f\u8ddd\u79bb\\(d_{ij}\\)\u3002<\/p>\n\n\n\n

\uff082\uff09\u5bf9\u4e8e\u6d89\u53ca\u5230\u4e09\u4e2a\u539f\u5b50i<\/em> \u3001j<\/em>\u548ck<\/em>\u7684\u89d2\u5ea6\uff0c\u8ba1\u7b97\u5b83\u4eec\u5f62\u6210\u7684\u89d2\\(\\alpha(k,j,i)\\)\u3002<\/p>\n\n\n\n

2.\u5f84\u5411\u57fa\u51fd\u6570\uff08RBF\uff09\u548c\u89d2\u5411\u57fa\u51fd\u6570\uff08SBF\uff09\uff1a<\/p>\n\n\n\n

\uff081\uff09\u4f7f\u7528\u5f84\u5411\u57fa\u51fd\u6570\\(e_{\\mathrm{RBF}}^{(j,i)}\\)\u8f6c\u6362\u8ddd\u79bb\\(\\boldsymbol{d}_{\\hat{y}}\\)\uff0c\u8fd9\u4e9b\u51fd\u6570\u5c06\u8ddd\u79bb\u6807\u91cf\u8f6c\u6362\u4e3a\u9ad8\u7ef4\u7684\u7279\u5f81\u5411\u91cf\uff0c\u4ee5\u4fbf\u795e\u7ecf\u7f51\u7edc\u80fd\u591f\u66f4\u597d\u5730\u5904\u7406\u3002<\/p>\n\n\n\n

\uff082\uff09\u4f7f\u7528\u89d2\u5411\u57fa\u51fd\u6570\\(a_{\\mathrm{SBF}}^{(k,j,i)}\\)\u8f6c\u6362\u89d2\u5ea6\\(\\alpha(k,j,i)\\)\uff0c\u8fd9\u4e9b\u51fd\u6570\u5e2e\u52a9\u7f51\u7edc\u6355\u6349\u539f\u5b50\u4e09\u5143\u7ec4\u4e4b\u95f4\u7684\u7a7a\u95f4\u5173\u7cfb\u3002<\/p>\n\n\n\n

3. Embedding \u5c42\uff1a<\/p>\n\n\n\n

\u521d\u59cb\u539f\u5b50\u5d4c\u5165z<\/strong><\/em>\uff08\u901a\u5e38\u662f\u901a\u8fc7\u539f\u5b50\u7c7b\u578b\u6216\u5176\u4ed6\u539f\u5b50\u5c5e\u6027\u83b7\u5f97\u7684\u7279\u5f81\u5411\u91cf\uff09\u4e0e\u7531RBF\u548cSBF\u5904\u7406\u8fc7\u7684\u8ddd\u79bb\u548c\u89d2\u5ea6\u4fe1\u606f\u7ed3\u5408\uff0c\u8fdb\u5165Embedding\u5c42\u3002\u8fd9\u4e2a\u5c42\u5c06\u521d\u59cb\u7684\u539f\u5b50\u7279\u5f81\u4e0e\u7ed3\u6784\u4fe1\u606f\u6574\u5408\u5728\u4e00\u8d77\uff0c\u4ea7\u751f\u66f4\u4e30\u5bcc\u7684\u7279\u5f81\u8868\u793a\u3002<\/p>\n\n\n\n

4. Interaction \u5c42\uff1a<\/p>\n\n\n\n

Embedding\u5c42\u7684\u8f93\u51fa\u7136\u540e\u901a\u8fc7\u4e00\u7cfb\u5217\u7684Interaction\u5c42\u3002\u8fd9\u4e9b\u5c42\u901a\u5e38\u5305\u542b\u4e86\u66f4\u6df1\u5c42\u6b21\u7684\u795e\u7ecf\u7f51\u7edc\u7ed3\u6784\uff0c\u7528\u4e8e\u5b66\u4e60\u539f\u5b50\u4e4b\u95f4\u590d\u6742\u7684\u76f8\u4e92\u4f5c\u7528\u3002<\/p>\n\n\n\n

5. \u8f93\u51fa\uff1a<\/p>\n\n\n\n

\u7ecf\u8fc7\u4e00\u7cfb\u5217\u7684Interaction\u5c42\u540e\uff0c\u6a21\u578b\u4ea7\u751f\u6700\u7ec8\u7684\u8f93\u51fa t<\/em><\/strong>\uff0c\u5b83\u662f\u4e00\u4e2a\u7efc\u5408\u4e86\u6240\u6709\u8f93\u5165\u4fe1\u606f\u7684\u5411\u91cf\uff0c\u8fd9\u4e2a\u5411\u91cf\u53ef\u4ee5\u7528\u4e8e\u9884\u6d4b\u5206\u5b50\u7684\u6027\u8d28\u6216\u4f5c\u4e3a\u5176\u4ed6\u4e0b\u6e38\u4efb\u52a1\u7684\u8f93\u5165\u3002<\/p>\n\n\n\n

\u4e0b\u9762\uff0c\u4f9d\u6b21\u89e3\u91ca\u6bcf\u4e2a\u5b50\u6a21\u5757\u7684\u8be6\u7ec6\u7ed3\u6784\u548c\u8fd0\u7b97\u8fc7\u7a0b\uff1a<\/p>\n\n\n\n

\u56fe\u795e\u7ecf\u7f51\u7edc\u4e2d\u7684\u5d4c\u5165\uff08Embedding\uff09\u90e8\u5206\u5982\u56fe6\u6240\u793a\uff0c\u5d4c\u5165\u6b65\u9aa4\u7684\u4f5c\u7528\u662f\u5c06\u539f\u5b50\u7684\u7c7b\u578b\u4fe1\u606f\u8f6c\u5316\u4e3a\u4e30\u5bcc\u7684\u7279\u5f81\u8868\u793a\uff0c\u4e3a\u8fdb\u4e00\u6b65\u7684\u6d88\u606f\u4f20\u9012\u548c\u76f8\u4e92\u4f5c\u7528\u5efa\u6a21\u5960\u5b9a\u57fa\u7840\uff0c\u56fe6\u5177\u4f53\u63cf\u8ff0\u4e86\u539f\u5b50\u95f4\u6d88\u606f\u7684\u5d4c\u5165\u751f\u6210\u6d41\u7a0b\u3002<\/p>\n\n\n

\n
\"\"
\u56fe6 \u5d4c\u5165\u90e8\u5206<\/figcaption><\/figure><\/div>\n\n\n

\\(z_{i}\\)\u548c\\(z_{j}\\)\u662f\u4e24\u4e2a\u539f\u5b50\u7684\u521d\u59cb\u5316\u7279\u5f81\u5411\u91cf\uff0c\u5b83\u4eec\u53ef\u4ee5\u662f\u6839\u636e\u539f\u5b50\u7684\u79cd\u7c7b\u6216\u5176\u4ed6\u5316\u5b66\u6027\u8d28\u968f\u673a\u521d\u59cb\u5316\u7684\uff0c\u5e76\u901a\u8fc7\u540e\u7eed\u7684\u5b66\u4e60\u5f97\u4ee5\u4f18\u5316\u7684\u8868\u793a\u3002\u8fd9\u4e9b\u5411\u91cf\u643a\u5e26\u4e86\u539f\u5b50\u7684\u521d\u59cb\u8eab\u4efd\u4fe1\u606f\uff0c\u662f\u5206\u5b50\u7279\u6027\u9884\u6d4b\u4efb\u52a1\u7684\u8d77\u70b9\u3002<\/p>\n\n\n\n

\u5bf9\u4e8e\u539f\u5b50j<\/em>\u5230\u539f\u5b50i<\/em>\u7684\u6bcf\u4e00\u5bf9\uff0c\u5b83\u4eec\u7684\u5d4c\u5165\\(h_j^{(0)}\\)\u548c\\(h_i^{(0)}\\)\u88ab\u5e76\u5217\u653e\u7f6e\uff0c\u4e0e\u8f6c\u6362\u540e\u7684\u8ddd\u79bb\u7279\u5f81\\(e_{\\mathrm{RBF}}^{(j,i)}\\)\u62fc\u63a5\u5728\u4e00\u8d77\uff0c\u5f62\u6210\u4e86\u4e00\u4e2a\u7efc\u5408\u7279\u5f81\u5411\u91cf\u3002\u7136\u540e\uff0c\u8fd9\u4e2a\u7efc\u5408\u7279\u5f81\u5411\u91cf\u901a\u8fc7\u4e00\u4e2a\u5e26\u6743\u91cd\u77e9\u9635W<\/em><\/strong>\u548c\u504f\u7f6eb<\/em>\u7684\u7ebf\u6027\u53d8\u6362\u5c42\uff0c\u6743\u91cd\u548c\u504f\u7f6e\u90fd\u662f\u53ef\u901a\u8fc7\u8bad\u7ec3\u5f97\u5230\u7684\u53c2\u6570\u3002\u901a\u8fc7\u8fd9\u4e2a\u53d8\u6362\uff0c\u539f\u5b50\u7c7b\u578b\u7684\u79bb\u6563\u7279\u5f81\u4e0e\u539f\u5b50\u95f4\u8ddd\u79bb\u7684\u8fde\u7eed\u7279\u5f81\u8054\u5408\u8d77\u6765\uff0c\u6620\u5c04\u5230\u4e86\u4e00\u4e2a\u65b0\u7684\u7279\u5f81\u7a7a\u95f4\u3002\u6700\u540e\uff0c\u6fc0\u6d3b\u51fd\u6570\\(\\sigma \\)\u88ab\u5e94\u7528\u4e8e\u7ebf\u6027\u5c42\u7684\u8f93\u51fa\uff0c\u5f15\u5165\u4e86\u975e\u7ebf\u6027\u56e0\u7d20\uff0c\u8fd9\u662f\u795e\u7ecf\u7f51\u7edc\u80fd\u591f\u6355\u6349\u590d\u6742\u6a21\u5f0f\u6240\u5fc5\u9700\u7684\u3002\u8fd9\u79cd\u975e\u7ebf\u6027\u6620\u5c04\u4f7f\u5f97\u4ece\u4e00\u4e2a\u539f\u5b50\u5230\u53e6\u4e00\u4e2a\u539f\u5b50\u7684\u6d88\u606f\\(m_{ji}^{(1)}\\)\u53d8\u5f97\u66f4\u52a0\u4e30\u5bcc\uff0c\u80fd\u591f\u5305\u542b\u539f\u5b50\u95f4\u66f4\u5fae\u5999\u7684\u76f8\u4e92\u4f5c\u7528\u4fe1\u606f\u3002<\/p>\n\n\n\n

\u8d1f\u8d23\u4fe1\u606f\u4f20\u9012\u7684\u4ea4\u4e92\u5c42\uff08Interaction\uff09\u7ed3\u6784\u5982\u56fe7\u6240\u793a\u3002<\/p>\n\n\n

\n
\"\"
\u56fe7 \u4ea4\u4e92\u5c42\u7ed3\u6784<\/figcaption><\/figure><\/div>\n\n\n

\u5148\u89e3\u91ca\u56fe7\u4e2d\u5de6\u4e0a\u89d2\u7684\u5b50\u90e8\u5206\uff0c\u5373\u65b9\u5411\u6027\u4fe1\u606f\u4f20\u9012\u90e8\u5206\uff08Directional Message Passing\uff09\u3002\u8f93\u5165\\(e_{\\mathrm{RBF}}^{(j,i)}\\)\u4ee3\u8868\u4f7f\u7528\u5f84\u5411\u57fa\u51fd\u6570\uff08RBF\uff09\u7f16\u7801\u7684\u539f\u5b50\u5bf9\u4e4b\u95f4\u7684\u8ddd\u79bb\u4fe1\u606f\uff0c\u6355\u6349\u539f\u5b50\u4e4b\u95f4\u7684\u8ddd\u79bb\u3002\u800c\\(a_{\\mathrm{SBF}}^{(k,j,i)}\\)\u5219\u7f16\u7801\u4e86\u4e09\u4e2a\u539f\u5b50\uff08i<\/em> \u3001j<\/em>\u548ck<\/em>\uff09\u4e4b\u95f4\u89d2\u5ea6\u4fe1\u606f\u7684\u7403\u5f62\u57fa\u51fd\u6570\uff08SBF\uff09\uff0c\u5b83\u6355\u6349\u5206\u5b50\u5185\u90e8\u7684\u4e09\u7ef4\u7a7a\u95f4\u5173\u7cfb\u3002\\(m_{kj}^{(l-1)}\\)\u662f\u4ece\u539f\u5b50k<\/em>\u5230\u539f\u5b50j<\/em>\u7684\u6d88\u606f\uff0c\u662f\u7f51\u7edc\u524d\u4e00\u5c42\uff08l<\/em>-1 \u5c42\uff09\u8ba1\u7b97\u5f97\u5230\u7684\u3002\u7ed3\u54081.3\u8282\u8bb2\u7684\u6d88\u606f\u4f20\u9012\uff0c i<\/em>\u662f\u5f53\u524d\u8981\u66f4\u65b0\u4fe1\u606f\u7684\u8282\u70b9\uff0cj<\/em>\u662fi<\/em> \u7684\u90bb\u5c45\uff0ck<\/em>\u662fj<\/em>\u7684\u90bb\u5c45\uff08 \u90bb\u5c45\u7684\u90bb\u5c45\uff09\u3002<\/p>\n\n\n\n

\u5f84\u5411\u57fa\u51fd\u6570\\( e_{\\mathrm{RBF}}^{(j,i)}\\)\u548c\u7403\u5f62\u57fa\u51fd\u6570(a_{\\mathrm{SBF}}^{(k,j,i)}\\)\u7684\u7f16\u7801\u7ed3\u679c\u5c06\u5206\u522b\u7ecf\u8fc7\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff0c\u8c03\u6574\u5176\u8868\u793a\uff0c\u4ee5\u4fbf\u66f4\u597d\u7684\u4e0e\u6d88\u606f\u7ed3\u5408, \u8fd9\u91cc\u5c06\u4e24\u4e2a\u7ebf\u6027\u53d8\u5316\u540e\u7684\u7ed3\u679c\u79f0\u4e3atmp1\u548ctmp2\u3002<\/a>\\(m_{kj}^{(l-1)}\\)\u5305\u542b\u66f4\u65b0\u8282\u70b9i<\/em>\u7684\u90bb\u5c45j<\/em>\u548c\u90bb\u5c45\u7684\u90bb\u5c45k<\/em>\u4e4b\u95f4\u7684\u89d2\u5ea6\u4fe1\u606f\u7684\uff08\u8be6\u89c11.3\u8282\uff09\uff0c\u5176\u7ed3\u679c\u5c06\u8fdb\u4e00\u6b65\u901a\u8fc7\u4e00\u4e2a\u5e26\u6fc0\u6d3b\u51fd\u6570\u7684\u795e\u7ecf\u7f51\u7edc\u5c42\u505a\u8fdb\u4e00\u6b65\u6620\u5c04\uff0c\u7ed3\u679c\u79f0\u4e3atmp3\u3002tmp3\u5c06\u4e0e\u5f84\u5411\u57fa\u51fd\u6570\\( e_{\\mathrm{RBF}}^{(j,i)}\\)\u7ebf\u6027\u53d8\u5316\u540e\u7684\u7ed3\u679ctmp1\uff08\u5305\u542b\u8ddd\u79bb\u4fe1\u606f\uff09\u8fdb\u884c\u5143\u7d20\u95f4\u7684\u4e58\u6cd5\u64cd\u4f5c\uff0c\u5f97\u5230\u7684\u7ed3\u679c\u79f0\u4e3atmp4\uff0ctmp4\u80fd\u591f\u6355\u6349\u4e24\u79cd\u7c7b\u578b\u7684\u7279\u5f81\u76f8\u7ed3\u5408\u540e\u7684\u590d\u6742\u6a21\u5f0f\u3002<\/p>\n\n\n\n

\u63a5\u4e0b\u6765\\(\\square^T\\mathbf{W}\\square \\)\u662f\u4e00\u4e2a\u53cc\u7ebf\u6027\u5c42\uff08Bilinear Layer\uff09\uff0c\u5b83\u901a\u8fc7\u5bf9tmp2\u548ctmp4\u8fdb\u884c\u72ec\u7acb\u7684\u7ebf\u6027\u53d8\u6362\u3002tmp2\u8868\u5f81\u7684\u662f\u57fa\u4e8e\u7403\u8c10\u51fd\u6570\u5bf9i<\/em> \u3001j<\/em>\u548ck<\/em>\u4e4b\u95f4\u89d2\u5ea6\u4fe1\u606f\u7684\u7279\u5f81\u7f16\u7801\uff0ctmp4\u8868\u5f81\u7684\u662f\u878d\u5408\u4e86\u8ddd\u79bb\u4fe1\u606f\u548c\u7ecf\u8fc7\u975e\u7ebf\u6027\u53d8\u6362\u7684\u90bb\u5c45\u4fe1\u606f\u7684\u7ed3\u679c\u3002\u5b83\u4eec\u4e4b\u95f4\u7684\u4ea4\u4e92\u4e0d\u4ec5\u4ec5\u662f\u7b80\u5355\u7684\u7279\u5f81\u878d\u5408\uff0c\u800c\u662f\u4e00\u79cd\u80fd\u591f\u8868\u793a\u7279\u5f81\u5bf9\u4e4b\u95f4\u590d\u6742\u5173\u7cfb\u7684\u65b9\u6cd5\u3002\u8fd9\u79cd\u9ad8\u9636\u7684\u7279\u5f81\u4ea4\u4e92\u5728\u7406\u89e3\u539f\u5b50\u95f4\u5982\u4f55\u76f8\u4e92\u4f5c\u7528\u65f6\u5c24\u4e3a\u91cd\u8981\uff0c\u56e0\u4e3a\u5b83\u4eec\u901a\u5e38\u4ee5\u590d\u6742\u7684\u65b9\u5f0f\u76f8\u4e92\u4f9d\u8d56\u3002\u6700\u540e\u2211k<\/sub><\/em>\u8868\u793a\u8981\u7efc\u5408\u8003\u8651\u6240\u6709\u90bb\u5c45\u7684\u90bb\u5c45\u4fe1\u606f\uff0c\u5373\u4e24\u8df3\uff082-hop\uff09\u7684\u6d88\u606f\u4f20\u9012\u673a\u5236\uff0c\u5f97\u5230\u7684\u4e2d\u95f4\u7ed3\u679c\u79f0\u4e3atmp5\u3002<\/p>\n\n\n\n

\u65b9\u5411\u6027\u4fe1\u606f\u4f20\u9012\u90e8\u5206\uff08Directional Message Passing\uff09\u4ee5\u5916\u7684\u6a21\u578b\u7ed3\u6784\u903b\u8f91\u5c31\u76f8\u5bf9\u6bd4\u8f83\u7b80\u5355\u6e05\u695a\u3002\u9996\u5148\uff0c\\(m_{ji}^{(l-1)}\\)\u5305\u542b\u66f4\u65b0\u8282\u70b9i<\/em> \u548c\u5176\u90bb\u5c45j<\/em>\u4e4b\u95f4\u7684\u89d2\u5ea6\u4fe1\u606f\uff0c\u5904\u7406\u65b9\u5f0f\u4e0e\\(m_{kj}^{(l-1)}\\)\u4e00\u6837\uff0c\u901a\u8fc7\u4e00\u4e2a\u5e26\u6fc0\u6d3b\u51fd\u6570\u7684\u795e\u7ecf\u7f51\u7edc\u5c42\u505a\u8fdb\u4e00\u6b65\u6620\u5c04\u3002\u6620\u5c04\u540e\u5f97\u5230\u7684\u7ed3\u679c\u4e0etmp5\u76f8\u52a0\uff0c\u8868\u5f81\u7efc\u5408\u8003\u8651\u66f4\u65b0\u8282\u70b9i<\/em> \u7684\u90bb\u57df\u4fe1\u606f\u548c\u90bb\u57df\u7684\u90bb\u57df\u4fe1\u606f\uff0c\u6700\u540e\u518d\u7ecf\u8fc7\u4e00\u4e9b\u795e\u7ecf\u7f51\u7edc\u5c42\u505a\u8fdb\u4e00\u6b65\u7684\u5b66\u4e60\u6620\u5c04\uff0c\u5f97\u5230\u7ed3\u679c\\(m_{ji}^{(l)}\\)\u3002\u5728\u8fd9\u4e9b\u795e\u7ecf\u7f51\u7edc\u5c42\u7684\u5b9e\u73b0\u4e2d\uff0c\u4e5f\u52a0\u5165\u4e86\u6b8b\u5dee\u8fde\u63a5\u7684\u5b9e\u73b0\uff0c\u7528\u4e8e\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u66f4\u597d\u7684\u68af\u5ea6\u4fe1\u606f\u4f20\u9012\uff0c\u5e2e\u52a9\u6a21\u578b\u6536\u655b\uff0c\u6b8b\u5dee\u6a21\u5757\u7ed3\u6784\u5982\u56fe8\u6240\u793a\u3002<\/p>\n\n\n

\n
\"\"
\u56fe8 \u6b8b\u5dee\u7ed3\u6784<\/figcaption><\/figure><\/div>\n\n\n

\u6700\u540e\uff0c\u4ea4\u4e92\u5c42\u7684\u7ed3\u679c\\(m_{ji}^{(l)}\\)\u4e0d\u4ec5\u4f5c\u4e3a\u4e0b\u4e00\u4e2a\u4ea4\u4e92\u6a21\u5757\u7684\u8f93\u5165\u4fe1\u606f\uff0c\u4e5f\u4e0e\\( e_{\\mathrm{RBF}}^{(j,i)}\\)\u4e00\u8d77\u9001\u5165\u8be5\u4ea4\u4e92\u6a21\u5757\u7684\u9876\u5c42\uff08Output\u5c42\uff09,\u5e76\u5c06\u9876\u5c42\u7684\u8ba1\u7b97\u7ed3\u679c\u4f5c\u4e3a\u5f53\u524d\u4ea4\u4e92\u6a21\u5757\u7684\u8f93\u51fa\\(t_i^{(l)}\\)\uff0c\u9876\u5c42\u7684\u6a21\u578b\u7ed3\u6784\u8bbe\u8ba1\u5982\u56fe9\u6240\u793a\u3002<\/p>\n\n\n

\n
\"\"
\u56fe9 \u9876\u5c42\u6a21\u578b\u7ed3\u6784<\/figcaption><\/figure><\/div>\n\n\n

\u5982\u56fe8\u6240\u793a\uff0c\u5728\u4ea4\u4e92\u6a21\u5757\u7684\u9876\u5c42\u8bbe\u8ba1\u4e2d\uff0c\u5305\u542b\u8282\u70b9\u5bf9\u4e4b\u95f4\u8ddd\u79bb\u4fe1\u606f\u7684\\(e_{\\mathrm{RBF}}^{(ji)}\\)\u9996\u5148\u7ecf\u8fc7\u7ebf\u6027\u6620\u5c04\uff0c\u7136\u540e\u4e0e\u5305\u542b\u8282\u70b9\u5bf9\u89d2\u5ea6\u4fe1\u606f\u7684\\(m_{ji}^{(l)}\\)\u8fdb\u884c\u76f8\u4e58\u6c42\u548c\uff0c\u5b8c\u6210\u4fe1\u606f\u7684\u4ea4\u4e92\u4e0e\u878d\u5408\uff0c\u6700\u540e\u518d\u7ecf\u8fc7\u4e00\u4e9b\u795e\u7ecf\u7f51\u7edc\u5c42\u7684\u5904\u7406\u8f93\u5165\u6700\u7ec8\u7ed3\u679c\u5373\u53ef\u3002<\/p>\n\n\n\n

\u56de\u987e\u603b\u7ed3\u6574\u4e2a\u6a21\u578b\u8bbe\u8ba1\uff0c\u5982\u56fe5\u6240\u793a\uff0c\u9488\u5bf9\u5b50\u6a21\u5757Embedding\u548cInteraction\u4e0a\u8ff0\u5df2\u7ecf\u505a\u4e86\u8be6\u7ec6\u7684\u4ecb\u7ecd\u5206\u6790\u3002DimeNet\u6a21\u578b\u4f1a\u53d6\u5230\u6bcf\u4e2a\u4ea4\u4e92\u6a21\u5757\u7684\u7ed3\u679c\\(t_{i}\\)\u8fdb\u884c\u6c42\u548c\u5f97\u5230\u6700\u7ec8\u7684\u7ed3\u679ct<\/strong><\/em>\u3002\u800c\u4e14\uff0c\u8ddd\u79bb\u4fe1\u606fRBF\u548c\u89d2\u5ea6\u4fe1\u606fSBF\u662f\u8d2f\u7a7f\u6574\u4e2aDimeNet\u6a21\u578b\u7684\uff0c\u9001\u5165\u4e86\u6bcf\u4e2a\u4ea4\u4e92\u6a21\u578b\uff0c\u4ee5\u5e2e\u52a9\u66f4\u597d\u7684\u8fdb\u884c\u8282\u70b9\u95f4\u7684\u4fe1\u606f\u4f20\u9012\uff0c\u4e0d\u4ec5\u4ec5\u662f\u4f5c\u4e3a\u6570\u636e\u7684\u9884\u5904\u7406\u65b9\u5f0f\u3002\u8fd9\u6837\u505a\u7684\u597d\u5904\u662f\u4e3a\u4e86\u4fdd\u6301\u7269\u7406\u6027\u8d28\u7684\u4e00\u81f4\u6027\uff0c\u5373\u5206\u5b50\u7684\u7269\u7406\u5c5e\u6027\u662f\u7531\u539f\u5b50\u95f4\u8ddd\u79bb\u548c\u89d2\u5ea6\u786e\u5b9a\u7684\u3002\u5982\u679c\u8ddd\u79bb\u548c\u89d2\u5ea6\u4fe1\u606f\u4ec5\u5728\u7b2c\u4e00\u5c42\u63d0\u4f9b\uff0c\u90a3\u4e48\u968f\u7740\u4fe1\u606f\u5728\u7f51\u7edc\u4e2d\u7684\u4f20\u9012\uff0c\u8fd9\u4e9b\u7269\u7406\u5c5e\u6027\u7684\u76f4\u63a5\u5f71\u54cd\u53ef\u80fd\u4f1a\u9010\u6e10\u51cf\u5f31\uff0c\u4ece\u800c\u5f71\u54cd\u6a21\u578b\u7684\u9884\u6d4b\u80fd\u529b\u3002\u5728\u6bcf\u4e2a\u4ea4\u4e92\u6a21\u5757\u4e2d\u91cd\u590d\u8fd9\u4e9b\u4fe1\u606f\u786e\u4fdd\u4e86\u7269\u7406\u4e00\u81f4\u6027\u3002\u4f7f\u5176\u53ef\u4ee5\u66f4\u5168\u9762\u5730\u6355\u6349\u548c\u5b66\u4e60\u5206\u5b50\u5185\u90e8\u590d\u6742\u7684\u4e09\u7ef4\u7a7a\u95f4\u7ed3\u6784\u548c\u76f8\u4e92\u4f5c\u7528\uff0c\u4ece\u800c\u63d0\u9ad8\u4e86\u6a21\u578b\u9884\u6d4b\u5316\u5b66\u5c5e\u6027\u548c\u53cd\u5e94\u7684\u51c6\u786e\u6027\u3002<\/p>\n\n\n\n

1.5 DimeNet++<\/h4>\n\n\n\n

DimeNet++\u662fDimeNet\u7684\u6539\u8fdb\u7248\u672c\uff0c\u5f15\u5165\u4e86\u4e00\u79cd\u66f4\u9ad8\u6548\u7684\u6d88\u606f\u4f20\u9012\u673a\u5236\u3002\u5177\u4f53\u6765\u8bf4\uff0c DimeNet \u901a\u8fc7\u5229\u7528\u65b9\u5411\u6027\u6d88\u606f\u4f20\u9012\u6765\u8003\u8651\u539f\u5b50\u5bf9\u4e4b\u95f4\u7684\u89d2\u5ea6\u4f9d\u8d56\u6027\uff0c\u8fd9\u4f7f\u5f97\u8ba1\u7b97\u53d8\u5f97\u76f8\u5bf9\u8f83\u6162\u548c\u590d\u6742\u3002DimeNet++\u5bf9\u6b64\u8fdb\u884c\u4e86\u6539\u8fdb\uff0c\u901a\u8fc7\u91cd\u65b0\u8bbe\u8ba1\u6d88\u606f\u4f20\u9012\u51fd\u6570\u6765\u51cf\u5c11\u8ba1\u7b97\u91cf\uff0c\u540c\u65f6\u4fdd\u6301\u4e86\u6a21\u578b\u7684\u8868\u73b0\u529b\u3002\u6b64\u5916\uff0cDimeNet++\u5bf9\u6a21\u578b\u7684\u6574\u4f53\u7ed3\u6784\u8fdb\u884c\u4e86\u7b80\u5316\uff0c\u964d\u4f4e\u4e86\u6a21\u578b\u590d\u6742\u5ea6\uff0c\u4f7f\u5f97\u6a21\u578b\u66f4\u6613\u4e8e\u8bad\u7ec3\uff0c\u540c\u65f6\u51cf\u5c11\u4e86\u8fc7\u62df\u5408\u7684\u98ce\u9669\uff0c\u4ee5\u63d0\u9ad8\u6a21\u578b\u7684\u6cdb\u5316\u80fd\u529b\u548c\u9884\u6d4b\u6027\u80fd\u3002<\/p>\n\n\n\n

\u76f8\u6bd4\u4e8e DimeNet\uff0cDimeNet++\u5728\u591a\u4e2a\u57fa\u51c6\u6570\u636e\u96c6\u4e0a\u663e\u793a\u51fa\u4e86\u66f4\u597d\u7684\u6027\u80fd\uff0c\u5305\u62ec\u66f4\u9ad8\u7684\u51c6\u786e\u7387\u548c\u66f4\u4f4e\u7684\u8bef\u5dee\u3002\u7b80\u800c\u8a00\u4e4b\uff0cDimeNet++\u5728\u4fdd\u6301DimeNet\u6838\u5fc3\u601d\u60f3\u7684\u57fa\u7840\u4e0a\uff0c\u901a\u8fc7\u5bf9\u6a21\u578b\u7ed3\u6784\u548c\u8ba1\u7b97\u8fc7\u7a0b\u7684\u4f18\u5316\uff0c\u5b9e\u73b0\u4e86\u66f4\u9ad8\u7684\u6548\u7387\u548c\u66f4\u597d\u7684\u6027\u80fd\uff0c\u4f7f\u5176\u6210\u4e3a\u5206\u5b50\u56fe\u8868\u793a\u548c\u5206\u5b50\u6027\u8d28\u9884\u6d4b\u9886\u57df\u7684\u4e00\u4e2a\u5f3a\u5927\u5de5\u5177\u3002<\/p>\n\n\n\n

DimeNet++\u7684\u4f18\u5316\u4e3b\u8981\u4f53\u73b0\u5728\u4ea4\u4e92\u6a21\u5757\uff08Interaction Block\uff09\u4e2d\uff0c\u56de\u987e\u4e00\u4e0bDimeNet\u7684\u6838\u5fc3\u90e8\u4ef6\uff1a\u65b9\u5411\u6027\u6d88\u606f\u4f20\u9012\u6a21\u5757\uff0c\u5982\u56fe10\u6240\u793a\u3002<\/p>\n\n\n

\n
\"\"
\u56fe10 \u65b9\u5411\u6027\u6d88\u606f\u4f20\u9012<\/figcaption><\/figure><\/div>\n\n\n

\u5176\u4e2d\u7684\u5173\u952e\u662f\u4f7f\u7528\u53cc\u7ebf\u6027\u5c42\\(\\square^T\\mathbf{W}\\square \\)\u6765\u8fdb\u884c\u5d4c\u5165\\(m_{k,j}\\)\u548c\u57fa\u7840\u8868\u793a\\(e_{\\mathrm{RBF}}^{(ji)}\\)\u4e0e\\(a_{\\mathrm{sBF}}^{(k,j,i)}\\)\u95f4\u7684\u4fe1\u606f\u4ea4\u4e92\uff0c\u6b63\u5982\u4e4b\u524d\u6240\u8bf4\uff0c\u8fd9\u4e2a\u5c42\u7684\u8ba1\u7b97\u975e\u5e38\u6602\u8d35\uff0c\u800c\u4e14\u88ab\u7528\u5728\u6a21\u578b\u6700\u6602\u8d35\u7684\u65b9\u5411\u6027\u6d88\u606f\u4f20\u9012\u7ec4\u4ef6\u4e2d\u3002<\/p>\n\n\n\n

DimeNet++\u7528\u4e00\u4e2a\u7b80\u5355\u7684\u54c8\u8fbe\u739b\u79ef\uff08Hadamard Product\uff0c\u5373\u5143\u7d20\u95f4\u7684\u4e58\u79ef\uff09\u548c\u4e24\u4e2a\u7b80\u5355\u7684\u795e\u7ecf\u7f51\u7edc\u5c42\u6765\u66ff\u4ee3\u539f\u6765\u7684\u53cc\u7ebf\u6027\u5c42\uff0c\u4ee5\u6b64\u6765\u964d\u4f4e\u8ba1\u7b97\u590d\u6742\u5ea6\u3002\u4e3a\u4e86\u8865\u507f\u7531\u4e8e\u7b80\u5316\u64cd\u4f5c\u800c\u53ef\u80fd\u5e26\u6765\u7684\u8868\u8fbe\u80fd\u529b\u635f\u5931\uff0c\u589e\u52a0\u4e86\u591a\u5c42\u611f\u77e5\u5668\uff08MLPs\uff09\u6765\u5904\u7406\u57fa\u7840\u8868\u793a\u3002\u4f8b\u5982\uff0c\u4e3a\u5f84\u5411\u57fa\u51fd\u6570\uff08RBF\uff09\u548c\u7403\u5f62\u57fa\u51fd\u6570\uff08SBF\uff09\u4f7f\u75282\u5c42\u591a\u5c42\u611f\u77e5\u5668\uff08MLP\uff09\u3002\u8fd9\u4e00\u6539\u53d8\u5728\u4fdd\u6301\u539f\u59cb\u7cbe\u5ea6\u7684\u540c\u65f6\u663e\u8457\u964d\u4f4e\u4e86\u8ba1\u7b97\u6210\u672c\uff0c\u6539\u8fdb\u540e\u7684\u65b9\u5411\u6027\u6d88\u606f\u4f20\u9012\u6a21\u5757\u548c\u4ea4\u4e92\u6a21\u5757\u5982\u56fe11\u6240\u793a\u3002<\/p>\n\n\n

\n
\"\"
\u56fe11 \u6a21\u578b\u6539\u8fdb<\/figcaption><\/figure><\/div>\n\n\n



\\(W_{\\uparrow}\\)\u548c\\(W_{\\downarrow}\\)\u662f\u6743\u91cd\u77e9\u9635\uff0c\u7528\u4e8e\u5728\u7f51\u7edc\u7684\u4e0d\u540c\u5c42\u4e4b\u95f4\u8c03\u6574\uff08\u4e0a\u91c7\u6837\u6216\u4e0b\u91c7\u6837\uff09\u7279\u5f81\u7684\u7ef4\u5ea6\u3002 \\(W_{\\downarrow}\\)\u901a\u5e38\u7528\u4e8e\u51cf\u5c11\u7279\u5f81\u7ef4\u5ea6\uff0c\u6216\u8005\u8bf4\u662f\u8fdb\u884c\u7279\u5f81\u7684\u4e0b\u91c7\u6837\uff08downsampling\uff09\u6216\u7ef4\u5ea6\u7684\u7f29\u51cf\u3002 \\(W_{\\uparrow}\\)\u5219\u76f8\u53cd\uff0c\u901a\u5e38\u7528\u4e8e\u589e\u52a0\u7279\u5f81\u7ef4\u5ea6\uff0c\u5373\u8fdb\u884c\u7279\u5f81\u7684\u4e0a\u91c7\u6837\uff08upsampling\uff09\u6216\u7ef4\u5ea6\u7684\u6269\u5c55\u3002\u8fd9\u6837\u7684\u5c42\u6b21\u7ed3\u6784\u53ef\u4ee5\u6709\u6548\u5730\u5904\u7406\u4e0d\u540c\u5927\u5c0f\u548c\u4e0d\u540c\u5206\u8fa8\u7387\u7684\u7279\u5f81\u8868\u793a\uff0c\u8fd9\u5728\u5904\u7406\u5177\u6709\u591a\u5c3a\u5ea6\u7ed3\u6784\u6216\u591a\u79cd\u7c7b\u578b\u4ea4\u4e92\u7684\u590d\u6742\u56fe\u65f6\u5c24\u5176\u91cd\u8981\u3002\u540c\u65f6\uff0c\u4e5f\u6709\u52a9\u4e8e\u5728\u51cf\u5c11\u8ba1\u7b97\u91cf\u548c\u907f\u514d\u8fc7\u5ea6\u62df\u5408\u4e4b\u95f4\u53d6\u5f97\u5e73\u8861\u3002<\/p>\n\n\n\n

\u6b64\u5916\uff0c DimeNet++\u4e5f\u505a\u4e86\u5176\u4ed6\u8c03\u53c2\u5b9e\u9a8c\uff0c\u4f8b\u5982\uff0c\u5c06\u4ea4\u4e92\u6a21\u5757\u4ece\u539f\u6765\u76846\u5c42\u51cf\u5c11\u52304\u5c42\u3002\u5c06\u4e09\u5143\u7ec4\u5d4c\u5165\u7684\u5927\u5c0f\u51cf\u5c11\u523064\u3002\u5c06\u8f93\u51fa\uff08\u539f\u5b50\uff09\u5d4c\u5165\u7684\u5927\u5c0f\u589e\u52a0\u5230256\u7b49\uff0c\u6d88\u878d\u5b9e\u9a8c\u7684\u7ed3\u679c\u5982\u88681\u6240\u793a\u3002<\/p>\n\n\n

\n
\"\"<\/figure><\/div>\n\n\n

\u88681\u4e2d\u5217\u51fa\u4e86DimeNet\u53ca\u5176\u5728DimeNet++\u4e2d\u5404\u9879\u6539\u8fdb\u540e\u5728QM9\u6570\u636e\u96c6\u4e0a\u9a8c\u8bc1\u7684\u5e73\u5747\u7edd\u5bf9\u8bef\u5dee\uff08MAE\uff09\u4ee5\u53ca\u6bcf\u4e2aepoch\u7684\u8bad\u7ec3\u65f6\u95f4\u3002\u8fd9\u4e9b\u6539\u8fdb\u5206\u522b\u662f\uff1a<\/p>\n\n\n\n

\uff081\uff09\u4f7f\u7528\u54c8\u8fbe\u739b\u79ef\u66ff\u6362\u539f\u6709\u64cd\u4f5c\u3002<\/p>\n\n\n\n

\uff082\uff09\u4e3a\u5f84\u5411\u57fa\u51fd\u6570\uff08RBF\uff09\u548c\u7403\u5f62\u57fa\u51fd\u6570\uff08SBF\uff09\u4f7f\u75282\u5c42\u591a\u5c42\u611f\u77e5\u5668\uff08MLP\uff09\u3002<\/p>\n\n\n\n

\uff083\uff09\u5c06\u6a21\u578b\u5c42\u6570\u51cf\u5c11\u52304\u5c42\u3002<\/p>\n\n\n\n

\uff084\uff09\u5c06\u4e09\u5143\u7ec4\u5d4c\u5165\u7684\u5927\u5c0f\u51cf\u5c11\u523064\u3002<\/p>\n\n\n\n

\uff085\uff09\u5c06\u8f93\u51fa\uff08\u539f\u5b50\uff09\u5d4c\u5165\u7684\u5927\u5c0f\u589e\u52a0\u5230256\u3002<\/p>\n\n\n\n

\u5728\u8868\u683c1\u4e2d\u7684\u7b2c\u4e00\u5217\u8868\u793a\u6a21\u578b\u5728\u6570\u636e\u96c6\u4e0a\u8bad\u7ec3\u4e00\u8f6e\u6240\u6d88\u8017\u7684\u65f6\u95f4\uff0c\u800c\u7b2c\u4e8c\u5217\u4e2d\u7684MAE\u662f\u4e00\u79cd\u8861\u91cf\u6a21\u578b\u9884\u6d4b\u5206\u5b50\u7684\u7535\u91cf\u4e0e\u771f\u5b9e\u7ed3\u679c\u4e4b\u95f4\u5dee\u8ddd\u7684\u4e00\u79cd\u65b9\u5f0f\uff0c\u5176\u503c\u8d8a\u5c0f\u4ee3\u8868\u6a21\u578b\u7684\u9884\u6d4b\u7ed3\u679c\u8d8a\u597d\u3002\u4ece\u8868\u683c\u4e2d\u7684\u6570\u636e\u53ef\u4ee5\u53d1\u73b0\uff1a<\/p>\n\n\n\n

\uff081\uff09\u539f\u59cb\u7684DimeNet\u6a21\u578b\u7684\u9884\u6d4b\u7ed3\u679c\u548c\u771f\u503c\u4e4b\u95f4\u7684MAE\u4e3a8.27\u6beb\u7535\u5b50\u4f0f\uff08meV\uff09\uff0c\u6bcf\u4e2aepoch\u7684\u8bad\u7ec3\u65f6\u95f4\u4e3a35.4\u5206\u949f\u3002\u5f15\u5165\u54c8\u8fbe\u739b\u79ef\u4e4b\u540e\uff0cMAE\u7565\u5fae\u589e\u52a0\u52309.45 meV\uff0c\u4f46\u6bcf\u4e2aepoch\u7684\u65f6\u95f4\u5927\u5e45\u51cf\u5c11\u81f36.6\u5206\u949f\u3002<\/p>\n\n\n\n

\uff082\uff09\u6dfb\u52a02\u5c42MLP\u540e\uff0cMAE\u964d\u4f4e\u52308.42 meV\uff0c\u6bcf\u4e2aepoch\u7684\u65f6\u95f4\u4e3a7.1\u5206\u949f\u3002<\/p>\n\n\n\n

\uff083\uff09\u6a21\u578b\u7b80\u5316\u4e3a4\u5c42\u540e\uff0cMAE\u7565\u6709\u4e0a\u5347\u81f38.54 meV\uff0c\u4f46\u6bcfepoch\u65f6\u95f4\u8fdb\u4e00\u6b65\u51cf\u5c11\u81f34.7\u5206\u949f\u3002<\/p>\n\n\n\n

\uff084\uff09\u5c06\u4e09\u5143\u7ec4\u5d4c\u5165\u5927\u5c0f\u51cf\u5c11\u81f364\u540e\uff0cMAE\u964d\u4f4e\u52307.60 meV\uff0c\u65f6\u95f4\u964d\u81f34.2\u5206\u949f\u3002<\/p>\n\n\n\n

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