{"id":5973,"date":"2025-06-12T10:58:06","date_gmt":"2025-06-12T02:58:06","guid":{"rendered":"https:\/\/nullthought.net\/?p=5973"},"modified":"2025-06-12T10:58:09","modified_gmt":"2025-06-12T02:58:09","slug":"tensorgrad%ef%bc%9a%e4%b8%ba%e7%a5%9e%e7%bb%8f%e7%ae%97%e5%ad%90%e7%9a%84%e8%ae%ad%e7%bb%83%e5%bc%95%e5%85%a5%e7%bb%93%e6%9e%84%e6%84%9f%e7%9f%a5%e7%9a%84%e5%bc%a0%e9%87%8f%e7%ba%a7%e6%a2%af%e5%ba%a6","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=5973","title":{"rendered":"TensorGRaD\uff1a\u4e3a\u795e\u7ecf\u7b97\u5b50\u7684\u8bad\u7ec3\u5f15\u5165\u7ed3\u6784\u611f\u77e5\u7684\u5f20\u91cf\u7ea7\u68af\u5ea6\u538b\u7f29\u673a\u5236\uff0c\u517c\u5177\u4f4e\u79e9\u4e0e\u7a00\u758f\u7ed3\u6784\u5efa\u6a21\u80fd\u529b"},"content":{"rendered":"\n<p>\u8bba\u6587<strong><a href=\"https:\/\/arxiv.org\/abs\/2501.02379\" target=\"_blank\" rel=\"noreferrer noopener\">TensorGRaD: Tensor Gradient Robust Decomposition for Memory-Efficient Neural Operator Training<\/a><\/strong>\u63d0\u51fa\u4e86\u4e00\u79cd\u5f20\u91cf\u7ea7\u7684\u68af\u5ea6\u9c81\u68d2\u5206\u89e3\u65b9\u6cd5\u2014\u2014TensorGRaD\u3002TensorGRaD\u4e3a\u795e\u7ecf\u7b97\u5b50\u7684\u8bad\u7ec3\u5f15\u5165\u4e86\u7ed3\u6784\u611f\u77e5\u7684\u5f20\u91cf\u7ea7\u68af\u5ea6\u538b\u7f29\u673a\u5236\uff0c\u5728\u7406\u8bba\u4e0e\u5b9e\u8df5\u4e0a\u5747\u5c55\u73b0\u51fa\u5353\u8d8a\u7684\u8868\u73b0\u3002\u5b83\u4e0d\u4ec5\u4fdd\u7559\u4e86\u5f20\u91cf\u7ed3\u6784\u4fe1\u606f\uff0c\u4e5f\u964d\u4f4e\u4e86\u5bf9\u663e\u5b58\u7684\u8981\u6c42\uff0c\u7279\u522b\u9002\u7528\u4e8e\u9ad8\u5206\u8fa8\u7387PDE\u7c7b\u79d1\u5b66\u4efb\u52a1\u7684\u9ad8\u6548\u8bad\u7ec3\u3002\u8be5\u65b9\u6cd5\u4e3a\u4eca\u540e\u66f4\u5e7f\u6cdb\u91c7\u7528\u5f20\u91cf\u795e\u7ecf\u7f51\u7edc\u548c\u4f4e\u8d44\u6e90\u573a\u666f\u4e2d\u7684\u8bad\u7ec3\u63d0\u4f9b\u4e86\u6280\u672f\u57fa\u7840\u3002<\/p>\n\n\n\n<p>\u8bba\u6587\u4f5c\u8005\u4e3aSebastian Loeschcke, David Pitt, Robert Joseph George, Jiawei Zhao, Cheng Luo, Yuandong Tian, Jean Kossaifi, Anima Anandkumar\uff0c\u6765\u81eaUniversity of Copenhagen, California Institute of Technology, Meta FAIR, NVIDIA AI\u3002<\/p>\n\n\n\n<p>\u4e00\u3001\u7814\u7a76\u80cc\u666f\u4e0e\u52a8\u673a<\/p>\n\n\n\n<p>\u968f\u7740\u6df1\u5ea6\u5b66\u4e60\u6a21\u578b\u4e0d\u65ad\u6269\u5c55\uff0c\u5c24\u5176\u662f\u5728\u79d1\u5b66\u8ba1\u7b97\u7b49\u5bf9\u9ad8\u7ef4\u6570\u636e\u5efa\u6a21\u8981\u6c42\u6781\u9ad8\u7684\u9886\u57df\uff0c<a href=\"https:\/\/nullthought.net\/?p=4270\" target=\"_blank\" rel=\"noreferrer noopener\">\u795e\u7ecf\u7b97\u5b50\uff08Neural Operators, NOs\uff09<\/a>\u6210\u4e3a\u89e3\u51b3\u504f\u5fae\u5206\u65b9\u7a0b\uff08PDEs\uff09\u95ee\u9898\u7684\u6709\u529b\u5de5\u5177\u3002\u7136\u800c\uff0cNO\u6a21\u578b\u7684\u6743\u91cd\u548c\u68af\u5ea6\u901a\u5e38\u4e3a\u9ad8\u9636\u5f20\u91cf\u7ed3\u6784\uff0c\u8fd9\u5728\u8bad\u7ec3\u8fc7\u7a0b\u4e2d\u5e26\u6765\u4e86\u5de8\u5927\u7684\u5185\u5b58\u6d88\u8017\uff0c\u5c24\u5176\u662f\u5728\u4f7f\u7528\u8bf8\u5982AdamW\u7b49\u81ea\u9002\u5e94\u4f18\u5316\u5668\u65f6\uff0c\u9700\u8981\u5b58\u50a8\u6bcf\u4e2a\u53c2\u6570\u7684\u68af\u5ea6\u4e00\u9636\u548c\u4e8c\u9636\u52a8\u91cf\uff0c\u8fdb\u4e00\u6b65\u52a0\u5267\u4e86\u95ee\u9898\u3002\u73b0\u6709\u5982GaLore\u3001GRASS\u7b49\u65b9\u6cd5\u591a\u805a\u7126\u4e8e\u4f4e\u79e9\u6216\u7a00\u758f\u7ed3\u6784\u7684\u77e9\u9635\u68af\u5ea6\u538b\u7f29\uff0c\u96be\u4ee5\u4fdd\u7559\u5f20\u91cf\u591a\u6a21\u6001\u7ed3\u6784\uff0c\u56e0\u6b64\u5bf9NO\u7c7b\u6a21\u578b\u6548\u679c\u4e0d\u4f73\u3002\u4e3a\u89e3\u51b3\u8fd9\u4e00\u96be\u9898\uff0c\u4f5c\u8005\u63d0\u51fa\u4e86\u4e00\u79cd\u5f20\u91cf\u7ea7\u7684\u68af\u5ea6\u9c81\u68d2\u5206\u89e3\u65b9\u6cd5\u2014\u2014TensorGRaD\uff0c\u517c\u5177\u4f4e\u79e9\u4e0e\u7a00\u758f\u7ed3\u6784\u5efa\u6a21\u80fd\u529b\uff0c\u5728\u4e0d\u635f\u5931\u7cbe\u5ea6\u7684\u524d\u63d0\u4e0b\u53ef\u663e\u8457\u8282\u7701\u5185\u5b58\u3002<\/p>\n\n\n\n<p>\u4e8c\u3001\u65b9\u6cd5\u6982\u8ff0\u4e0e\u6838\u5fc3\u601d\u60f3<\/p>\n\n\n\n<p>TensorGRaD\u7684\u6838\u5fc3\u601d\u60f3\u662f\u5c06\u68af\u5ea6\u5f20\u91cf GGG \u5206\u89e3\u4e3a\u4f4e\u79e9\u5f20\u91cf L \u4e0e\u7a00\u758f\u5f20\u91cf S \u7684\u548c\uff0c\u5373\uff1aG=L+S<\/p>\n\n\n\n<p>\u5176\u4e2d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>\u4f4e\u79e9\u90e8\u5206 L<\/strong> \u91c7\u7528 Tucker \u5206\u89e3\u83b7\u53d6\u6838\u5fc3\u5f20\u91cf\u4e0e\u6b63\u4ea4\u77e9\u9635\u7684\u4e58\u79ef\uff0c\u4fdd\u6301\u68af\u5ea6\u5f20\u91cf\u7684\u591a\u6a21\u7ed3\u6784\uff0c\u7c7b\u4f3c\u77e9\u9635\u7684SVD\u3002<\/li>\n\n\n\n<li><strong>\u7a00\u758f\u90e8\u5206 S<\/strong> \u91c7\u7528\u975e\u7ed3\u6784\u5316\u7684COO\u683c\u5f0f\uff0c\u4ec5\u4fdd\u7559Top-k\u68af\u5ea6\u5143\u7d20\u7684\u4fe1\u606f\uff0c\u53ef\u9ad8\u6548\u8868\u793a\u5c16\u9510\u7684\u3001\u5c40\u90e8\u7684\u53d8\u5316\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u8be5\u5206\u89e3\u7ed3\u6784\u5177\u6709\u5982\u4e0b\u7279\u6027\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u4fdd\u7559\u4e86\u5f20\u91cf\u7ed3\u6784\u4e2d\u7684\u6a21\u6001\u4fe1\u606f\uff0c\u907f\u514d\u4fe1\u606f\u4e22\u5931\uff1b<\/li>\n\n\n\n<li>\u7a00\u758f\u90e8\u5206\u53ef\u5bb9\u7eb3\u5f02\u5e38\u503c\u3001\u79bb\u7fa4\u70b9\uff0c\u63d0\u5347\u68af\u5ea6\u8868\u793a\u9c81\u68d2\u6027\uff1b<\/li>\n\n\n\n<li>\u4f4e\u79e9\u90e8\u5206\u53ef\u6355\u6349\u5f20\u91cf\u4e2d\u5168\u5c40\u7684\u5e73\u6ed1\u6a21\u5f0f\uff1b<\/li>\n\n\n\n<li>\u5728\u8bad\u7ec3\u4e2d\u5206\u522b\u5bf9 L \u4e0e S \u8fdb\u884cAdamW\u66f4\u65b0\uff0c\u7136\u540e\u91cd\u6784\u68af\u5ea6\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u4e09\u3001\u65b9\u6cd5\u5b9e\u73b0\u7ec6\u8282<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>\u68af\u5ea6\u538b\u7f29\u6d41\u7a0b<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u9996\u5148\u62bd\u53d6\u7a00\u758f\u90e8\u5206 G<sub>S<\/sub>\uff0c\u5269\u4f59\u4f5c\u4e3a\u6b8b\u5dee\uff1b<\/li>\n\n\n\n<li>\u5bf9\u6b8b\u5dee\u8fdb\u884cTucker\u4f4e\u79e9\u5206\u89e3\uff0c\u5f97\u51fa\u4f4e\u79e9\u90e8\u5206 G<sub>L\u200b<\/sub>\uff1b<\/li>\n\n\n\n<li>\u4e24\u90e8\u5206\u5206\u522b\u8fdb\u884cAdam\u66f4\u65b0\uff0c\u6700\u540e\u5408\u5e76\u5f97\u5230\u91cd\u6784\u68af\u5ea6\uff1b<\/li>\n\n\n\n<li>\u4e3a\u8282\u7701\u5185\u5b58\uff0c\u4f4e\u79e9\u90e8\u5206\u91cd\u6784\u540e\u76f4\u63a5\u5c06\u7a00\u758f\u90e8\u5206\u4ee5scatter\u65b9\u5f0f\u53e0\u52a0\uff0c\u907f\u514d\u4e24\u4efd\u5b8c\u6574\u5f20\u91cf\u540c\u65f6\u5b58\u5728\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u6df7\u5408\u7cbe\u5ea6\u8bad\u7ec3<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u6743\u91cd\u3001\u6fc0\u6d3b\u3001\u68af\u5ea6\u4f7f\u7528\u534a\u7cbe\u5ea6\uff08FP16\uff09\u5b58\u50a8\u548c\u8fd0\u7b97\uff1b<\/li>\n\n\n\n<li>\u4f18\u5316\u5668\u72b6\u6001\uff08\u5373Adam\u7684\u52a8\u91cf\uff09\u4ecd\u4fdd\u7559\u4e3a\u5168\u7cbe\u5ea6\uff08FP32\uff09\uff0c\u9632\u6b62\u7cbe\u5ea6\u4e22\u5931\u5bfc\u81f4\u6a21\u578b\u6027\u80fd\u4e0b\u964d\uff1b<\/li>\n\n\n\n<li>\u9a8c\u8bc1\u4e86\u5728\u4fdd\u7559\u7cbe\u5ea6\u7684\u540c\u65f6\uff0c\u6df7\u5408\u7cbe\u5ea6\u7b56\u7565\u53ef\u989d\u5916\u8282\u7701\u663e\u5b58\u3002<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>\u7b97\u6cd5\u4f2a\u4ee3\u7801<\/strong>\uff1a\n<ul class=\"wp-block-list\">\n<li>\u63d0\u4f9b\u4e86\u5b8c\u6574\u7684TensorGRaD\u8bad\u7ec3\u6d41\u7a0b\uff0c\u5305\u62ec\u68af\u5ea6\u538b\u7f29\u3001\u4f4e\u79e9\u5206\u89e3\u3001\u7a00\u758f\u62bd\u6837\u4e0e\u4f18\u5316\u5668\u66f4\u65b0\u7b49\uff1b<\/li>\n\n\n\n<li>\u6bcfT\u6b65\u91cd\u65b0\u9009\u62e9\u7a00\u758f\u7d22\u5f15\u4e0e\u4f4e\u79e9\u5b50\u7a7a\u95f4\uff0c\u7f13\u89e3\u8ba1\u7b97\u5f00\u9500\u3002<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n<p>\u56db\u3001\u7406\u8bba\u5206\u6790<\/p>\n\n\n\n<p>\u4f5c\u8005\u5728\u7406\u8bba\u4e0a\u8bc1\u660e\u4e86TensorGRaD\u5728\u5404\u4e2a\u6a21\u6001\u4e0a\u7684\u68af\u5ea6\u6b8b\u5dee\u4f1a\u9010\u6b65\u6536\u655b\u81f3\u96f6\uff0c\u6761\u4ef6\u662f\u6bcf\u4e2a\u6a21\u6001\u7684\u6295\u5f71\u7b97\u5b50\u5177\u6709\u8db3\u591f\u7684\u201c\u6700\u5c0f\u5947\u5f02\u503c\u201d\u4fdd\u969c\u3002\u8fd9\u4e00\u7ed3\u679c\u610f\u5473\u7740\u5373\u4f7f\u4f7f\u7528\u538b\u7f29\u68af\u5ea6\u8868\u793a\uff0c\u4e5f\u80fd\u4fdd\u8bc1\u8bad\u7ec3\u8fc7\u7a0b\u6536\u655b\u3002\u8be5\u7406\u8bba\u62d3\u5c55\u4e86GaLore\u7684\u6536\u655b\u6027\u5206\u6790\uff0c\u4ece\u77e9\u9635\u7a7a\u95f4\u63a8\u5e7f\u81f3\u5f20\u91cf\u7a7a\u95f4\u3002<\/p>\n\n\n\n<p>\u4e94\u3001\u5b9e\u9a8c\u8bbe\u7f6e\u4e0e\u7ed3\u679c\u5206\u6790<\/p>\n\n\n\n<p>\u4f5c\u8005\u5728\u591a\u4e2a\u5177\u6709\u6311\u6218\u6027\u7684PDE\u4efb\u52a1\u4e0a\u6d4b\u8bd5\u4e86TensorGRaD\u65b9\u6cd5\uff0c\u5305\u62ec\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Navier-Stokes \u65b9\u7a0b<\/strong>\uff1a\u4e8c\u7ef4Kolmogorov\u6d41\uff0cReynolds\u6570\u8fbe 10<sup>5<\/sup>\uff0c\u6781\u5177\u6e4d\u6d41\u7279\u6027\uff1b<\/li>\n\n\n\n<li><strong>Darcy Flow<\/strong>\uff1a\u975e\u7ebf\u6027\u692d\u5706\u578bPDE\uff0c\u6d4b\u8bd5\u5b54\u9699\u4ecb\u8d28\u4e2d\u6d41\u4f53\u4f20\u64ad\uff1b<\/li>\n\n\n\n<li><strong>Burgers Equation<\/strong>\uff1a\u4e00\u7ef4\u975e\u7ebf\u6027\u7c98\u6027\u5b88\u6052\u5f8b\uff1b<\/li>\n\n\n\n<li><strong>\u7535\u78c1\u6ce2\u4f20\u64ad<\/strong>\uff1a\u975e\u7ebf\u6027Schr\u00f6dinger\u65b9\u7a0b\uff0c\u5177\u6709\u590d\u6570\u7279\u5f81\u548c\u4e8c\u6b21\u8c10\u6ce2\u751f\u6210\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u7ed3\u679c\u663e\u793a\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5728Navier-Stokes 1024\u00d71024\u4efb\u52a1\u4e2d\uff0cTensorGRaD\u5728\u4f7f\u7528\u4ec525%\u5185\u5b58\uff085%\u7a00\u758f + 20%\u4f4e\u79e9\uff09\u7684\u60c5\u51b5\u4e0b\uff0c\u8fbe\u5230\u6bd4\u5168\u7cbe\u5ea6Adam\u66f4\u4f4e\u7684\u6d4b\u8bd5L2\u8bef\u5dee\uff0816.82e-2 vs 17.02e-2\uff09\uff1b<\/li>\n\n\n\n<li>\u76f8\u8f83\u5355\u72ec\u4f7f\u7528\u7a00\u758f\u6216\u4f4e\u79e9\u65b9\u6cd5\uff0cTensorGRaD\u7ec4\u5408\u7ed3\u6784\u663e\u8457\u63d0\u5347\u7cbe\u5ea6\uff0c\u7a00\u758f\u4f18\u5148\u5206\u89e3\u4f18\u4e8e\u4f4e\u79e9\u4f18\u5148\uff1b<\/li>\n\n\n\n<li>\u5728\u6df7\u5408\u7cbe\u5ea6\u4e0b\uff0c\u4fdd\u6301\u4f18\u5316\u5668\u72b6\u6001\u4e3a\u5168\u7cbe\u5ea6\u81f3\u5173\u91cd\u8981\uff0c\u5426\u5219\u6a21\u578b\u6027\u80fd\u4e25\u91cd\u4e0b\u964d\uff1b<\/li>\n\n\n\n<li>\u4e0eGaLore\u65b9\u6cd5\u7684\u76f4\u63a5\u5f20\u91cf\u5316\u62d3\u5c55\u76f8\u6bd4\uff0cTensorGRaD\u663e\u8457\u4f18\u4e8e\u5176\u5728\u6027\u80fd\u4e0e\u5185\u5b58\u6d88\u8017\u4e24\u65b9\u9762\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u516d\u3001\u76f8\u5173\u5de5\u4f5c\u6bd4\u8f83<\/p>\n\n\n\n<p>TensorGRaD\u7684\u4f18\u52bf\u5728\u4e8e\uff1a<\/p>\n\n\n\n<ul 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