Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense
具有ReLU、漏ReLU和softplus激活函数的深度神经网络能够证明性地克服Kolmogorov偏微分方程中的维度诅咒
机构 * Department of Mathematics & Informatics, University of Wuppertal(乌尔姆大学数学与信息学系) ; School of Data Science and Shenzhen Research Institute of Big Data, The Chinese University of Hong Kong, Shenzhen (CUHK-Shenzhen)(香港中文大学(深圳)数据科学学院及深圳大数据研究院) ; Applied Mathematics: Institute for Analysis and Numerics, University of Münster(穆恩斯特大学应用数学系及分析与数值分析研究所) ; Department of Mathematical Sciences, University of Arkansas(阿肯色大学数学科学系) ; Center for Astrophysics, Space Physics, and Engineering Research, Baylor University(贝勒大学天体物理学、空间物理学与工程研究中心)
AI总结 本文证明了使用ReLU、leaky ReLU和softplus激活函数的深度神经网络在L^p意义下能克服高维Kolmogorov偏微分方程的维度诅咒,扩展了此前在L^2意义下的结果。
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