Approximations with deep neural networks in Sobolev time-space
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉东计算与应用数学研究所）；Faculty of Mathematics, University of Vienna（维也纳大学数学学院）
Ahmed Abdeljawad, Philipp Grohs
备注：
34 pages. This is the first version of the paper. It is expected that some changes will be performed for the next version. arXiv admin note: text overlap with arXiv:1902.07896
发展方程的解一般位于一定的Bochner-Sobolev空间中，在该空间中，解对于时间变量可能具有正则性和可积性，而对于空间变量可能不同。因此，在本文中，我们开发了一个框架，表明深度神经网络可以近似关于Bochner-Sobolev空间的Sobolev正则函数。在我们的工作中，我们使用所谓的Rectified Cubic Unit（ReCU）作为网络中的激活函数，这使我们能够推导出神经网络的近似结果，同时避免了最常用的Rectified Linear Unit（ReLU）激活函数的非正则性所引起的问题。
英文摘要：
Solutions of evolution equation generally lies in certain Bochner-Sobolev spaces, in which the solution may has regularity and integrability properties for the time variable that can be different for the space variables. Therefore, in this paper, we develop a framework shows that deep neural networks can approximate Sobolev-regular functions with respect to Bochner-Sobolev spaces. In our work we use the so-called Rectified Cubic Unit (ReCU) as an activation function in our networks, which allows us to deduce approximation results of the neural networks while avoiding issues caused by the non regularity of the most commonly used Rectivied Linear Unit (ReLU) activation function.
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Wuhan University（武汉大学）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；National Center for Applied Mathematics in Hubei（湖北国家应用数学中心）；Hubei Key Laboratory of Computational Science, Wuhan University（武汉大学湖北省计算科学重点实验室）
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Research Institute of Mathematics, Seoul National University（首尔国立大学数学研究所）；Department of Mathematics, Inha University（inha大学数学系）；Department of Mathematical Sciences, Seoul National University（首尔国立大学数学科学系）
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