小波
计算
非线性系统
人工神经网络
数学
应用数学
算法
计算机科学
理论计算机科学
离散数学
人工智能
量子力学
物理
作者
Yuwei Fan,Cindy Orozco Bohorquez,Lexing Ying
标识
DOI:10.1016/j.jcp.2019.02.002
摘要
This paper proposes a novel neural network architecture inspired by the nonstandard form proposed by Beylkin, Coifman, and Rokhlin in [Communications on Pure and Applied Mathematics, 44(2), 141-183]. The nonstandard form is a highly effective wavelet-based compression scheme for linear integral operators. In this work, we first represent the matrix-vector product algorithm of the nonstandard form as a linear neural network where every scale of the multiresolution computation is carried out by a locally connected linear sub-network. In order to address nonlinear problems, we propose an extension, called BCR-Net, by replacing each linear sub-network with a deeper and more powerful nonlinear one. Numerical results demonstrate the efficiency of the new architecture by approximating nonlinear maps that arise in homogenization theory and stochastic computation.
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