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This paper proposes a novel neural network architecture inspired by thenonstandard form proposed by Beylkin, Coifman, and Rokhlin in [Communicationson Pure and Applied Mathematics, 44(2), 141-183]. The nonstandard form is ahighly effective wavelet-based compression scheme for linear integraloperators. In this work, we first represent the matrix-vector product algorithmof the nonstandard form as a linear neural network where every scale of themultiresolution computation is carried out by a locally connected linearsub-network. In order to address nonlinear problems, we propose an extension,called BCR-Net, by replacing each linear sub-network with a deeper and morepowerful nonlinear one. Numerical results demonstrate the efficiency of the newarchitecture by approximating nonlinear maps that arise in homogenizationtheory and stochastic computation.

BCR-Net: a neural network based on the nonstandard wavelet form