  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.
