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

TL;DR

BCR-Net introduces a neural network architecture inspired by wavelet-based nonstandard forms, effectively approximating nonlinear maps in complex computational problems.

Contribution

The paper presents a novel neural network architecture based on the nonstandard wavelet form, extending linear schemes to nonlinear problems with improved efficiency.

Findings

Efficient approximation of nonlinear maps in homogenization theory

Extension of wavelet-based linear algorithms to nonlinear neural networks

Demonstrated effectiveness in stochastic computation tasks

Abstract

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.