A multiscale neural network based on hierarchical matrices

TL;DR

This paper introduces a multiscale neural network inspired by hierarchical matrices, capable of efficiently approximating nonlinear maps from discretized PDEs, with potential applications in physics and quantum chemistry.

Contribution

It extends hierarchical matrix structures to nonlinear neural networks by integrating local deep neural networks at multiple scales.

Findings

Efficient approximation of nonlinear PDE discretizations.

Successful application to nonlinear Schrödinger equations.

Effective modeling of Kohn-Sham density functional theory.

Abstract

In this work we introduce a new multiscale artificial neural network based on the structure of $\mathcal{H}$-matrices. This network generalizes the latter to the nonlinear case by introducing a local deep neural network at each spatial scale. Numerical results indicate that the network is able to efficiently approximate discrete nonlinear maps obtained from discretized nonlinear partial differential equations, such as those arising from nonlinear Schr\"odinger equations and the Kohn-Sham density functional theory.