Augmenting Basis Sets by Normalizing Flows

TL;DR

This paper explores combining traditional basis sets with normalizing flows to enhance nonlinear approximation capabilities, especially for high-dimensional and oscillatory functions, demonstrating promising convergence in quantum eigenfunction simulations.

Contribution

It introduces a novel approach of augmenting basis sets with normalizing flows, maintaining density properties while improving approximation power for complex functions.

Findings

Enhanced approximation spaces with normalizing flows.

Convergence observed in quantum harmonic oscillator eigenfunction simulations.

Maintains well-posedness and convergence properties of traditional methods.

Abstract

Approximating functions by a linear span of truncated basis sets is a standard procedure for the numerical solution of differential and integral equations. Commonly used concepts of approximation methods are well-posed and convergent, by provable approximation orders. On the down side, however, these methods often suffer from the curse of dimensionality, which limits their approximation behavior, especially in situations of highly oscillatory target functions. Nonlinear approximation methods, such as neural networks, were shown to be very efficient in approximating high-dimensional functions. We investigate nonlinear approximation methods that are constructed by composing standard basis sets with normalizing flows. Such models yield richer approximation spaces while maintaining the density properties of the initial basis set, as we show. Simulations to approximate eigenfunctions of a perturbed quantum harmonic oscillator indicate convergence with respect to the size of the basis set.