Approximation of Functionals by Neural Network without Curse of Dimensionality

TL;DR

This paper introduces a neural network approach for approximating functionals from infinite to finite dimensions, achieving an error rate that overcomes the curse of dimensionality by leveraging a new Barron spectral space.

Contribution

It proposes a novel neural network approximation method for functionals, establishing an error bound that avoids the curse of dimensionality using Barron spectral spaces.

Findings

Neural network approximation error is O(1/√m)

Method overcomes curse of dimensionality

Uses Barron spectral space for functionals

Abstract

In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of networks, which overcomes the curse of dimensionality. The key idea of the approximation is to define a Barron spectral space of functionals.