Approximation Rates for Neural Networks with Encodable Weights in Smoothness Spaces

TL;DR

This paper investigates the complexity of neural networks with encodable weights in approximating functions from smoothness spaces, establishing bounds and frameworks that advance PDE solution approximation theory.

Contribution

It provides lower bounds for network weights in approximation and introduces a unifying framework for constructing neural network partitions of unity.

Findings

Lower bounds for nonzero encodable weights in Besov and Sobolev spaces

A framework for neural network approximation of localized Taylor polynomials

Almost optimal upper bounds in higher-order Sobolev norms

Abstract

We examine the necessary and sufficient complexity of neural networks to approximate functions from different smoothness spaces under the restriction of encodable network weights. Based on an entropy argument, we start by proving lower bounds for the number of nonzero encodable weights for neural network approximation in Besov spaces, Sobolev spaces and more. These results are valid for all sufficiently smooth activation functions. Afterwards, we provide a unifying framework for the construction of approximate partitions of unity by neural networks with fairly general activation functions. This allows us to approximate localized Taylor polynomials by neural networks and make use of the Bramble-Hilbert Lemma. Based on our framework, we derive almost optimal upper bounds in higher-order Sobolev norms. This work advances the theory of approximating solutions of partial differential equations by neural networks.