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September 17, 2025

On the Universal Approximation Property of Deep Fully Convolutional Neural Networks

QLQianxiao LiNational University of Singapore TLTing LinMing Chi University of Technology ZSZuowei ShenNational University of Singapore

Key Points

Deep residual fully convolutional networks can universally approximate certain symmetric functions at constant channel width.
Networks with at least two channels and a convolutional kernel size of at least 2 achieve universal approximation, highlighting specific requirements.
The study examines the role of dynamical systems in understanding approximation capabilities of neural networks.
Meeting channel and kernel size specifications is necessary; networks with fewer channels or smaller kernels fail to achieve universality.

Abstract

.We study the approximation of shift-invariant or equivariant functions by deep fully convolutional networks from the dynamical systems perspective. We prove that deep residual fully convolutional networks and their continuous-layer counterparts can achieve universal approximation of these symmetric functions at constant channel width. Moreover, we show that the same can be achieved by nonresidual variants with at least two channels in each layer and convolutional kernel size of at least 2. In addition, we show that these requirements are necessary in the sense that networks with fewer channels or smaller kernels fail to be universal approximators.Keywordsapproximation theoryconvolutional networksdeep learninguniversal approximationinvarianceMSC codes68T0541A99

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