Universal approximation results for neural networks with non-polynomial activation function over non-compact domains

• URL: http://arxiv.org/abs/2410.14759v3
• Date: Mon, 24 Feb 2025 13:12:26 GMT
• Title: Universal approximation results for neural networks with non-polynomial activation function over non-compact domains
• Authors: Ariel Neufeld, Philipp Schmocker,
• Abstract summary: We extend the universal approximation property of single-hidden-layer feedforward neural networks beyond compact domains.<n>By assuming that the activation function is non-polynomial, we establish universal approximation results within function spaces defined over non-compact subsets of a Euclidean space.
• Score: 3.3379026542599934
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper extends the universal approximation property of single-hidden-layer feedforward neural networks beyond compact domains, which is of particular interest for the approximation within weighted $C^k$-spaces and weighted Sobolev spaces over unbounded domains. More precisely, by assuming that the activation function is non-polynomial, we establish universal approximation results within function spaces defined over non-compact subsets of a Euclidean space, including $L^p$-spaces, weighted $C^k$-spaces, and weighted Sobolev spaces, where the latter two include the approximation of the (weak) derivatives. Moreover, we provide some dimension-independent rates for approximating a function with sufficiently regular and integrable Fourier transform by neural networks with non-polynomial activation function.

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