Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms

• URL: http://arxiv.org/abs/2311.04779v1
• Date: Wed, 8 Nov 2023 15:59:10 GMT
• Title: Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms
• Authors: Yahong Yang and Yulong Lu
• Abstract summary: This paper establishes the nearly optimal rate of approximation for deep neural networks (DNNs) when applied to Korobov functions. Our achieved approximation rate demonstrates a remarkable "super-convergence" rate, outperforming traditional methods and any continuous function approximator.
• Score: 8.249180979158819
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper establishes the nearly optimal rate of approximation for deep neural networks (DNNs) when applied to Korobov functions, effectively overcoming the curse of dimensionality. The approximation results presented in this paper are measured with respect to $L_p$ norms and $H^1$ norms. Our achieved approximation rate demonstrates a remarkable "super-convergence" rate, outperforming traditional methods and any continuous function approximator. These results are non-asymptotic, providing error bounds that consider both the width and depth of the networks simultaneously.

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