On the approximation capability of shallow and deep neural networks having smooth activations with respect to the Sobolev norm.
basic_science · Level V
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- Record sourced from PubMed, PMID 41946150.
- Also identified by DOI 10.1016/j.neunet.2026.108935.
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Abstract
In this paper, we investigate the simultaneous approximation of the functions and their derivatives by neural networks having smooth non-polynomial activation functions and a fixed depth, which is motivated by the physics-informed machine learning. We start by proving that the neural networks with smooth non-polynomial activation functions and with only one hidden layer having width O(N<sup>d</sup>) can approximate any W<sup>s,p</sup>-regular function with rate O(N<sup>k-s</sup>) in the W<sup>k,p</sup>-norm. We then extend this result to the networks having more than one hidden layers by using the mathematical induction.