AI RESEARCH
Deep Neural Networks with General Activations: Super-Convergence in Sobolev Norms
arXiv CS.LG
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ArXi:2508.05141v2 Announce Type: replace This paper establishes a comprehensive approximation result for deep fully-connected neural networks with commonly-used and general activation functions in Sobole spaces $W^{n,\infty}$, with errors measured in the $W^{m,p}$-norm for $m < n$ and $1\le p \le \infty$. The derived rates surpass those of classical numerical approximation techniques, such as finite element and spectral methods, exhibiting a phenomenon we refer to as \emph{super-convergence.