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Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation — arXiv:2610.09338

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.09338
License (as archived): CC0-1.0
Archived: 2026-10-08T12:43:37.147Z
SHA-256 of the archived text: f781f0f19b59bde21be4d34ba7c20be680a06f6bae488e4e4fa9da2cd87934ae
Stable URL of this page: https://metraipolis.com/documento/dc_aa23a16331ae

How to cite

Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation — arXiv:2610.09338 — arXiv (en), https://arxiv.org/abs/2610.09338 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_aa23a16331ae (Metraipolis archive dc_aa23a16331ae, sha256 f781f0f19b59bde21be4d34ba7c20be680a06f6bae488e4e4fa9da2cd87934ae)

BibTeX

@misc{dcaa23a16331ae,
  title = {Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation — arXiv:2610.09338},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.09338},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_aa23a16331ae; sha256 f781f0f19b59bde21be4d34ba7c20be680a06f6bae488e4e4fa9da2cd87934ae},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.09338. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We construct shallow neural networks with fixed Gaussian and tanh-difference activations on bounded Lipschitz domains $\Omega\subset\mathbb{R}^d$, $d\ge2$. For each $s>0$, explicit deterministic rules prescribe all hidden parameters independently of the target $u\in H^s(\Omega)$. At frequency resolution $M$, the $N_M\asymp M^d$ prescribed features span a linear space $V_M$. For every sufficiently large integer $N$, we choose $M\asymp N^{1/d}$ with $N_M\le N$ and construct a bounded linear operator $A_N:H^s(\Omega)\to V_M$ such that \[ \|u-A_Nu\|_{H^m(\Omega)} \le C N^{-(s-m)/d}\|u\|_{H^s(\Omega)} \] for every $u\in H^s(\Omega)$ and every integer $0\le m<s$, with $C$ independent of $u$, $N$, and $m$. Sobolev width lower bounds show that this rate is optimal among linear spaces of dimension at most $N$ chosen independently of the target. We use explicit spherical cubature nodes for the directions and Gauss-Legendre nodes or equispaced midpoints for the offsets. We show that the exterior offset margin can shrink to zero as $M\to\infty$ while preserving the approximation rate. A margin beyond $\sup_{x\in\Omega}|x|$ of order $M^{-1}\log M$ for the Gaussian and $M^{-1}(\log M)^2$ for the tanh difference suffices to truncate the ridge integral and remove exterior midpoint nodes. Numerical experiments in two and three dimensions examine $L^2$ and $H^1$ approximation.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.NA, nuovi lavori del giorno).
Categorie arXiv: Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2610.09338 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-08T12:16:45.208Z dalla pagina https://arxiv.org/list/math.NA/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).