Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation
Abstract
We construct shallow neural networks with fixed Gaussian and tanh-difference activations on bounded Lipschitz domains Ω⊂ℝd, d≥2. For each s>0, explicit deterministic rules prescribe all hidden parameters independently of the target u∈ Hs(Ω). At frequency resolution M, the NMasymp Md prescribed features span a linear space VM. For every sufficiently large integer N, we choose Masymp N1/d with NM≤ N and construct a bounded linear operator AN:Hs(Ω)→ VM such that |u−ANu|Hm(Ω) ≤ C N−(s−m)/d|u|Hs(Ω) for every u∈ Hs(Ω) and every integer 0≤ 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→∞ while preserving the approximation rate. A margin beyond supx∈Ω|x| of order M−1log 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 L2 and H1 approximation.
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BibTeX
@misc{bi2026fully,
title = {{Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation}},
author = {Ran Bi and Weibing Deng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09338},
eprint = {2610.09338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09338},
}
RIS
TY - UNPB
TI - Fully Deterministic Shallow Neural Networks with Localized Activations. Part I: Construction and Optimal Sobolev Approximation
AU - Ran Bi
AU - Weibing Deng
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09338
UR - https://arxiv.org/abs/2610.09338
AB - We construct shallow neural networks with fixed Gaussian and tanh-difference activations on bounded Lipschitz domains Ω⊂ℝ^d, d≥2. For each s>0, explicit deterministic rules prescribe all hidden parameters independently of the target u∈ H^s(Ω). At frequency resolution M, the N_Masymp M^d prescribed features span a linear space V_M. For every sufficiently large integer N, we choose Masymp N^1/d with N_M≤ N and construct a bounded linear operator A_N:H^s(Ω)→ V_M such that |u−A_Nu|_H^m(Ω) ≤ C N^−(s−m)/d|u|_H^s(Ω) for every u∈ H^s(Ω) and every integer 0≤ 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→∞ while preserving the approximation rate. A margin beyond sup_x∈Ω|x| of order M^−1log 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.
ER -