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

arXiv:2610.09338v1 Announce Type: new Abstract: 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