  The paper contains approximation guarantees for neural networks that are
trained with gradient flow, with error measured in the continuous
$L_2(\mathbb{S}^{d-1})$-norm on the $d$-dimensional unit sphere and targets
that are Sobolev smooth. The networks are fully connected of constant depth and
increasing width. Although all layers are trained, the gradient flow
convergence is based on a neural tangent kernel (NTK) argument for the
non-convex second but last layer. Unlike standard NTK analysis, the continuous
error norm implies an under-parametrized regime, possible by the natural
smoothness assumption required for approximation. The typical
over-parametrization re-enters the results in form of a loss in approximation
rate relative to established approximation methods for Sobolev smooth
functions.
