Exponential Expressivity of ReLU$^k$ Neural Networks on Gevrey Classes with Point Singularities

arXiv:2403.02035v1 Announce Type: cross
Abstract: We analyze deep Neural Network emulation rates of smooth functions with point singularities in bounded, polytopal domains $\mathrm{D} \subset \mathbb{R}^d$, $d=2,3$. We prove exponential emulation rates in Sobolev spaces in terms of the number of neurons and in terms of the number of nonzero coefficients for Gevrey-regular solution classes defined in terms of weighted Sobolev scales in $\mathrm{D}$, comprising the countably-normed spaces of I.M. Babu\v{s}ka and B.Q. Guo.
As intermediate result, we prove that continuous, …

abstract analyze arxiv cs.lg cs.na deep neural network domains functions math.na network networks neural network neural networks neurons prove relu spaces terms type