  We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs)
in $H^1(\Omega)$ for weighted analytic function classes in certain polytopal
domains $\Omega$, in space dimension $d=2,3$. Functions in these classes are
locally analytic on open subdomains $D\subset \Omega$, but may exhibit isolated
point singularities in the interior of $\Omega$ or corner and edge
singularities at the boundary $\partial \Omega$. The exponential expression
rate bounds proved here imply uniform exponential expressivity by ReLU NNs of
solution families for several elliptic boundary and eigenvalue problems with
analytic data. The exponential approximation rates are shown to hold in space
dimension $d = 2$ on Lipschitz polygons with straight sides, and in space
dimension $d=3$ on Fichera-type polyhedral domains with plane faces. The
constructive proofs indicate in particular that NN depth and size increase
poly-logarithmically with respect to the target NN approximation accuracy
$\varepsilon>0$ in $H^1(\Omega)$. The results cover in particular solution sets
of linear, second order elliptic PDEs with analytic data and certain nonlinear
elliptic eigenvalue problems with analytic nonlinearities and singular,
weighted analytic potentials as arise in electron structure models. In the
latter case, the functions correspond to electron densities that exhibit
isolated point singularities at the positions of the nuclei. Our findings
provide in particular mathematical foundation of recently reported, successful
uses of deep neural networks in variational electron structure algorithms.

Numerical Analysis;Machine Learning;Numerical Analysis