Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities

Abstract In certain polytopal domains $$\varOmega $$ ? , in space dimension $$d=2,3$$ d = 2 , 3 we prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) $$H^1(\varOmega )$$ H 1 ( ) for weighted analytic function classes. These classes comprise particular solution sets of source and eigenvalue problems elliptic PDEs data. Functions these are locally on open subdomains $$D\subset \varOmega D ? but may exhibit isolated point singularities the interior or corner edge at boundary $$\partial ? . The approximation rates shown to hold $$d = 2$$ Lipschitz polygons straight sides, $$d=3$$ Fichera-type polyhedral plane faces. constructive proofs indicate that NN depth size increase poly-logarithmically respect target accuracy $$\varepsilon >0$$ ? > 0 results cover linear, second-order data nonlinear nonlinearities singular, potentials as arise electron structure models. Here, functions correspond densities nuclei.