First Order System Least Squares Neural Networks

We introduce a conceptual framework for numerically solving linear elliptic, parabolic, and hyperbolic PDEs on bounded, polytopal domains in euclidean spaces by deep neural networks. The PDEs are recast as minimization of a least-squares (LSQ for short) residual of an equivalent, well-posed first-order system, over parametric families of deep neural networks. The associated LSQ residual is a) equal or proportional to a weak residual of the PDE, b) additive in terms of contributions from localized subnetworks, indicating locally ``out-of-equilibrium'' of neural networks with respect to the PDE residual, c) serves as numerical loss function for neural network training, and d) constitutes, even with incomplete training, a computable, (quasi-)optimal numerical error estimator in the context of adaptive LSQ finite element methods. In addition, an adaptive neural network growth strategy is proposed which, assuming exact numerical minimization of the LSQ loss functional, yields sequences of neural networks with realizations that converge rate-optimally to the exact solution of the first order system LSQ formulation.

Exponential ReLU Neural Network Approximation Rates for Point and Edge Singularities

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.