Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

The approximation of solutions to partial differential equations (PDEs) in high dimensions by classical algorithms such as finite difference or finite element methods is burdened by the so called curse of dimension. This means that the computational cost to achieve a certain accuracy depends exponentially on the dimension of the domain with respect to the reciprocal of the accuracy as base. This is for example improved in the case of so called sparse tensor discretizations. There the logarithm of the reciprocal of the accuracy is the base, but the dependence with respect to the dimension is still exponential [33]. This curse of dimension does not appear in Monte Carlo methods, which are stochastic methods and converge in the root mean squared sense. These methods are however typically restricted to evaluating the solution of a given PDE at a single point rather than the full computational domain. The approximation of solutions to PDEs in high dimensions on the full computational domain hence remains a challenging problem. Deep neural networks (DNNs) emerge as an approximation architecture with application in various areas of function approximation theory, which are in many cases as good as the established state of the art method, cf.