A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

In this work, we analyze the suitability of deep neural networks (DNNs) for the numerical solution of parametric problems. Such problems connect a parameter space with a solution state space via a so-called parametric map, [53]. One special case of such a parametric problem arises when the parametric map results from solving a partial differential equation (PDE) and the parameters describe physical or geometrical constraints of the PDE such as, for example, the shape of the physical domain, boundary conditions, or a source term. Applications that lead to these problems include modeling unsteady and steady heat and mass transfer, acoustics, fluid mechanics, or electromagnetics, [34]. Solving a parametric PDE for every point in the parameter space of interest individually typically leads to two types of problems.