DVM: a deep learning algorithm for minimizing functionals

Abstract

The use of data-driven techniques to solve PDEs is a rapidly developing field. Current deep learning methods can find solutions to high-dimensional PDEs with great accuracy and efficiency. However, for certain classes of problems these techniques may be inefficient. We focus on PDEs with a so-called 'variational formulation'. Here the solution to the PDE is represented as a minimizer or maximizer to a functional. We propose a family of novel deep learning algorithms to find these minimizers with similar accuracy and greater efficiency than techniques using the PDE formulation. These algorithms can be also be used to minimize functionals which do not have an equivalent PDE formulation. We call these algorithms 'Deep Variational Methods' (DVM).