The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems

Deep learning has had great success in computer vision and other artificial intelligence tasks [1]. Underlying this success is a new way to approximate functions, from an additive construction commonly used in approximation theory to a compositional construction used in deep neural networks. The compositional construction seems to be particularly powerful in high dimensions. This suggests that deep neural network based models can be of use in other contexts that involve constructing functions. This includes solving partial differential equations, molecular modeling, model reduction, etc. These aspects have been explored recently in [2, 3, 4, 5, 6, 7]. 1 In this paper, we continue this line of work and propose a new algorithm for solving variational problems. We call this new algorithm the Deep Ritz method since it is based on using the neural network representation of functions in the context of the Ritz method. The Deep Ritz method has a number of interesting and promising features, which we explore later in the paper.