Generalization Error in the Deep Ritz Method with Smooth Activation Functions

Deep Ritz method is a deep learning paradigm to solve partial differential equations. In this article we study the generalization error of the Deep Ritz method. We focus on the quintessential problem which is the Poisson’s equation. We show that generalization error of the Deep Ritz method converges to zero with rate $\frac{C}{\sqrt{n}},$ and we discuss about the constant $C.$ Results are obtained for shallow and residual neural networks with smooth activation functions.