The deep Ritz method as partition of unity finite element method

Abstract

During the last decade, various methods based on artificial neural networks (ANN) were developed in order to solve partial differential equations numerically. In particular, the DeepRitz Method (DRM) employs the variational formulation of such a differential equation with given boundary conditions in order to formulate it as a minimization problem. Afterwards,the solution of this minimization problem is approximated by a suitable neural network.The DRM also uses the penalty method to ensure that this network satisfies the boundary conditions. We restrict ourself to Poisson problems with homogeneous Dirichlet boundary conditions and pursue a similar approach. First, we triangulate the underlying domainand use the nodal basis functions associated with the triangulation to formulate a trialfunction. For each interior node of the triangulation, we multiply the corresponding nodalbasis function with an ANN and than sum up all such products. Since we don’t do this for the nodes on the boundary of the triangulation, our trial function naturally satisfies the boundary conditions and we don’t need a penalty term. Here, the nodal basis functions actas a partition of unity on the underlying domain. As in the Deep Ritz Method, we insert our trial function into the variational formulation of our chosen problem and minimize the term obtained that way with respect to the parameters of the ANN’s associated with the nodes of the triangulation. In this thesis, we implement both methods and compare their accuracy. Specifically, we compute the numerical approximations of the solution of a Poisson problem on the unit square and of a Laplace problem on an L-shaped domain. Our numerical results show that both methods perform with similar accuracy with respect tothe L2-error, but also that our method has a lower accuracy with respect to the H1-error.