[2022/08/12] Analysis of deep Ritz method for Poisson equation with Dirichlet boundary condition: gamma convergence and boundary penalty method

Title : Analysis of deep Ritz method for Poisson equation with Dirichlet boundary condition: gamma convergence and boundary penalty method

Time : Aug. 12 (Fri), 3:00 pm - 4:00 pm

Place : 32356 SKKU, Suwon

Speaker : Seungchan Ko (BRL)

Abstract :

Following the generalization error analysis of DRM discussed last week, we will study further convergence analysis of DRM for the Poisson equation with Dirichlet boundary data. We shall use the notion of Gamma convergence and boundary penalty method to show that ReLU networks of growing architecture that are trained with respect to suitably regularized Dirichlet energies converge to the true solution of the given problem. Furthermore, we will derive a quantitative error estimate which can be applied to arbitrary and in general non-linear ansatz classes. We then discuss the implications for ansatz classes which are given through ReLU networks and the relation to existing estimates for finite element functions. Main references are "Johannes Müller, Marius Zeinhofer (2020): Deep Ritz revisited, ICLR workshop on Integration of Deep Neural Models and Differential Equations" and "Johannes Müller, Marius Zeinhofer (2022): Error Estimates for the Variational Training of Neural Networks with Boundary Penalty, 3rd conference on Mathematical and Scientific Machine Learning (MSML2022)".