Error Analysis of Mixed Residual Methods for Elliptic Equations

Abstract

We present a rigorous theoretical analysis of the convergence rate of the deep mixed residual method (MIM) when applied to a linear elliptic equation with various types of boundary conditions. The MIM method has been proposed as a more effective numerical approximation method compared to the deep Galerkin method (DGM) and deep Ritz method (DRM) in various cases. Our analysis shows that MIM outperforms DRM and deep Galerkin method for weak solution (DGMW) in the Dirichlet case due to its ability to enforce the boundary condition. However, for the Neumann and Robin cases, MIM demonstrates similar performance to the other methods. Our results provides valuable insights into the strengths of MIM and its comparative performance in solving linear elliptic equations with different boundary conditions.