Error Analysis of the Mixed Residual Method for Elliptic Equations

We present a rigorous analysis of the convergence rate of the deep mixed residual method (MIM) when applied to a linear elliptic equation with different types of boundary conditions. The MIM has been proposed to solve high-order partial differential equations in high dimensions. Our analysis shows that MIM outperforms deep Ritz method and deep Galerkin method for weak solution 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 provide valuable insights into the strengths of MIM and its comparative performance in solving linear elliptic equations with different boundary conditions.