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.

翻译：我们针对混合残差方法（MIM）应用于具有各种边界条件的线性椭圆方程时的收敛速度，给出了严格的理论分析。MIM方法已被提出作为一种比深度Galerkin方法（DGM）和深度Ritz方法（DRM）在多种情形下更有效的数值逼近方法。我们的分析表明，在Dirichlet边界条件下，MIM因其能够强制执行边界条件而优于DRM和用于弱解的深度Galerkin方法（DGMW）。然而，对于Neumann和Robin情形，MIM展现出与其它方法相似的表现。我们的结果为MIM的优势及其在求解具有不同边界条件的线性椭圆方程时的比较性能提供了宝贵见解。