TY - JOUR
T1 - Error Analysis of the Mixed Residual Method for Elliptic Equations
AU - Gu , Kai
AU - Fang , Peng
AU - Sun , Zhiwei
AU - Du , Rui
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 534
EP - 554
PY - 2024
DA - 2024/05
SN - 17
DO - http://doi.org/10.4208/nmtma.OA-2023-0136
UR - https://global-sci.org/intro/article_detail/nmtma/23111.html
KW - Deep mixed residual method, deep neural network, error analysis, elliptic equations, Rademacher complexity
AB - <p style="text-align: justify;">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.</p>