TY - JOUR
T1 - Error Analysis of the Deep Mixed Residual Method for High-Order Elliptic Equations
AU - Bai , Mengjia
AU - Chen , Jingrun
AU - Du , Rui
AU - Sun , Zhiwei
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 2
SP - 395
EP - 436
PY - 2025
DA - 2025/05
SN - 18
DO - http://doi.org/10.4208/nmtma.OA-2024-0134
UR - https://global-sci.org/intro/article_detail/nmtma/24070.html
KW - Neural network approximation, deep mixed residual method, high-order elliptic equation.
AB - <p style="text-align: justify;">This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous
boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM with two types of loss functions, referred to as first-order and second-order least squares systems. By providing boundedness and coercivity analysis, we
leverage Céa’s Lemma to decompose the total error into the approximation, generalization, and optimization errors. Utilizing the Barron space theory and Rademacher
complexity, an a priori error is derived regarding the training samples and network
size that are exempt from the curse of dimensionality. Our results reveal that MIM
significantly reduces the regularity requirements for activation functions compared
to the deep Ritz method, implying the effectiveness of MIM in solving high-order
equations.</p>