Year: 2025
Author: Mengjia Bai, Jingrun Chen, Rui Du, Zhiwei Sun
Numerical Mathematics: Theory, Methods and Applications, Vol. 18 (2025), Iss. 2 : pp. 395–436
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2024-0134
Numerical Mathematics: Theory, Methods and Applications, Vol. 18 (2025), Iss. 2 : pp. 395–436
Published online: 2025-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 42
Keywords: Neural network approximation deep mixed residual method high-order elliptic equation.