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 C\'{e}a'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.

翻译：本文针对具有非齐次边界条件（包括Dirichlet、Neumann和Robin条件）的高阶椭圆方程求解，提出了深度混合残差方法（MIM）的先验误差分析。我们研究了采用两种损失函数形式的MIM，分别称为一阶和二阶最小二乘系统。通过有界性和强制性分析，我们利用Céa引理将总误差分解为近似误差、泛化误差和优化误差。结合Barron空间理论和Rademacher复杂度，推导出关于训练样本和网络规模的先验误差估计，该估计不受维度灾难的影响。我们的结果表明，与深度Ritz方法相比，MIM显著降低了对激活函数正则性的要求，这体现了MIM在求解高阶方程方面的有效性。