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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., Comment: 40 pages, none figures

This study investigates the atomistic spin system in $\rm CrCl_{3}$, which exhibits topologically nontrivial meron structures within its layered hexagonal lattice framework. We analyze the complete model of discrete spin dynamics on a two-dimensional hexagonal lattice and demonstrate its convergence to the continuum Landau-Lifshitz-Gilbert equation in the weak sense. The primary challenge lies in defining appropriate difference quotient and interpolation operators for the hexagonal lattice since the loss of symmetry. To address these, we utilized a one-step difference quotient for the 2nd nearest neighbors and introduced novel multi-step difference quotients for the 1st and 3rd nearest neighbors, enabling the integration by parts formula. Additionally, we generalized Ladysenskaya's interpolation operator for hexagonal lattices and provided an alternative strategy for the convergence procedure by applying an isometric mapping property. This work provides necessary tools for analyzing weak convergence in other atomistic nonlinear problems on hexagonal lattices towards the continuum limit., Comment: 28 pages, 4 figures