arXiv subjects

Mengjia Bai

Publications and source records attributed to Mengjia Bai.

4 recordsLinked to original sources

arXiv2026

A mixed residual method for biharmonic equations in spectral Barron spaces

We propose a mixed residual method (MIM) for numerically solving the biharmonic equation with nonhomogeneous clamped boundary conditions. By establishing the well-posedness of the biharmonic equation in spectral Barron spaces, we derive an error bound for MIM that relates shallow neural network approximations to the exact solution and overcomes the curse of dimensionality. This error bound consists of two components: the first corresponds to the approximation error of the neural network, while the second represents the generalization error arising from randomly sampled training data. Several numerical experiments are presented to demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2025

Uniformly accurate structure-preserving neural surrogates for radiative transfer

In this work, we propose a uniformly accurate, structure-preserving neural surrogate for the radiative transfer equation with periodic boundary conditions based on a multiscale parity decomposition framework. The formulation introduces a refined decomposition of the particle distribution into macroscopic, odd, and higher-order even components, leading to an asymptotic-preserving neural network system that remains stable and accurate across all parameter regimes. By constructing key higher-order correction functions, we establish rigorous uniform error estimates with respect to the scale parameter $\varepsilon$, which ensures $\varepsilon$-independent accuracy. Furthermore, the neural architecture is designed to preserve intrinsic physical structures such as parity symmetry, conservation, and positivity through dedicated architectural constraints. The framework extends naturally from one to two dimensions and provides a theoretical foundation for uniformly accurate neural solvers of multiscale kinetic equations. Numerical experiments confirm the effectiveness of our approach.

math.NA↗

arXiv2024

Error Analysis of the Deep Mixed Residual Method for High-order Elliptic Equations

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.

math.NA↗

arXiv2024

Continuum Limit of Spin Dynamics on Hexagonal Lattice

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.

math-ph↗