arXiv subjects

Yulei Liao

Publications and source records attributed to Yulei Liao.

8 recordsLinked to original sources

arXiv2025

A concurrent global-local numerical method for multiscale parabolic equations

This paper presents a concurrent global-local numerical method for solving multiscale parabolic equations in divergence form. The proposed method employs hybrid coefficient to provide accurate macroscopic information while preserving essential microscopic details within specified local defects. Both the macroscopic and microscopic errors have been improved compared to existing results, eliminating the factor of $\Delta t^{-1/2}$ when the diffusion coefficient is time-independent. Numerical experiments demonstrate that the proposed method effectively captures both global and local solution behaviors.

math.NA↗

arXiv2025

Sharp uniform approximation for spectral Barron functions by deep neural networks

This work explores the neural network approximation capabilities for functions within the spectral Barron space $\mathscr{B}^s$, where $s$ is the smoothness index. We demonstrate that for functions in $\mathscr{B}^{1/2}$, a shallow neural network (a single hidden layer) with $N$ units can achieve an $L^p$-approximation rate of $\mathcal{O}(N^{-1/2})$. This rate also applies to uniform approximation, differing by at most a logarithmic factor. Our results significantly reduce the smoothness requirement compared to existing theory, which necessitate functions to belong to $\mathscr{B}^1$ in order to attain the same rate. Furthermore, we show that increasing the network's depth can notably improve the approximation order for functions with small smoothness. Specifically, for networks with $L$ hidden layers, functions in $\mathscr{B}^s$ with $0 < sL \le 1/2$ can achieve an approximation rate of $\mathcal{O}(N^{-sL})$. The rates and prefactors in our estimates are dimension-free. We also confirm the sharpness of our findings, with the lower bound closely aligning with the upper, with a discrepancy of at most one logarithmic factor.

math.NA↗

arXiv2025

A broken Hardy inequality on finite element space and application to strain gradient elasticity

We illustrate a broken Hardy inequality on discontinuous finite element spaces, blowing up with a logarithmic factor with respect to the meshes size. This is motivated by numerical analysis for the strain gradient elasticity with natural boundary conditions. A mixed finite element pair is employed to solve this model with nearly incompressible materials. This pair is quasi-stable with a logarithmic factor, which is not significant in the approximation error, and converges robustly in the incompressible limit and uniformly in the microscopic material parameter. Numerical results back up that the theoretical predictions are nearly optimal. Moreover, the regularity estimates for the model over a smooth domain have been proved with the aid of the Agmon-Douglis-Nirenberg theory.

math.NA↗

arXiv2025

Heterogeneous multiscale methods for fourth-order singular perturbations

We develop a numerical homogenization method for fourth-order singular perturbation problems within the framework of heterogeneous multiscale method. These problems arise from heterogeneous strain gradient elasticity and elasticity models for architectured materials. We establish an error estimate for the homogenized solution applicable to general media and derive an explicit convergence for the locally periodic media with the fine-scale $\varepsilon$. For cell problems of size $\delta=\mathbb{N}\varepsilon$, the classical resonance error $\mathcal{O}(\varepsilon/\delta)$ can be eliminated due to the dominance of the higher-order operator. Despite the occurrence of boundary layer effects, discretization errors do not necessarily deteriorate for general boundary conditions. Numerical simulations corroborate these theoretical findings.

math.NA↗

arXiv2024

A pre-training deep learning method for simulating the large bending deformation of bilayer plates

We propose a deep learning based method for simulating the large bending deformation of bilayer plates. Inspired by the greedy algorithm, we propose a pre-training method on a series of nested domains, which accelerate the convergence of training and find the absolute minimizer more effectively. The proposed method exhibits the capability to converge to an absolute minimizer, overcoming the limitation of gradient flow methods getting trapped in the local minimizer basins. We showcase better performance with fewer numbers of degrees of freedom for the relative energy errors and relative $L^2$-errors of the minimizer through numerical experiments. Furthermore, our method successfully maintains the $L^2$-norm of the isometric constraint, leading to an improvement of accuracy.

math.NA↗

arXiv2023

Spectral Barron space for deep neural network approximation

We prove the sharp embedding between the spectral Barron space and the Besov space with embedding constants independent of the input dimension. Given the spectral Barron space as the target function space, we prove a dimension-free convergence result that if the neural network contains $L$ hidden layers with $N$ units per layer, then the upper and lower bounds of the $L^2$-approximation error are $\mathcal{O}(N^{-sL})$ with $0 < sL\le 1/2$, where $s\ge 0$ is the smoothness index of the spectral Barron space.

math.NA↗

arXiv2023

Taylor-Hood like finite elements for nearly incompressible strain gradient elasticity problems

We propose a family of mixed finite elements that are robust for the nearly incompressible strain gradient model, which is a fourth-order singular perturbed elliptic system. The element is similar to [C. Taylor and P. Hood, Comput. & Fluids, 1(1973), 73-100] in the Stokes flow. Using a uniform discrete B-B inequality for the mixed finite element pairs, we show the optimal rate of convergence that is robust in the incompressible limit. By a new regularity result that is uniform in both the materials parameter and the incompressibility, we prove the method converges with $1/2$ order to the solution with strong boundary layer effects. Moreover, we estimate the convergence rate of the numerical solution to the unperturbed second-order elliptic system. Numerical results for both smooth solutions and the solutions with sharp layers confirm the theoretical prediction.

math.NA↗

arXiv2021

Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions

We propose a new method to deal with the essential boundary conditions encountered in the deep learning-based numerical solvers for partial differential equations. The trial functions representing by deep neural networks are non-interpolatory, which makes the enforcement of the essential boundary conditions a nontrivial matter. Our method resorts to Nitsche's variational formulation to deal with this difficulty, which is consistent, and does not require significant extra computational costs. We prove the error estimate in the energy norm and illustrate the method on several representative problems posed in at most 100 dimension.

math.NA↗