Publications (28)
Stability of a force-based hybrid method with planar sharp interface
Jianfeng Lu, Pingbing Ming
We study a force-based hybrid method that couples atomistic model with Cauchy-Born elasticity model with sharp transition interface. We identify stability conditions that guarantee…
The TRUNC element in any dimension and application to a modified Poisson equation
Hongliang Li, Pingbing Ming, Yinghong Zhou
We introduce a novel TRUNC finite element in n dimensions, encompassing the traditional TRUNC triangle as a particular instance. By establishing the weak continuity identity, we id…
Error Estimate of Multiscale Finite Element Method for Periodic Media Revisited
Pingbing Ming, Siqi Song
We derive the optimal energy error estimate for multiscale finite element method with oversampling technique applying to elliptic system with rapidly oscillating periodic coefficie…
Sharp uniform approximation for spectral Barron functions by deep neural networks
Yulei Liao, Pingbing Ming, Hao Yu
This work explores the neural network approximation capabilities for functions within the spectral Barron space , where is the smoothness index. We demonstrate t…
Heterogeneous multiscale methods for fourth-order singular perturbations
Yulei Liao, Pingbing Ming
We develop a numerical homogenization method for fourth-order singular perturbation problems within the framework of heterogeneous multiscale method. These problems arise from hete…
On the effect of ghost force in the quasicontinuum method: dynamic problems in one dimension
Xiantao Li, Pingbing Ming
Numerical error induced by the "ghost forces" in the quasicontinuum method is studied in the context of dynamic problems. The error in the ({W}^{1,\infty}) norm is analyzed for the…
Deep Nitsche Method: Deep Ritz Method with Essential Boundary Conditions
Yulei Liao, Pingbing Ming
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 functi…
Two robust nonconforming Helements for linear strain gradient elasticity
Hongliang Li, Pingbing Ming, Zhong-ci Shi
We propose two nonconforming finite elements to approximate a boundary value problem arising from strain gradient elasticity, which is a higher-order perturbation of the linearized…
An Efficient Online-Offline Method for Elliptic Homogenization Problems
Yufang Huang, Pingbing Ming, Siqi Song
We present a new numerical method for solving the elliptic homogenization problem. The main idea is that the missing effective matrix is reconstructed by solving the local least-sq…
Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schrödinger Eigenvalue Problems
Hao Yu, Yixiao Guo, Pingbing Ming
We propose a machine learning method for computing eigenvalues and eigenfunctions of the Schrödinger operator on a -dimensional hypercube with Dirichlet boundary conditions. Th…
An Arbitrary-Order Discontinuous Galerkin Method with One Unknown Per Element
Ruo Li, Pingbing Ming, Zhiyuan Sun +1
We propose an arbitrary-order discontinuous Galerkin method for second-order elliptic problem on general polygonal mesh with only one degree of freedom per element. This is achieve…
A pre-training deep learning method for simulating the large bending deformation of bilayer plates
Xiang Li, Yulei Liao, Pingbing Ming
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 seri…
A Concurrent Global-local Numerical Method for Multiscale PDEs
Yufang Huang, Jianfeng Lu, Pingbing Ming
We present a new hybrid numerical method for multiscale partial differential equations, which simultaneously captures the global macroscopic information and resolves the local micr…
Spectral Barron space for deep neural network approximation
Yulei Liao, Pingbing Ming
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 t…
A concurrent global-local numerical method for multiscale parabolic equations
Yulei Liao, Yang Liu, Pingbing Ming
This paper presents a concurrent global-local numerical method for solving multiscale parabolic equations in divergence form. The proposed method employs hybrid coefficient to prov…
Spectral connvergece of random feature method in one dimension
Pingbing Ming, Hao Yu
Among the various machine learning methods solving partial differential equations, the Random Feature Method (RFM) stands out due to its accuracy and efficiency. In this paper, we…
From atomistic model to the Peierls-Nabarro model with -surface for dislocations
Tao Luo, Pingbing Ming, Yang Xiang
The Peierls-Nabarro (PN) model for dislocations is a hybrid model that incorporates the atomistic information of the dislocation core structure into the continuum theory. In this p…
A broken Hardy inequality on finite element space and application to strain gradient elasticity
Yulei Liao, Pingbing Ming
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…
A study on the quasiconinuum approximations of a one-dimensional fracture model
Xiantao Li, Pingbing Ming
We study three quasicontinuum approximations of a lattice model for crack propagation. The influence of the approximation on the bifurcation patterns is investigated. The estimate…
H Korn's Inequality and the Nonconforming Elements for The Strain Gradient Elastic Model
Hongliang Li, Pingbing Ming, Huiyu Wang
We establish a new H2 Korn's inequality and its discrete analog, which greatly simplify the construction of nonconforming elements for a linear strain gradient elastic model. The S…
A Discontinuous Galerkin Method by Patch Reconstruction for Biharmonic Problem
Ruo Li, Pingbing Ming, Zhiyuan Sun +2
We propose a new discontinuous Galerkin method based on the least-squares patch reconstruction for the biharmonic problem. We prove the optimal error estimate of the proposed metho…
Convergence of a force-based hybrid method for atomistic and continuum models in three dimension
Jianfeng Lu, Pingbing Ming
We study a force-based hybrid method that couples atomistic models with nonlinear Cauchy-Born elasticity models. We show that the proposed scheme converges quadratically to the sol…
New Nonconforming Elements for Linear Strain Gradient Elastic Model
Hongliang Li, Pingbing Ming, Zhong-ci Shi
Based on a new HKorn's inequality, we propose new nonconforming elements for the linear strain gradient elastic model. The first group of elements are Hconforming but H$^…
Ghost Force Influence of a Quasicontinuum Method in Two Dimension
Jingrun Chen, Pingbing Ming
We derive an analytical expression for the solution of a two-dimensional quasicontinuum method with a planar interface. The expression is used to prove that the ghost force may lea…
A Deep Learning Method for Computing Eigenvalues of the Fractional Schrödinger Operator
Yixiao Guo, Pingbing Ming
We present a novel deep learning method for computing eigenvalues of the fractional Schrödinger operator. Our approach combines a newly developed loss function with an innovative…
Barron regularity of many particle Schrödinger eigenfunctions
Pingbing Ming, Hao Yu
This work investigates the regularity of Schrödinger eigenfunctions and the solvability of Schrödinger equations in spectral Barron space , wher…
Taylor-Hood like finite elements for nearly incompressible strain gradient elasticity problems
Yulei Liao, Pingbing Ming, Yun Xu
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 el…
A Nitsche Hybrid multiscale method with non-matching grids
Pingbing Ming, Siqi Song
We propose a Nitsche method for multiscale partial differential equations, which retrieves the macroscopic information and the local microscopic information at one stroke. We prove…