Hehu Xie

Toward Pólya's Conjecture: Improving the Individual Li-Yau Bound via Energy Orthogonality

Yifan Wang, Hehu Xie

We establish two complementary lower-bound mechanisms for individual eigenvalues of the Dirichlet Laplacian. First, energy orthogonality yields a frequency-dependent cap on the Fou…

Mini mixed finite element method for nearly incompressible linear elasticity problems

Zhijin Guan, Yue Feng, Hehu Xie +1

This paper addresses the numerical solution of nearly incompressible linear elasticity boundary value problems and their associated eigenvalue problems. A mixed finite element form…

A Tensor Neural Network Method for High-Order Homogenization of Locally Periodic Elliptic Problems

Huaijia Zhang, Haochen Liu, Xia Ji +1

We develop a high-order tensor neural network (TNN) method for locally periodic elliptic multiscale problems of the form $- \nabla \cdot (A (x, x / ε) \nabla u_{ε}) = f$ . B…

Quadrature-Enhanced Monte Carlo fPINN Method for High-Dimensional Fractional PDEs

Qingkui Ma, Hehu Xie, Xiaobo Yin

Fractional PDEs involving the fractional Laplacian on bounded domains are challenging because of hypersingular nonlocal kernels, exterior Dirichlet constraints, reduced boundary re…

Regularity Analysis and Tensor Neural Network Methods for Quasiperiodic Elliptic Equations

Jingze Ren, Yifan Wang, Hehu Xie +1

This paper investigates quasiperiodic elliptic equations using a well-established projection method, by which the original problem is reformulated as a degenerate periodic variatio…

Continuous Finite Element Method For Maxwell Eigenvalue Problems With Regular Decomposition Technique

Feiyi Liao, Haochen Liu, Hehu Xie

With the regular decomposition technique, we decompose the space $H_{0}^{s} (curl; Ω)$ into the sum of a vector potential space and the gradient of a scalar space, bot…

A Robust GPU-Accelerated Kernel Compensation Solver with Novel Discretization for Photonic Crystals in Anisotropic Media

Chenhao Jin, Hehu Xie

This paper develops a robust solver for the Maxwell eigenproblem in 3D photonic crystals with anisotropic media. The solver employs the kernel compensation technique under the fram…

The Weak Galerkin and Crouzeix-Raviart element method for elastic eigenvalue problems

Wei Lu, Hehu Xie, Qilong Zhai

In this paper, we first introduce an abstract framework to solve the eigenvalue problem by weak Galerkin (WG) method. By the application of the framework, WG method is proved to be…

Neural Network Element Method for Partial Differential Equations

Yifan Wang, Zhongshuo Lin, Hehu Xie

In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are design…

PASE: A Massively Parallel Augmented Subspace Eigensolver for Large Scale Eigenvalue Problems

Yangfei Liao, Haochen Liu, Hehu Xie +1

In this paper, we present a novel parallel augmented subspace method and build a package Parallel Augmented Subspace Eigensolver (PASE) for solving large scale eigenvalue problems…

math.NA2024★ 1 cited

Adaptive Neural Network Subspace Method for Solving Partial Differential Equations with High Accuracy

Zhongshuo Lin, Yifan Wang, Hehu Xie

Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopt…

FieldTNN-based machine learning method for Maxwell eigenvalue problems

Jiantao Jiang, Yanli Wang, Yifan Wang +1

The aim of this paper is to introduce a FieldTNN-based machine learning method for solving the Maxwell eigenvalue problem in both 2D and 3D domains, including both tensor and non-t…

Error estimates of finite element methods for nonlocal problems using exact or approximated interaction neighborhoods

Qiang Du, Hehu Xie, Xiaobo Yin +1

We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. Th…

Enhanced Error Estimates for Augmented Subspace Method with Crouzeix-Raviart Element

Zhijin Guan, Yifan Wang, Hehu Xie +1

In this paper, we present some enhanced error estimates for augmented subspace methods with the nonconforming Crouzeix-Raviart (CR) element. Before the novel estimates, we derive t…

A Nonnested Augmented Subspace Method for Kohn-Sham Equation

Guanghui Hu, Hehu Xie, Fei Xu +1

In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented sub…

math.NA2024★ 2 cited

Tensor Neural Network Interpolation and Its Applications

Yongxin Li, Zhongshuo Lin, Yifan Wang +1

Based on tensor neural network, we propose an interpolation method for high dimensional non-tensor-product-type functions. This interpolation scheme is designed by using the tensor…

Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems

Zhongshuo Lin, Haochen Liu, Hehu Xie

In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direc…

Solving High-dimensional Parametric Elliptic Equation Using Tensor Neural Network

Hongtao Chen, Rui Fu, Yifan Wang +1

In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded phys…