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IEEE Transactions on Artificial Intelligence 2024年第3期5卷 1067-1076页

作者： Xing, Baixue Liu, Hongliang Tang, Xiao Shi, Long School of Mathematics and Computational Science Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Xiangtan411105 China School of Computational Science and Electronics Hunan Institute of Engineering Xiangtan411105 China

In view of the fact that impulsive differential equations have the discreteness due to the impulse phenomenon, this article proposes a single hidden layer neural networkmethod-based extreme learning machine and a physics-informed neural network method combined with learning rate attenuation strategy to solve linear impulsive differential equations and nonlinear impulsive differential equations, respectively. For the linear impulsive differential equations, first, the interval is segmented according to the impulse points, and a single hidden layer neural network model is constructed, the weight parameters of the hidden layer are randomly set, the optimal output parameters, and solution of the first segment are obtained by the extreme learning machine algorithm, then we calculate the initial value of the second segment according to the jumping equation and the remaining segments are solved in turn in the same way. Although the single hidden layer neural network method proposed can solve linear equations with high accuracy, it is not suitable for solving nonlinear equations. Therefore, we propose the physics-informed neural network combined with a learning rate attenuation strategy to solve the nonlinear impulsive differential equations, then the Adam algorithm and L-BFGS algorithm are combined to find the optimal approximate solution of each segment. Numerical examples show that the single hidden layer neural network method with Legendre polynomials as the activation function and the physics-informed neural network method combined with learning rate attenuation strategy can solve linear and nonlinear impulsive differential equations with higher accuracy. Impact Statement-It is difficult to obtain the analytical solutions of impulsive differential equations because of the existence of impulse points, and the current numerical methods are complicated and demanding. In recent years, artificial neural network methods have been widely used due to its simplicity and efficie

关键词： Approximation algorithms

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A Deep Learning Method for Computing Eigenvalues of the Fractional Schrödinger Operator

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Journal of Systems Science & Complexity 2024年第2期37卷 391-412页

作者： GUO Yixiao MING Pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

The authors present a novel deep learning method for computing eigenvalues of the fractional Schrödinger *** proposed approach combines a newly developed loss function with an innovative neural network architecture that incorporates prior knowledge of the *** improvements enable the proposed method to handle both high-dimensional problems and problems posed on irregular bounded *** authors successfully compute up to the first 30 eigenvalues for various fractional Schrödinger *** an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

关键词： Eigenvalue problem deep learning fractional Schrödinger operator isospectral problem

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CONVERGENCE ANALYSIS OF SOME FINITE ELEMENT PARALLEL ALGORITHMS FOR THE STATIONARY INCOMPRESSIBLE MHD EQUATIONS

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Journal of computational mathematics 2024年第1期42卷 49-70页

作者： Xiaojing Dong Yinnian He Hunan Key Laboratory for Computation and Simulation in Science and Engineering Key Laboratory of Intelligent Computing&Information Processing of Ministry of EducationSchool of Mathematics and Computational ScienceXiangtan UniversityXiangtan 411105China Institute of Applied Physics and Computational Mathematics Beijing 100088China School of Mathematics and Statistics Xi'an Jiaotong UniversityXi'an 710049China

By combination of iteration methods with the partition of unity method(PUM),some finite element parallel algorithms for the stationary incompressible magnetohydrodynamics(MHD)with different physical parameters are presented and *** algorithms are highly *** first,a global solution is obtained on a coarse grid for all approaches by one of the iteration *** parallelized residual schemes,local corrected solutions are calculated on finer meshes with overlapping *** subdomains can be achieved flexibly by a class of *** proposed algorithm is proved to be uniformly stable and ***,one numerical example is presented to confirm the theoretical findings.

关键词： Partition of unity method Local and parallel algorithm Finite element method Iteration methods Magnetohydrodynamics

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A Derivative-Free Optimization Algorithm Combining Line-Search and Trust-Region Techniques

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Chinese Annals of mathematics,Series B 2023年第5期44卷 719-734页

作者： Pengcheng XIE Ya-xiang YUAN State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematicsand Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesUniversity of Chinese Academy of SciencesBeijing 100190China

The speeding-up and slowing-down(SUSD)direction is a novel direction,which is proved to converge to the gradient descent direction under some *** authors propose the derivative-free optimization algorithm SUSD-TR,which combines the SUSD direction based on the covariance matrix of interpolation points and the solution of the trust-region subproblem of the interpolation model function at the current iteration *** analyze the optimization dynamics and convergence of the algorithm *** of the trial step and structure step are *** results show their algorithm’s efficiency,and the comparison indicates that SUSD-TR greatly improves the method’s performance based on the method that only goes along the SUSD *** algorithm is competitive with state-of-the-art mathematical derivative-free optimization algorithms.

关键词： Nonlinear optimization Derivative-Free Quadratic model Line-Search Trust-Region

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STOCHASTIC TRUST-REGION METHODS WITH TRUST-REGION RADIUS DEPENDING ON PROBABILISTIC MODELS

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Journal of computational mathematics 2022年第2期40卷 294-334页

作者： Xiaoyu Wang Ya-xiang Yuan Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China University of Chinese Academy of Sciences Beijing 100049China State Key Laboratory of Scientific/Engineering Computing Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China

We present a stochastic trust-region model-based framework in which its radius is related to the probabilistic ***,we propose a specific algorithm termed STRME,in which the trust-region radius depends linearly on the gradient used to define the latest *** complexity results of the STRME method in nonconvex,convex and strongly convex settings are presented,which match those of the existing algorithms based on probabilistic *** addition,several numerical experiments are carried out to reveal the benefits of the proposed methods compared to the existing stochastic trust-region methods and other relevant stochastic gradient methods.

关键词： Trust-region methods,Stochastic optimization Probabilistic models Trust-region radius Global convergence

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Solving Time Dependent Fokker-Planck Equations via Temporal Normalizing Flow

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Communications in computational Physics 2022年第7期32卷 401-423页

作者： Xiaodong Feng Li Zeng Tao Zhou LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesBeijingChina

In this work,we propose an adaptive learning approach based on temporal normalizing flows for solving time-dependent Fokker-Planck(TFP)*** is well known that solutions of such equations are probability density functions,and thus our approach relies on modelling the target solutions with the temporal normalizing *** temporal normalizing flow is then trained based on the TFP loss function,without requiring any labeled *** a machine learning scheme,the proposed approach is mesh-free and can be easily applied to high dimensional *** present a variety of test problems to show the effectiveness of the learning approach.

关键词： Temporal normalizing flow Fokker-Planck equations adaptive density approximation

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Convergent and Orthogonality Preserving Schemes for Approximating the Kohn-Sham Orbitals

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Numerical mathematics(Theory,Methods and Applications) 2023年第1期16卷 1-25页

作者： Xiaoying Dai Liwei Zhang Aihui Zhou LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure *** this paper,we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model,with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given *** addition,we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our theory.

关键词： Gradient flow based model density functional theory orthogonality preserving scheme convergence temporal discretization

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MC-Nonlocal-PINNs:Handling Nonlocal Operators in PINNs Via Monte Carlo Sampling

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Numerical mathematics(Theory,Methods and Applications) 2023年第3期16卷 769-791页

作者： Xiaodong Feng Yue Qian Wanfang Shen Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina Shandong Key Laboratory of Blockchain Finance Shandong University of Finance and EconomicsJinan 250014China

We propose Monte Carlo Nonlocal physics-informed neural networks(MC-Nonlocal-PINNs),which are a generalization of MC-fPINNs in *** et al.(*** ***.400(2022),115523)for solving general nonlocal models such as integral equations and nonlocal *** to MC-fPINNs,our MC-Nonlocal-PINNs handle nonlocal operators in a Monte Carlo way,resulting in a very stable approach for high dimensional *** present a variety of test problems,including high dimensional Volterra type integral equations,hypersingular integral equations and nonlocal PDEs,to demonstrate the effectiveness of our approach.

关键词： Nonlocal models PINNs Monte Carlo sampling deep neural networks

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A Stochastic Gradient Descent Method for computational Design of Random Rough Surfaces in Solar Cells

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Communications in computational Physics 2023年第10期34卷 1361-1390页

作者： Qiang Li Gang Bao Yanzhao Cao Junshan Lin Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing 100190China Department of Mathematics Zhejiang UniversityChina Department of Mathematics and Statistics Auburn UniversityAuburnAL 36849USA

In this work,we develop a stochastic gradient descent method for the computational optimal design of random rough surfaces in thin-film solar *** formulate the design problems as random PDE-constrained optimization problems and seek the optimal statistical parameters for the random *** optimizations at fixed frequency as well as at multiple frequencies and multiple incident angles are *** evaluate the gradient of the objective function,we derive the shape derivatives for the interfaces and apply the adjoint state method to perform the *** stochastic gradient descent method evaluates the gradient of the objective function only at a few samples for each iteration,which reduces the computational cost *** numerical experiments are conducted to illustrate the efficiency of the method and significant increases of the absorptance for the optimal random *** also examine the convergence of the stochastic gradient descent algorithm theoretically and prove that the numerical method is convergent under certain assumptions for the random interfaces.

关键词： Optimal design random rough surface solar cell Helmholtz equation stochastic gradient descent method

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An Augmented Two-Scale Finite Element Method for Eigenvalue Problems

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Communications on Applied mathematics and computation 2025年第2期7卷 663-688页

作者： Xiaoying Dai Yunyun Du Fang Liu Aihui Zhou LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of SciencesBeijing100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing100049China School of Statistics and Mathematics Central University of Finance and EconomicsBeijing102206China

In this paper,an augmented two-scale finite element method is proposed for a class of linear and nonlinear eigenvalue problems on tensor-product *** a correction step,the augmented two-scale finite element solution is obtained by solving an eigenvalue problem on a low-dimensional augmented *** analysis and numerical experiments show that the augmented two-scale finite element solution achieves the same order of accuracy as the standard finite element solution on a fine grid,but the computational cost required by the former solution is much lower than that demanded by the *** augmented two-scale finite element method also improves the approximation accuracy of eigenfunctions in the L^(2)(Ω)norm compared with the two-scale finite element method.

关键词： Two-scale Finite element Augmented subspace method Eigenvalue problem Partial differential equation

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