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A Deep Learning Method for Computing Eigenvalues of the Fractional Schr?dinger Operator

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Journal of Systems Science & Complexity 2024年

作者： GUO Yixiao ming pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of Sciences School of Mathematical Sciences University of Chinese Academy of Sciences

The authors present a novel deep learning method for computing eigenvalues of the fractional Schr?dinger *** 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 Schrodinger *** an application,the authors share a conjecture to the fractional order isospectral problem that has not yet been studied.

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A New Function Space from Barron Class and Application to Neural Network Approximation

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Communications in Computational Physics 2022年第10期32卷 1361-1400页

作者： Yan Meng pingbing ming School of Mathematics Renmin University of ChinaBeijing 100872China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesNo.55East Road Zhong-Guan-CunBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China.

We introduce a new function space,dubbed as the Barron spectrum space,which arises from the target function space for the neural network *** give a Bernstein type sufficient condition for functions in this space,and clarify the embedding among the Barron spectrum space,the Bessel potential space,the Besov space and the Sobolev ***,the unexpected smoothness and the decaying behavior of the radial functions in the Barron spectrum space have been *** an application,we prove a dimension explicit L^(q) error bound for the two-layer neural network with the Barron spectrum space as the target function space,the rate is dimension independent.

关键词： Fourier transform Besov space Sobolev space radial function neural network

GHOST FORCE INFLUENCE OF A QUASICONTINUUM METHOD IN TWO DIMENSION

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Journal of Computational Mathematics 2012年第6期30卷 657-683页

作者： Jingrun Chen pingbing ming South Hall 6705 Mathematics Department University of California Santa Barbara CA93106 USA ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China LSEC ICMSEC Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 China

The error caused by the ghost force is studied for a quasicontinuum method with planar interface in two dimension. For a special case, we derive an analytical expression of the error, which is exploited to prove that the ghost force may lead to a finite size error for the gradient of the solution. The pointwise estimate of the error shows that the error decays algebraically away from the interface, which is much slower than that of the one- dimensional problem, for which the error decays exponentially away from the interface.

关键词： Quasicontinuum method Ghost force Interracial layer.

Specht Triangle Approximation of Large Bending Isometries

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Annals of Applied Mathematics 2023年第4期39卷 544-569页

作者： Xiang Li pingbing ming LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesNo.55East Road Zhong-Guan-CunBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

We propose a Specht triangle discretization for a geometrically nonlinear Kirchhoff plate model with large bending isometry.A combination of an adaptive time-stepping gradient flow and a Newton’s method is employed to solve the ensuing nonlinear minimization problem.Γ−convergence of the Specht triangle discretization and the unconditional stability of the gradient flow algorithm are *** present several numerical examples to demonstrate that our approach significantly enhances both the computational efficiency and accuracy.

关键词： Specht triangle plate bending isometry constraint adaptive time-stepping gradient flow

On the Effect of Ghost Force in the Quasicontinuum Method:Dynamic Problems in One Dimension

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Communications in Computational Physics 2014年第3期15卷 647-676页

作者： Xiantao Li pingbing ming Department of Mathematics the Pennsylvania State UniversityUniversity ParkPennsylvania16802USA LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesNo.55East Road Zhong-Guan-CunBeijing100190P.R.China

Numerical error caused by“ghost forces”in a quasicontinuum method is studied in the context of dynamic *** error in the discrete W^(1,∞)norm is analyzed for the time scale O(ε)and the time scale O(1)withεbeing th... 详细信息

关键词： Quasicontinuum methods ghost force dynamic problems

An Efficient Multigrid Method for Molecular Mechanics Modeling in Atomic Solids

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Communications in Computational Physics 2011年第6期10卷 70-89页

作者： Jingrun Chen pingbing ming Institute of Computational Mathematics and Scientific/Engineering Computing AMSSChinese Academy of SciencesBeijing 100190China LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesBeijing 100190China

We propose a multigrid method to solve the molecular mechanics model(molecular dynamics at zero temperature).The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state as the initial guess of the molecular mechanics *** efficiency of the algorithm is demonstrated by three examples with homogeneous deformation,namely,one dimensional chain under tensile deformation and aluminum under tension and shear *** method exhibits linear-scaling computational complexity,and is insensitive to parameters arising from iterative *** addition,we study two examples with inhomogeneous deformation:vacancy and nanoindentation of *** results are still satisfactory while the linear-scaling property is lost for the latter example.

关键词： Multigrid method linear-scaling algorithm Cauchy-Born rule nanoindentation

Cascadic multigrid methods for parabolic problems

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Science China Mathematics 2008年第8期51卷 1415-1439页

作者： DU Qiang ming pingbing Department of Mathematics Pennsylvania State UniversityUniversity ParkPA 16802USA LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAMSSChinese Academy of SciencesBeijing 100190China

In this paper,we consider the cascadic multigrid method for a parabolic type *** Euler approximation in time and linear finite element approximation in space are employed.A stability result is established under some conditions on the *** new and sharper estimates for the smoothers that reflect the precise dependence on the time step and the spatial mesh parameter,these conditions are verified for a number of popular *** error bound sare derived for both smooth and non-smooth *** strategies guaranteeing both the optimal accuracy and the optimal complexity are presented.

关键词： cascadic multigrid method parabolic problem finite element methods backward Euler scheme smoother stability optimal error order optimal complexity

The efect of ghost forces for a quasicontinuum method in three dimension

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Science China Mathematics 2013年第12期56卷 2571-2589页

作者： CUI Long ming pingbing LSEC Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems ScienceChinese Academy of Sciences

We study the efect of"ghost forces"for a quasicontinuum method in three dimension with a planar interface."Ghost forces"are the inconsistency of the quasicontinuum method across the interface between the atomistic region and the continuum *** results suggest that"ghost forces"may lead to a negilible error on the solution,while lead to a fnite size error on the gradient of the *** error has a layer-like profle,and the interfacial layer width is of O(ε).The error in certain component of the displacement gradient decays algebraically from O(1)to O(ε)away from the interface.A surrogate model is proposed and analyzed,which suggests the same scenario for the efect of"ghost forces".Our analysis is based on the explicit solution of the surrogate model.

关键词：连续介质方法维度生日尺寸误差位移梯度代理模型计算结果界面层

A Family of Nonconforming Rectangular Elements for Strain Gradient Elasticity

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Advances in Applied Mathematics and Mechanics 2019年第6期11卷 1263-1286页

作者： Yulei Liao pingbing ming School of Mathematics Renmin University of ChinaNo.59Zhongguancun StreetBeijing 100872China LSEC Institute of Computational MathematicsAMSS Chinese Academy of SciencesNo.55East Road ZhongguancunBeijing 100190Chinaand School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijing 100049China

We propose a family of nonconforming rectangular elements for the linear strain gradient elastic *** error estimates uniformly with respect to the small material parameter have been *** results confirm the theoretical... 详细信息

关键词： Nonconforming finite elements strain gradient elasticity uniform error estimate