This article reviews the application and progress of deep learning in efficient numerical computing methods.Deep learning,as an important branch of machine learning,provides new ideas for numerical computation by cons...This article reviews the application and progress of deep learning in efficient numerical computing methods.Deep learning,as an important branch of machine learning,provides new ideas for numerical computation by constructing multi-layer neural networks to simulate the learning process of the human brain.The article explores the application of deep learning in solving partial differential equations,optimizing problems,and data-driven modeling,and analyzes its advantages in computational efficiency,accuracy,and adaptability.At the same time,this article also points out the challenges faced by deep learning numerical computation methods in terms of computational efficiency,interpretability,and generalization ability,and proposes strategies and future development directions for integrating with traditional numerical methods.展开更多
We propose a high-order conservative method for the nonlinear Sehodinger/Gross-Pitaevskii equation with time- varying coefficients in modeling Bose Einstein condensation （BEC）. This scheme combined with the sixth-or...We propose a high-order conservative method for the nonlinear Sehodinger/Gross-Pitaevskii equation with time- varying coefficients in modeling Bose Einstein condensation （BEC）. This scheme combined with the sixth-order compact finite difference method and the fourth-order average vector field method, finely describes the condensate wave function and physical characteristics in some small potential wells. Numerical experiments are presented to demonstrate that our numerical scheme is efficient by the comparison with the Fourier pseudo-spectral method. Moreover, it preserves several conservation laws well and even exactly under some specific conditions.展开更多
In the field of discretization-based meshfree/meshless methods,the improvements in the higher-order consistency,stability,and computational efficiency are of great concerns in computational science and numerical solut...In the field of discretization-based meshfree/meshless methods,the improvements in the higher-order consistency,stability,and computational efficiency are of great concerns in computational science and numerical solutions to partial differential equations.Various alternative numerical methods of the finite particle method(FPM)frame have been extended from mathematical theories to numerical applications separately.As a comprehensive numerical scheme,this study suggests a unified resolved program for numerically investigating their accuracy,stability,consistency,computational efficiency,and practical applicability in industrial engineering contexts.The high-order finite particle method(HFPM)and corrected methods based on the multivariate Taylor series expansion are constructed and analyzed to investigate the whole applicability in different benchmarks of computational fluid dynamics.Specifically,four benchmarks are designed purposefully from statical exact solutions to multifaceted hydrodynamic tests,which possess different numerical performances on the particle consistency,numerical discretized forms,particle distributions,and transient time evolutional stabilities.This study offers a numerical reference for the current unified resolved program.展开更多
Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. However, the computation cost generally increases linearly with ...Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. However, the computation cost generally increases linearly with increased order of accuracy. Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite-difference formulas for the first-order derivative, we examine the variation of finite-difference coefficients with accuracy order and note that there exist some very small coefficients. With the order increasing, the number of these small coefficients increases, however, the values decrease sharply. An error analysis demonstrates that omitting these small coefficients not only maintain approximately the same level of accuracy of finite difference but also reduce computational cost significantly. Moreover, it is easier to truncate for the high-order finite-difference formulas than for the pseudospectral for- mulas. Thus this study proposes a truncated high-order finite-difference method, and then demonstrates the efficiency and applicability of the method with some numerical examples.展开更多
We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equa...We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.展开更多
The smoothed particle hydrodynamics（SPH） method is usually expected to be an efficient numerical tool for calculating the fluid-structure interactions in compressors; however, an endogenetic restriction is the probl...The smoothed particle hydrodynamics（SPH） method is usually expected to be an efficient numerical tool for calculating the fluid-structure interactions in compressors; however, an endogenetic restriction is the problem of low-order consistency. A high-order SPH method by introducing inverse kernels, which is quite easy to be implemented but efficient, is proposed for solving this restriction. The basic inverse method and the special treatment near boundary are introduced with also the discussion of the combination of the Least-Square（LS） and Moving-Least-Square（MLS） methods. Then detailed analysis in spectral space is presented for people to better understand this method. Finally we show three test examples to verify the method behavior.展开更多
The time-dependent Schrodinger equation(TDSE) is usually treated in the real space in the textbook.However,it makes the numerical simulations of strong-field processes difficult due to the wide dispersion and fast osc...The time-dependent Schrodinger equation(TDSE) is usually treated in the real space in the textbook.However,it makes the numerical simulations of strong-field processes difficult due to the wide dispersion and fast oscillation of the electron wave packets under the interaction of intense laser fields.Here we demonstrate that the TDSE can be efficiently solved in the momentum space.The high-order harmonic generation and above-threshold ionization spectra obtained by numerical solutions of TDSE in momentum space agree well with previous studies in real space,but significantly reducing the computation cost.展开更多
In this paper,we concentrate on high-order boundary treatments for finite volume methods solving hyperbolic conservation laws.The complex geometric physical domain is covered by a Cartesian mesh,resulting in the bound...In this paper,we concentrate on high-order boundary treatments for finite volume methods solving hyperbolic conservation laws.The complex geometric physical domain is covered by a Cartesian mesh,resulting in the boundary intersecting the grids in various fashions.We propose two approaches to evaluate the cell averages on the ghost cells near the boundary.Both of them start from the inverse Lax-Wendroff(ILW)procedure,in which the normal spatial derivatives at inflow boundaries can be obtained by repeatedly using the governing equations and boundary conditions.After that,we can get an accurate evaluation of the ghost cell average by a Taylor expansion joined with high-order extrapolation,or by a Hermite extrapolation coupling with the cell averages on some“artificial”inner cells.The stability analysis is provided for both schemes,indicating that they can avoid the so-called“small-cell”problem.Moreover,the second method is more efficient under the premise of accuracy and stability.We perform numerical experiments on a collection of examples with the physical boundary not aligned with the grids and with various boundary conditions,indicating the high-order accuracy and efficiency of the proposed schemes.展开更多
文摘This article reviews the application and progress of deep learning in efficient numerical computing methods.Deep learning,as an important branch of machine learning,provides new ideas for numerical computation by constructing multi-layer neural networks to simulate the learning process of the human brain.The article explores the application of deep learning in solving partial differential equations,optimizing problems,and data-driven modeling,and analyzes its advantages in computational efficiency,accuracy,and adaptability.At the same time,this article also points out the challenges faced by deep learning numerical computation methods in terms of computational efficiency,interpretability,and generalization ability,and proposes strategies and future development directions for integrating with traditional numerical methods.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11571366 and 11501570the Open Foundation of State Key Laboratory of High Performance Computing of China+1 种基金the Research Fund of National University of Defense Technology under Grant No JC15-02-02the Fund from HPCL
文摘We propose a high-order conservative method for the nonlinear Sehodinger/Gross-Pitaevskii equation with time- varying coefficients in modeling Bose Einstein condensation （BEC）. This scheme combined with the sixth-order compact finite difference method and the fourth-order average vector field method, finely describes the condensate wave function and physical characteristics in some small potential wells. Numerical experiments are presented to demonstrate that our numerical scheme is efficient by the comparison with the Fourier pseudo-spectral method. Moreover, it preserves several conservation laws well and even exactly under some specific conditions.
基金supported by the National Natural Science Foundation of China(No.12002290)。
文摘In the field of discretization-based meshfree/meshless methods,the improvements in the higher-order consistency,stability,and computational efficiency are of great concerns in computational science and numerical solutions to partial differential equations.Various alternative numerical methods of the finite particle method(FPM)frame have been extended from mathematical theories to numerical applications separately.As a comprehensive numerical scheme,this study suggests a unified resolved program for numerically investigating their accuracy,stability,consistency,computational efficiency,and practical applicability in industrial engineering contexts.The high-order finite particle method(HFPM)and corrected methods based on the multivariate Taylor series expansion are constructed and analyzed to investigate the whole applicability in different benchmarks of computational fluid dynamics.Specifically,four benchmarks are designed purposefully from statical exact solutions to multifaceted hydrodynamic tests,which possess different numerical performances on the particle consistency,numerical discretized forms,particle distributions,and transient time evolutional stabilities.This study offers a numerical reference for the current unified resolved program.
基金supported by China Scholarship Council and partially by the National "863" Program of China under contract No. 2007AA06Z218.
文摘Finite-difference methods with high-order accuracy have been utilized to improve the precision of numerical solution for partial differential equations. However, the computation cost generally increases linearly with increased order of accuracy. Upon examination of the finite-difference formulas for the first-order and second-order derivatives, and the staggered finite-difference formulas for the first-order derivative, we examine the variation of finite-difference coefficients with accuracy order and note that there exist some very small coefficients. With the order increasing, the number of these small coefficients increases, however, the values decrease sharply. An error analysis demonstrates that omitting these small coefficients not only maintain approximately the same level of accuracy of finite difference but also reduce computational cost significantly. Moreover, it is easier to truncate for the high-order finite-difference formulas than for the pseudospectral for- mulas. Thus this study proposes a truncated high-order finite-difference method, and then demonstrates the efficiency and applicability of the method with some numerical examples.
基金supported by National Science Foundation of China(52171251)Liao Ning Revitalization Talents Program(XLYC1907014)+2 种基金the Fundamental Research Funds for the Central Universities(DUT21ZD205)Ministry of Industry and Information Technology(2019-357)the Project of State Key Laboratory of Satellite Ocean Environment Dynamics,Second Institute of Oceanography,MNR(QNHX2112)。
文摘We propose an effective scheme of the deep learning method for high-order nonlinear soliton equations and explore the influence of activation functions on the calculation results for higherorder nonlinear soliton equations. The physics-informed neural networks approximate the solution of the equation under the conditions of differential operator, initial condition and boundary condition. We apply this method to high-order nonlinear soliton equations, and verify its efficiency by solving the fourth-order Boussinesq equation and the fifth-order Korteweg–de Vries equation. The results show that the deep learning method can be used to solve high-order nonlinear soliton equations and reveal the interaction between solitons.
基金funding from the European Community’s Seventh Framework Program (FP7/2007-2013) under grant agreement 225967 ‘‘Next Mu SE”supported by the National Natural Science Foundation of China (Nos. 11202013, 11572025 and 51420105008)
文摘The smoothed particle hydrodynamics（SPH） method is usually expected to be an efficient numerical tool for calculating the fluid-structure interactions in compressors; however, an endogenetic restriction is the problem of low-order consistency. A high-order SPH method by introducing inverse kernels, which is quite easy to be implemented but efficient, is proposed for solving this restriction. The basic inverse method and the special treatment near boundary are introduced with also the discussion of the combination of the Least-Square（LS） and Moving-Least-Square（MLS） methods. Then detailed analysis in spectral space is presented for people to better understand this method. Finally we show three test examples to verify the method behavior.
基金Project supported by the National Key Research and Development Program of China(Grant No.2019YFA0307702)the National Natural Science Foundation of China(Grants Nos.91850121 and 11674363)+1 种基金the Science Fund of Educational Department of Henan Province of China(Grant No.2011C140001)the Ninth Group of Key Disciplines in Henan Province,China(Grant No.2018119).
文摘The time-dependent Schrodinger equation(TDSE) is usually treated in the real space in the textbook.However,it makes the numerical simulations of strong-field processes difficult due to the wide dispersion and fast oscillation of the electron wave packets under the interaction of intense laser fields.Here we demonstrate that the TDSE can be efficiently solved in the momentum space.The high-order harmonic generation and above-threshold ionization spectra obtained by numerical solutions of TDSE in momentum space agree well with previous studies in real space,but significantly reducing the computation cost.
基金supported in part by the NSFC Grant 12271499supported in part by the R&D project of Pazhou Lab(Huangpu)under Grant 2023K0609 and the NSFC Grant 12126604.
文摘In this paper,we concentrate on high-order boundary treatments for finite volume methods solving hyperbolic conservation laws.The complex geometric physical domain is covered by a Cartesian mesh,resulting in the boundary intersecting the grids in various fashions.We propose two approaches to evaluate the cell averages on the ghost cells near the boundary.Both of them start from the inverse Lax-Wendroff(ILW)procedure,in which the normal spatial derivatives at inflow boundaries can be obtained by repeatedly using the governing equations and boundary conditions.After that,we can get an accurate evaluation of the ghost cell average by a Taylor expansion joined with high-order extrapolation,or by a Hermite extrapolation coupling with the cell averages on some“artificial”inner cells.The stability analysis is provided for both schemes,indicating that they can avoid the so-called“small-cell”problem.Moreover,the second method is more efficient under the premise of accuracy and stability.We perform numerical experiments on a collection of examples with the physical boundary not aligned with the grids and with various boundary conditions,indicating the high-order accuracy and efficiency of the proposed schemes.
