In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equa...In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equations and four-order Lagrangian equations are obtained from the high-order Hamilton＇s principle. Finally, the Hamilton＇s principle of high-order Lagrangian function is given.展开更多
An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstr...An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, f^om the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.展开更多
A multifractal model is developed to connect the Lagrangian multifractal dimensions with their Eulerian counterparts. We propose that the characteristic time scale of a Lagrangian quantity should be the Lagrangian tim...A multifractal model is developed to connect the Lagrangian multifractal dimensions with their Eulerian counterparts. We propose that the characteristic time scale of a Lagrangian quantity should be the Lagrangian time scale, and it should not be the Eulerian time scale which was widely used in previous studies on Lagrangian statistics. Using the present model, we can obtain the scaling exponents of Lagrangian velocity structure functions from the existing data or models of scaling exponents of Eulerian velocity structure functions. This model is validated by comparing its prediction with the results of experiments, direct numerical simulations, and the previous theoretical models. The comparison shows that the proposed model can better predict the scaling exponents of Lagrangian velocity structure functions, especially for orders larger than 6.展开更多
In this paper, a new augmented Lagrangian penalty function for constrained optimization problems is studied. The dual properties of the augmented Lagrangian objective penalty function for constrained optimization prob...In this paper, a new augmented Lagrangian penalty function for constrained optimization problems is studied. The dual properties of the augmented Lagrangian objective penalty function for constrained optimization problems are proved. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker (KKT) condition. Especially, when the KKT condition holds for convex programming its saddle point exists. Based on the augmented Lagrangian objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions.展开更多
Tensor robust principal component analysis（TRPCA） problem aims to separate a low-rank tensor and a sparse tensor from their sum. This problem has recently attracted considerable research attention due to its wide ra...Tensor robust principal component analysis（TRPCA） problem aims to separate a low-rank tensor and a sparse tensor from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications in computer vision and pattern recognition. In this paper, we propose a new model to deal with the TRPCA problem by an alternation minimization algorithm along with two adaptive rankadjusting strategies. For the underlying low-rank tensor, we simultaneously perform low-rank matrix factorizations to its all-mode matricizations; while for the underlying sparse tensor,a soft-threshold shrinkage scheme is applied. Our method can be used to deal with the separation between either an exact or an approximate low-rank tensor and a sparse one. We established the subsequence convergence of our algorithm in the sense that any limit point of the iterates satisfies the KKT conditions. When the iteration stops, the output will be modified by applying a high-order SVD approach to achieve an exactly low-rank final result as the accurate rank has been calculated. The numerical experiments demonstrate that our method could achieve better results than the compared methods.展开更多
物理信息神经网络(PINNs)作为人工智能助力科学研究(AI for Science)求解偏微分方程(PDEs)的一种无网格化求解框架,近年来受到广泛关注.然而,传统PINNs存在局限性:一方面,PINNs网络结构使用单向信息传递的多层感知机(MLPs),难以有效聚...物理信息神经网络(PINNs)作为人工智能助力科学研究(AI for Science)求解偏微分方程(PDEs)的一种无网格化求解框架,近年来受到广泛关注.然而,传统PINNs存在局限性:一方面,PINNs网络结构使用单向信息传递的多层感知机(MLPs),难以有效聚焦序列数据中蕴含的关键特征,信息表征能力弱;另一方面,PINNs的损失函数为嵌入物理约束的二次罚函数,其未受约束而无限膨胀的惩罚因子影响模型训练寻优效率.为应对上述挑战,本文提出一种基于信息表征-损失优化改进的PINNs——allaPINNs,旨在增强模型关键特征提取和训练寻优能力,提升其求解PDEs数值解的准确性和泛化能力.在信息表征方面,allaPINNs引入高效线性注意力(LA)增强模型关键特征识别能力,同时降低权重动态加权的计算复杂度.在损失优化方面,allaPINNs通过引入增广拉格朗日(AL)函数重构目标损失函数,利用可学习的拉格朗日乘子和惩罚因子有效调控各损失残差项的相互作用.由Helmholtz,Black-Scholes,Burgers和非线性Schrödinger四个基准方程验证allaPINNs的有效性.结果表明,allaPINNs能够有效求解不同类型PDEs,并展现出卓越的数值解预测精度与泛化能力.相较于当前先进PINNs,其预测精度提升一至两个数量级.展开更多
Path integral technique is discussed using Hamilton Jacobi method. The Hamilton Jacobi function of non-natural Lagrangian is obtained using separation of variables method. This function makes an important role in path...Path integral technique is discussed using Hamilton Jacobi method. The Hamilton Jacobi function of non-natural Lagrangian is obtained using separation of variables method. This function makes an important role in path integral quantization. The path integral is obtained as integration over the canonical phase space coordinates, which contains the generalized coordinate q and the generalized momentum p. One illustrative example is considered to explain the application of our formalism.展开更多
A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution functi...A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution function consists of macroscopic Lagrangian variables at time steps n and n + 1. It is different from the standard lattice Boltzmann method. In this method the element, instead of each particle, is required to satisfy the basic law. The element is considered as one large particle, which results in simpler version than the corresponding Eulerian one, because the advection term disappears here. Our numerical examples successfully reproduce the classical results.展开更多
基金the Natural Science Foundation of Jiangxi Provincethe Foundation of Education Department of Jiangxi Province under Grant No.[2007]136
文摘In this paper, based on the theorem of the high-order velocity energy, integration and variation principle, the high-order Hamilton＇s principle of general holonomic systems is given. Then, three-order Lagrangian equations and four-order Lagrangian equations are obtained from the high-order Hamilton＇s principle. Finally, the Hamilton＇s principle of high-order Lagrangian function is given.
文摘An exact augmented Lagrangian function for the nonlinear nonconvex programming problems with inequality constraints was discussed. Under suitable hypotheses, the relationship was established between the local unconstrained minimizers of the augmented Lagrangian function on the space of problem variables and the local minimizers of the original constrained problem. Furthermore, under some assumptions, the relationship was also established between the global solutions of the augmented Lagrangian function on some compact subset of the space of problem variables and the global solutions of the constrained problem. Therefore, f^om the theoretical point of view, a solution of the inequality constrained problem and the corresponding values of the Lagrange multipliers can be found by the well-known method of multipliers which resort to the unconstrained minimization of the augmented Lagrangian function presented.
基金supported by the National Natural Science Foundation of China(11072247,11021262,and 11232011)National Natural Science Associate Foundation of China(NSAF)(U1230126)973 program of China(2013CB834100)
文摘A multifractal model is developed to connect the Lagrangian multifractal dimensions with their Eulerian counterparts. We propose that the characteristic time scale of a Lagrangian quantity should be the Lagrangian time scale, and it should not be the Eulerian time scale which was widely used in previous studies on Lagrangian statistics. Using the present model, we can obtain the scaling exponents of Lagrangian velocity structure functions from the existing data or models of scaling exponents of Eulerian velocity structure functions. This model is validated by comparing its prediction with the results of experiments, direct numerical simulations, and the previous theoretical models. The comparison shows that the proposed model can better predict the scaling exponents of Lagrangian velocity structure functions, especially for orders larger than 6.
文摘In this paper, a new augmented Lagrangian penalty function for constrained optimization problems is studied. The dual properties of the augmented Lagrangian objective penalty function for constrained optimization problems are proved. Under some conditions, the saddle point of the augmented Lagrangian objective penalty function satisfies the first-order Karush-Kuhn-Tucker (KKT) condition. Especially, when the KKT condition holds for convex programming its saddle point exists. Based on the augmented Lagrangian objective penalty function, an algorithm is developed for finding a global solution to an inequality constrained optimization problem and its global convergence is also proved under some conditions.
基金Supported by the National Natural Science Foundation of China(Grant Nos.6157209961320106008+2 种基金91230103)National Science and Technology Major Project(Grant Nos.2013ZX040050212014ZX04001011)
文摘Tensor robust principal component analysis（TRPCA） problem aims to separate a low-rank tensor and a sparse tensor from their sum. This problem has recently attracted considerable research attention due to its wide range of potential applications in computer vision and pattern recognition. In this paper, we propose a new model to deal with the TRPCA problem by an alternation minimization algorithm along with two adaptive rankadjusting strategies. For the underlying low-rank tensor, we simultaneously perform low-rank matrix factorizations to its all-mode matricizations; while for the underlying sparse tensor,a soft-threshold shrinkage scheme is applied. Our method can be used to deal with the separation between either an exact or an approximate low-rank tensor and a sparse one. We established the subsequence convergence of our algorithm in the sense that any limit point of the iterates satisfies the KKT conditions. When the iteration stops, the output will be modified by applying a high-order SVD approach to achieve an exactly low-rank final result as the accurate rank has been calculated. The numerical experiments demonstrate that our method could achieve better results than the compared methods.
文摘物理信息神经网络(PINNs)作为人工智能助力科学研究(AI for Science)求解偏微分方程(PDEs)的一种无网格化求解框架,近年来受到广泛关注.然而,传统PINNs存在局限性:一方面,PINNs网络结构使用单向信息传递的多层感知机(MLPs),难以有效聚焦序列数据中蕴含的关键特征,信息表征能力弱;另一方面,PINNs的损失函数为嵌入物理约束的二次罚函数,其未受约束而无限膨胀的惩罚因子影响模型训练寻优效率.为应对上述挑战,本文提出一种基于信息表征-损失优化改进的PINNs——allaPINNs,旨在增强模型关键特征提取和训练寻优能力,提升其求解PDEs数值解的准确性和泛化能力.在信息表征方面,allaPINNs引入高效线性注意力(LA)增强模型关键特征识别能力,同时降低权重动态加权的计算复杂度.在损失优化方面,allaPINNs通过引入增广拉格朗日(AL)函数重构目标损失函数,利用可学习的拉格朗日乘子和惩罚因子有效调控各损失残差项的相互作用.由Helmholtz,Black-Scholes,Burgers和非线性Schrödinger四个基准方程验证allaPINNs的有效性.结果表明,allaPINNs能够有效求解不同类型PDEs,并展现出卓越的数值解预测精度与泛化能力.相较于当前先进PINNs,其预测精度提升一至两个数量级.
基金This work is supported by National Science Foundation of PRC grant under 10 1710 5 5 and Nature Science Foundation of Shandong province granted under (Q98A0 8116 )
文摘Path integral technique is discussed using Hamilton Jacobi method. The Hamilton Jacobi function of non-natural Lagrangian is obtained using separation of variables method. This function makes an important role in path integral quantization. The path integral is obtained as integration over the canonical phase space coordinates, which contains the generalized coordinate q and the generalized momentum p. One illustrative example is considered to explain the application of our formalism.
文摘A Lagrangian lattice Boltzmann method for solving Euler equations is proposed. The key step in formulating this method is the introduction of the displacement distribution function. The equilibrium distribution function consists of macroscopic Lagrangian variables at time steps n and n + 1. It is different from the standard lattice Boltzmann method. In this method the element, instead of each particle, is required to satisfy the basic law. The element is considered as one large particle, which results in simpler version than the corresponding Eulerian one, because the advection term disappears here. Our numerical examples successfully reproduce the classical results.
