Fourier三角基神经元网络的权值直接确定法

根据Fourier变换理论,本文构造出一类基于三角正交基的前向神经网络模型。该模型由输入层、隐层、输出层构成,其输入层和输出层采用线性激励函数,以一组三角正交基为其隐层神经元的激励函数。依据误差回传算法(即BP算法),推导了权值修正的迭代公式。针对BP迭代法收敛速度慢、逼近目标函数精度较低的缺点,进一步提出基于伪逆的权值直接确定法,该方法避免了权值反复迭代的冗长过程。仿真和预测结果表明,该方法比传统的BP迭代法具有更快的计算速度和更高的仿真与测试精度。     

基函数神经网络权值直接确定的图像复原

给出了基函数神经网络图像复原的模型,该神经网络模型是由三层构成的前向神经网络,以一组正交基为隐层神经元的激励函数。为了避免反复迭代权值修正的冗长BP训练过程,提出了一种权值直接确定的算法。实验结果表明,该种权值直接确定算法不仅能一步确定权值而获得更快的运算速度,而且能达到更高的精度。     

多输入La}uerre正交多项式前向神经网络权值与结构确定法

为克服邵神经网络模型及其学习算法中的固有缺陷,根据多项式插值和逼近理论,构造出一种以工;agucrre正交多项式作为隐层神经元激励函数的多输入前向神经网络模型。针对该网络模型,提出了权值与结构确定法,以便快速、自动地确定该网络的最优权值和最优结构。计算机仿真与实验结果显示:该算法是有效的,并且通过该算法所得到的网络具有较优的逼近性能和良好的去噪能力。     

切比雪夫正交基神经网络的权值直接确定法

经典的BP神经网络学习算法是基于误差回传的思想.而对于特定的网络模型,采用伪逆思想可以直接确定权值进而避免以往的反复迭代修正的过程.根据多项式插值和逼近理论构造一个切比雪夫正交基神经网络,其模型采用三层结构并以一组切比雪夫正交多项式函数作为隐层神经元的激励函数.依据误差回传(BP)思想可以推导出该网络模型的权值修正迭代公式,利用该公式迭代训练可得到网络的最优权值.区别于这种经典的做法,针对切比雪夫正交基神经网络模型,提出了一种基于伪逆的权值直接确定法,从而避免了传统方法通过反复迭代才能得到网络权值的冗长训练过程.仿真结果表明该方法具有更快的计算速度和至少相同的工作精度,从而验证了其优越性.     

SIMO傅里叶三角基神经网络的权值直接确定法和结构自确定算法

根据傅里叶级数逼近理论,将正交三角函数系作为隐层神经元激励函数,合理选取这些激励函数的周期参数,构造单输入多输出(SIMO)傅里叶三角基神经网络模型.根据该网络的特点,推导出一种基于伪逆的权值直接确定法,从而1步计算出网络最优权值,并在此基础上设计出隐层结构自确定算法.仿真结果表明,与传统BP(反向传播)神经网络及基于最小二乘法的SIMO傅里叶神经网络模型相比,本网络模型具有更高的计算精度和更快的计算速度.     

基于权值与结构确定法的单极Sigmoid神经网络分类器

构造了以单极Sigmoid函数作为隐层神经元激励函数的神经网络分类器,网络中输入层到隐层的权值和隐层神经元的阈值均为随机生成。同时,结合利用伪逆思想一步计算出隐层和输出层神经元之间连接权值的权值直接确定(WDD)法,进一步提出了具有边增边删和二次删除策略的网络结构自确定法,用来确定神经网络最优权值和结构。数值实验结果表明,该算法能够快速有效地确定单极Sigmoid激励函数神经网络分类器的最优网络结构; 分类器的分类性能良好。     

幂激励前向神经网络最优结构确定算法

针对一种以幂函数序列为各隐神经元激励函数的前向神经网络,提出了一种基于权值直接确定方法的网络最优结构确定算法。计算机仿真与验证结果表明,该算法能自动、快速、有效地确定网络的最优隐神经元数,达到网络的最佳逼近能力,从而实现网络结构的最优化。     

基于Hermite神经网络的动态手势学习和识别

为提高动态手势学习速度和识别准确率,本文提出一种基于Hermite正交基前向神经网络的动态手势识别方法。利用Camshift算法实时跟踪手势运动轨迹,提取手势特征向量作为神经网络的输入;以Hermite正交基函数作为隐含层激励函数构造三层前向神经网络,并给出一种基于伪逆的直接计算权值方法和根据网络目标精度要求自适应确定隐含节点数目方法;运用训练好的Hermite神经网络识别动态手势。测试结果表明:Hermite神经网络能够提高网络的学习训练速度和精度,提高手势学习速度和识别准确率,而且在手势识别方面具有较好的鲁棒性和泛化能力。     

基于Fourier神经网络的图像复原算法*

由于退化图像的点扩散函数难以准确确定,提出一种基于Fourier正交基函数的前向神经网络图像复原模型,该模型以一组Fourier正交基为隐层神经元的激励函数,根据误差传递算法进行权值修正,达到收敛目标。给出Fourier神经网络及其相应的衍生算法的图像恢复实现步骤。实验表明,该方法能较好地实现图像的复原。     

基于快速确定隐层神经元数的BP神经网络算法

根据多项式理论构造一种以正交多项式作为隐层神经元激活函数的PP神经网络模型。针对该网络提出一种算法,即一种隐层的激励函数为正交多项式及其神经元数目可快速确定的算法。首先通过数学证明从理论上验证了该算法的有效性。然后利用计算机对该算法进行仿真与校验,并与传统的PP算法进行比较。结果表明该算法不仅突破了传统PP神经网络的局限性,如收敛速率慢、最佳隐神经元数难确定等,而且能够达到更高的工作精度,从而从实验上验证了该算法的有效性。     

Application of Taylor series in obtaining the orthogonal operational matrix

In this research first we explicitly obtain the relation between the coefficients of the Taylor series and Jacobi polynomial expansions. Then we present a new method for computing classical operational matrices (derivative, integral and product) for general Jacobi orthogonal functions (polynomial and rational). This method can be used for many classes of orthogonal functions.     

Note on Jacobi’s method for approximating dominant roots

In 1834 Jacobi gave a method for approximating dominant roots of a polynomial. In 2002 Mignotte and Stefanescu showed that Jacobi’s method works only when the dominant roots are simple. In this note, we show that Jacobi’s method can still be useful even when the dominant roots are not simple, if we use it for approximating the “distinct” dominant roots.     

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi’s elliptic functions.For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi’s sn or cn functions. Examples illustrate key steps of the algorithms.The new algorithms are implemented in Mathematica. The package PDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed.A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.     

三角域上双变量Chebyshev 多项式及其与Bernstein 基的转换

为了更好的解决三角域上的Bézier 曲面在CAGD 中的最佳一致逼近问题, 构造出了三角域上的双变量Chebyshev 正交多项式,研究了与单变量Chebyshev 多项式相类 似的性质,并且给出了三角域上双变量Chebyshev 基和Bernstein 基的相互转换矩阵。通过 实例比较双变量Chebyshev 多项式与双变量Bernstein 多项式以及双变量Jacobi 多项式的最 小零偏差的大小,阐述了双变量Chebyshev 多项式的最小零偏差性。     

On a multivariable extension of Jacobi matrix polynomials

The classical Jacobi matrix polynomials only for commutative matrices were first studied by Defez et al. [E. Defez, L. Jódar, A. Law. Jacobi matrix differential equation, polynomial solutions and their properties, Comput. Math. Appl. 48 (2004) 789–803]. The main aim of this paper is to construct a multivariable extension with the help of the classical Jacobi matrix polynomials (JMPs). Generating matrix functions and recurrence relations satisfied by these multivariable matrix polynomials are derived. Furthermore, general families of multilinear and multilateral generating matrix functions are obtained and their applications are presented.     

Fractional Jacobi Galerkin spectral schemes for multi-dimensional time fractional advection–diffusion–reaction equations

One of the ongoing issues with time fractional diffusion models is the design of efficient high-order numerical schemes for the solutions of limited regularity. We construct in this paper two efficient Galerkin spectral algorithms for solving multi-dimensional time fractional advection–diffusion–reaction equations with constant and variable coefficients. The model solution is discretized in time with a spectral expansion of fractional-order Jacobi orthogonal functions. For the space discretization, the proposed schemes accommodate high-order Jacobi Galerkin spectral discretization. The numerical schemes do not require imposition of artificial smoothness assumptions in time direction as is required for most methods based on polynomial interpolation. We illustrate the flexibility of the algorithms by comparing the standard Jacobi and the fractional Jacobi spectral methods for three numerical examples. The numerical results indicate that the global character of the fractional Jacobi functions makes them well-suited to time fractional diffusion equations because they naturally take the irregular behavior of the solution into account and thus preserve the singularity of the solution.

     

Numerical solution of fractional delay differential equation by shifted Jacobi polynomials

In this paper, the fractional delay differential equation (FDDE) is considered for the purpose to develop an approximate scheme for its numerical solutions. The shifted Jacobi polynomial scheme is used to solve the results by deriving operational matrix for the fractional differentiation and integration in the Caputo and Riemann–Liouville sense, respectively. In addition to it, the Jacobi delay coefficient matrix is developed to solve the linear and nonlinear FDDE numerically. The error of the approximate solution of proposed method is discussed by applying the piecewise orthogonal technique. The applicability of this technique is shown by several examples like a mathematical model of houseflies and a model based on the effect of noise on light that reflected from laser to mirror. The obtained numerical results are tabulated and displayed graphically.     

Sparse shape functions for tetrahedral <Emphasis Type="Italic">p</Emphasis>-FEM using integrated Jacobi polynomials

Summary In this paper, we investigate the discretization of an elliptic boundary value problem in 3D by means of the hp-version of the finite element method using a mesh of tetrahedrons. We present several bases based on integrated Jacobi polynomials in which the element stiffness matrix has nonzero entries, where p denotes the polynomial degree. The proof of the sparsity requires the assistance of computer algebra software. Several numerical experiments show the efficiency of the proposed bases for higher polynomial degrees p.      

Stochastic differential games and inverse optimal control and stopper policies

In this paper, we consider a two-player stochastic differential game problem over an infinite time horizon where the players invoke controller and stopper strategies on a nonlinear stochastic differential game problem driven by Brownian motion. The optimal strategies for the two players are given explicitly by exploiting connections between stochastic Lyapunov stability theory and stochastic Hamilton–Jacobi–Isaacs theory. In particular, we show that asymptotic stability in probability of the differential game problem is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Isaacs equation, and hence, guaranteeing both stochastic stability and optimality of the closed-loop control and stopper policies. In addition, we develop optimal feedback controller and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the stochastic differential game problem. These results are then used to provide extensions of the linear feedback controller and stopper policies obtained in the literature to nonlinear feedback controllers and stoppers that minimise and maximise general polynomial and multilinear performance criteria.