神经网络中的逼近问题

本文主要讨论径向基神经网络对函数，连续泛函及连续算子的逼近．     

径向基函数神经网络在股票走势模式分类中的应用

探讨了径向基函数神经网络在个股走势模式分类中的应用问题。提出了一种可调基宽度的计算方法和若干数据预处理的措施。实例计算表明，该方法取得了较好的效果     

FA-RBF复合神经网络模型及其在地下水位预测中的应用

地下水动态变化过程呈现出高度复杂的非线性特征,增加了地下水位预测的难度.为充分反映地下水位变化过程中自变量和因变量之间的非线性映射关系,克服在获取水文地质参数与查明水文地质条件方面的困难,避免部分智能方法实现繁琐复杂、计算效率低、限制条件多等不足,提出将因子分析方法与RBF神经网络算法构成复合模型,用于地下水位预测.结果表明,复合模型可以用于地下水位预测,模型计算结果可靠,网络训练时间缩短,计算精度有所提高;而且有成熟算法,实现简单.     

基于径向基函数神经网络的流程企业供应链预测仿真

本文在比较预测方法的基础上，采用径向基函数（RBF）神经网络技术建立流程企业供应链预测模型，进行了实例预测仿真，并将预测结果与BP网络的预测结果进行了比较。结果表明，RBF网络误差小于BP网络，其中平方根RBF网络的预测仿真误差最小，而BP网络的误差最大。     

DENSENESS OF RADIAL-BASIS FUNCTIONS IN L^2（R^n） AND ITS APPLICATIONS IN NEURAL NETWORKS

The authors discuss problems of approximation to functions in L2(Rn) and operators fromL2(Rn1) to L2(Rn2) by Radial-Basis Functions. The results obtained solve the problem ofcapability of RBF neural networks, a basic problem in neural networks.     

基于模糊径向基函数神经网络的模糊数据建模研究

提出将模糊径向基函数神经网络(FRBFN)用于模糊数据的建模,并提出融和圆锥模糊向量的聚类方法和模糊线性回归的学习算法。仿真研究表明．FRBFN及其算法在模糊数据建模方面有一定的优势。     

关于径向基函数插值的收敛性

本文在n维空间给出了径向基函数插值及逼近的收敛性质,并给出了收敛阶。     

拟径向基函数法在公司债券风险衡量中的应用

本文应用拟径向基函数法(Q-RBFS)求解基于风险债券定价的Black-Scholes方程,并采用了特殊的方法降低系数矩阵的条件数来解决由于不断循环求解方程组所积累的误差,得到精确度较高的数值近似解,实现了债券风险定价.     

导弹武器系统费用预测的径向基函数网络方法

提出采用径向基函数网络理论来估算导弹武器系统的费用,武器系统的费用与武器特征参数的关系可通过神经网络的阈值和权值来表现,并且对几种用于导弹武器系统费用分析的数据分析结果进行比较分析.通过实例说明了应用径向基函数网络进行导弹武器系统费用分析不但算法可行性好、拟合精度高,而且具有运算简单,结果可靠的特点.     

关于径向基函数插值的收敛性

本文在 n 维空间给出了径向基函数插值及逼近的收敛性质,并给出了收敛阶.     

On the solution of the non-local parabolic partial differential equations via radial basis functions

In this paper, the problem of solving the one-dimensional parabolic partial differential equation subject to given initial and non-local boundary conditions is considered. The approximate solution is found using the radial basis functions collocation method. There are some difficulties in computing the solution of the time dependent partial differential equations using radial basis functions. If time and space are discretized using radial basis functions, the resulted coefficient matrix will be very ill-conditioned and so the corresponding linear system cannot be solved easily. As an alternative method for solution, we can use finite-difference methods for discretization of time and radial basis functions for discretization of space. Although this method is easy to use but an accurate solution cannot be provided. In this work an efficient collocation method is proposed for solving non-local parabolic partial differential equations using radial basis functions. Numerical results are presented and are compared with some existing methods.     

Multiscale RBF collocation for solving PDEs on spheres

In this paper, we discuss multiscale radial basis function collocation methods for solving certain elliptic partial differential equations on the unit sphere. The approximate solution is constructed in a multi-level fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. Two variants of the collocation method are considered (sometimes called symmetric and unsymmetric, although here both are symmetric). A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using compactly supported radial basis functions.     

The Embedding Theory of Native Spaces

In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L²(Ω) is also solved by the author.     

A meshless Galerkin method for Dirichlet problems using radial basis functions

In this paper, a numerical method is given for partial differential equations, which combines the use of Lagrange multipliers with radial basis functions. It is a new method to deal with difficulties that arise in the Galerkin radial basis function approximation applied to Dirichlet (also mixed) boundary value problems. Convergence analysis results are given. Several examples show the efficiency of the method using TPS or Sobolev splines.     

Solving differential equations with radial basis functions: multilevel methods and smoothing

Some of the meshless radial basis function methods used for the numerical solution of partial differential equations are reviewed. In particular, the differences between globally and locally supported methods are discussed, and for locally supported methods the important role of smoothing within a multilevel framework is demonstrated. A possible connection between multigrid finite elements and multilevel radial basis function methods with smoothing is explored. Various numerical examples are also provided throughout the paper. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Embedding Theory of Native Spaces

In the theory of radial basis functions, mathematicians use linear combinations of the translates of the radial basis functions as interpolants. The set of these linear combinations is a normed vector space. This space can be completed and become a Hilbert space, called native space, which is of great importance in the last decade. The native space then contains some abstract elements which are not linear combinations of radial basis functions. The meaning of these abstract elements is not fully known. This paper presents some interpretations for the these elements. The native spaces are embedded into some well-known spaces. For example, the Sobolev-space is shown to be a native space. Since many differential equations have solutions in the Sobolev-space, we can therefore approximate the solutions by linear combinations of radial basis functions. Moreover, the famous question of the embedding of the native space into L²() is also solved by the author.     

Approximation in rough native spaces by shifts of smooth kernels on spheres

Within the conventional framework of a native space structure, a smooth kernel generates a small native space, and “radial basis functions” stemming from the smooth kernel are intended to approximate only functions from this small native space. Therefore their approximation power is quite limited. Recently, Narcowich et al. (J. Approx. Theory 114 (2002) 70), and Narcowich and Ward (SIAM J. Math. Anal., to appear), respectively, have studied two approaches that have led to the empowerment of smooth radial basis functions in a larger native space. In the approach of [NW], the radial basis function interpolates the target function at some scattered (prescribed) points. In both approaches, approximation power of the smooth radial basis functions is achieved by utilizing spherical polynomials of a (possibly) large degree to form an intermediate approximation between the radial basis approximation and the target function. In this paper, we take a new approach. We embed the smooth radial basis functions in a larger native space generated by a less smooth kernel, and use them to approximate functions from the larger native space. Among other results, we characterize the best approximant with respect to the metric of the larger native space to be the radial basis function that interpolates the target function on a set of finite scattered points after the action of a certain multiplier operator. We also establish the error bounds between the best approximant and the target function.     

Semi-infinite cardinal interpolation with multiquadrics and beyond

This paper concerns the interpolation with radial basis functions on half-spaces, where the centres are multi-integers restricted to half-spaces as well. The existence of suitable Lagrange functions is shown for multiquadrics and inverse multiquadrics radial basis functions, as well as the decay rate and summability of its coefficients. The main technique is a so-called Wiener–Hopf factorisation of the symbol of the radial basis function and the careful study of the smoothness of its 2π-periodic factors. Dedicated to Charles A. Micchelli on his 60th Birthday Mathematics subject classifications (2000) 41A05, 41A15, 41A63, 47A68, 65D05, 65D07.     

Multivariate interpolation with increasingly flat radial basis functions of finite smoothness

In this paper, we consider multivariate interpolation with radial basis functions of finite smoothness. In particular, we show that interpolants by radial basis functions in ℝ ^d with finite smoothness of even order converge to a polyharmonic spline interpolant as the scale parameter of the radial basis functions goes to zero, i.e., the radial basis functions become increasingly flat.     

Coupling three-field formulation and meshless mixed Galerkin methods using radial basis functions

In this work, we solve the elliptic partial differential equation by coupling the meshless mixed Galerkin approximation using radial basis function with the three-field domain decomposition method. The formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function. Convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper.