Particular solutions of Laplace and bi-harmonic operators using Matérn radial basis functions

In this paper, we derive closed-form particular solutions of Matérn radial basis functions for the Laplace and biharmonic operator in 2D and Laplace operator in 3D. These derived particular solutions are essential for the implementation of the method of particular solutions for solving various types of partial differential equations. Four numerical examples in 2D and 3D are given to demonstrate the effectiveness of the derived particular solutions.     

Helmholtz方程的楔形基区域分解法

基于楔形基函数和无网格配点法,提出了一种求解Helmholtz型方程区域分解法。该方法克服了在求解大规模问题时用一般的全域配点法所带来的配置矩阵为非对称满阵,且高度病态的问题。通过数值结果表明,该算法在求解Helmholtz型方程降低系数矩阵条件数的同时,也能够降低误差,并达到满意的收敛效果。     

Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey

Since neural networks have universal approximation capabilities, therefore it is possible to postulate them as solutions for given differential equations that define unsupervised errors. In this paper, we present a wide survey and classification of different Multilayer Perceptron (MLP) and Radial Basis Function (RBF) neural network techniques, which are used for solving differential equations of various kinds. Our main purpose is to provide a synthesis of the published research works in this area and stimulate further research interest and effort in the identified topics. Here, we describe the crux of various research articles published by numerous researchers, mostly within the last 10 years to get a better knowledge about the present scenario.     

Numerical solutions of 3D Cauchy problems of elliptic operators in cylindrical domain using local weak equations and radial basis functions

This paper is concerned with the numerical solutions of 3D Cauchy problems of elliptic differential operators in the cylindrical domain. We assume that the measurements are only available on the outer boundary while the interior boundary is inaccessible and the solution should be obtained from the measurements from the outer layer. The proposed discretization approach uses the local weak equations and radial basis functions. Since the Cauchy problem is known to be ill-posed, the Thikhonov regularization strategy is employed to solve effectively the discrete ill-posed resultant linear system of equations. Numerical results of a different kind of test problems reveal that the method is very effective.     

Speech recognition using pulse-coupled neural networks with a radial basis function

We have developed a novel pulse-coupled neural network (PCNN) for speech recognition. One of the advantages of the PCNN is in its biologically based neural dynamic structure using feedback connections. To recall the memorized pattern, a radial basis function (RBF) is incorporated into the proposed PCNN. Simulation results show that the PCNN with a RBF can be useful for phoneme recognition. This work was presented in part at the 7th International Symposium on Artificial Life and Robotics, Oita, Japan, January 16–18, 2002     

The sample solution approach for determination of the optimal shape parameter in the Multiquadric function of the Kansa method

The Kansa method with the Multiquadric-radial basis function (MQ-RBF) is inherently meshfree and can achieve an exponential convergence rate if the optimal shape parameter is available. However, it is not an easy task to obtain the optimal shape parameter for complex problems whose analytical solution is often a priori unknown. This has long been a bottleneck for the MQ-Kansa method application to practical problems. In this paper, we present a novel sample solution approach (SSA) for achieving a reasonably good shape parameter of the MQ-RBF in the Kansa method for the solution of problems whose analytical solution is unknown. The basic assumption behind the SSA is that the optimal shape parameter is considered to be largely depended on the shape of computational domain, the type of the boundary conditions, the number and distribution of nodes, and the governing equation. In the procedure of the SSA, we set up a pseudo-problem as the sample solution whose solution is known. It is not difficult to obtain the optimal parameter of the MQ-RBF in the numerical solution of the pseudo-problem. The SSA suggests that the optimal shape parameter of the pseudo-problem can also achieve an approximately optimal accuracy in the solution of the original problem. Numerical examples and comparisons are provided to verify the proposed SSA in terms of accuracy and stability in solving homogeneous problems and non-homogeneous modified Helmholtz problems in several complex domains even using chaotic distribution of collocation points.     

Prediction of chenille yarn and fabric abrasion resistance using radial basis function neural network models

The abrasion resistance of chenille yarn is crucially important in particular because the effect sought is always that of the velvety feel of the pile. Thus, various methods have been developed to predict chenille yarn and fabric abrasion properties. Statistical models yielded reasonably good abrasion resistance predictions. However, there is a lack of study that encompasses the scope for predicting the chenille yarn abrasion resistance with artificial neural network (ANN) models. This paper presents an intelligent modeling methodology based on ANNs for predicting the abrasion resistance of chenille yarns and fabrics. Constituent chenille yarn parameters like yarn count, pile length, twist level and pile yarn material type are used as inputs to the model. The intelligent method is based on a special kind of ANN, which uses radial basis functions as activation functions. The predictive power of the ANN model is compared with different statistical models. It is shown that the intelligent model improves prediction performance with respect to statistical models.     

A forecasting system for car fuel consumption using a radial basis function neural network

A predictive system for car fuel consumption using a radial basis function (RBF) neural network is proposed in this paper. The proposed work consists of three parts: information acquisition, fuel consumption forecasting algorithm and performance evaluation. Although there are many factors affecting the fuel consumption of a car in a practical drive procedure, in the present system the relevant factors for fuel consumption are simply decided as make of car, engine style, weight of car, vehicle type and transmission system type which are used as input information for the neural network training and fuel consumption forecasting procedure. In fuel consumption forecasting, to verify the effect of the proposed RBF neural network predictive system, an artificial neural network with a back-propagation (BP) neural network is compared with an RBF neural network for car fuel consumption prediction. The prediction results demonstrated the proposed system using the neural network is effective and the performance is satisfactory in terms of fuel consumption prediction.     

Control of chaotic dynamical systems using radial basis function network approximators

This paper presents a general control method based on radial basis function networks (RBFNs) for chaotic dynamical systems. For many chaotic systems that can be decomposed into a sum of a linear and a nonlinear part, under some mild conditions the RBFN can be used to well approximate the nonlinear part of the system dynamics. The resulting system is then dominated by the linear part, with some small or weak residual nonlinearities due to the RBFN approximation errors. Thus, a simple linear state-feedback controller can be devised, to drive the system response to a desirable set-point. In addition to some theoretical analysis, computer simulations on two representative continuous-time chaotic systems (the Duffing and the Lorenz systems) are presented to demonstrate the effectiveness of the proposed method.     

Time series prediction using evolving radial basis function networks with new encoding scheme

This paper presents a new encoding scheme for training radial basis function (RBF) networks by genetic algorithms (GAs). In general, it is very difficult to select the proper input variables and the exact number of nodes before training an RBF network. In the proposed encoding scheme, both the architecture (numbers and selections of nodes and inputs) and the parameters (centres and widths) of the RBF networks are represented in one chromosome and evolved simultaneously by GAs so that the selection of nodes and inputs can be achieved automatically. The performance and effectiveness of the presented approach are evaluated using two benchmark time series prediction examples and one practical application example, and are then compared with other existing methods. It is shown by the simulation tests that the developed evolving RBF networks are able to predict the time series accurately with the automatically selected nodes and inputs.     

Localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for nonlocal diffusion problems

Spectral/pseudo-spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations. In this paper, we will present the use of a localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for solving 2D nonlocal problems with radial nonlocal kernels. The basic idea of the LRBF-PSM is to construct a set of orthogonal functions by RBFs on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. Numerical implementation indicates that the proposed LRBF-PSM is simple to use, efficient and robust to solve various nonlocal problems.     

Numerical solution of system of first-order delay differential equations using polynomial spline functions

The use of two constructed polynomial spline functions to approximate the solution of a system of first-order delay differential equations is described. The first spline function is a polynomial with an undetermined constant coefficient in the last term. The other has a polynomial spline form. The error analysis and stability of the second function are theoretically investigated and a test example is given. A comparison of the two forms is carried out to illustrate the pertinent features of the proposed techniques.     

Delta basis functions and their applications to systems of integral equations

In the present paper, delta functions (DFs) are proposed as a new set of basis functions. Their properties and relations to well-known triangular functions (TFs) are described. The simplicity and useful properties of newly proposed sets led us to use them with more accuracy and less computational burden.Furthermore, DFs are applied to propose an efficient method for approximating the solution of integral equations systems. Convergence analysis and the rate of convergence have been considered as well. Some numerical examples are provided to illustrate the computational efficiency and accuracy of the method.     

Discretization error due to the identity operator in surface integral equations

We consider the accuracy of surface integral equations for the solution of scattering and radiation problems in electromagnetics. In numerical solutions, second-kind integral equations involving well-tested identity operators are preferable for efficiency, because they produce diagonally-dominant matrix equations that can be solved easily with iterative methods. However, the existence of the well-tested identity operators leads to inaccurate results, especially when the equations are discretized with low-order basis functions, such as the Rao-Wilton-Glisson functions. By performing a computational experiment based on the nonradiating property of the tangential incident fields on arbitrary surfaces, we show that the discretization error of the identity operator is a major error source that contaminates the accuracy of the second-kind integral equations significantly.     

Online identification using radial basis function neural network coupled with KPCA

In this paper, we are going to propose an online radial basis function (RBF) neural network algorithm without any preprocessing step. Then a kernel principal component analysis (KPCA) is coupled with the proposed online RBF neural network algorithm. Indeed, the KPCA method is used as a preprocessing step to reduce the feature dimension which fed to the RBF neural network. Reducing memory requirements of the models makes RBF neural network training efficient and fast. These two proposed algorithms are applied, with success, for identification of a mobile robot position. The simulation results present that the used sigmoid function as a kernel, compared to other kernel functions, which gives an excellent model and a minimum mean square error.     

Topologically guaranteed bivariate solutions of under-constrained multivariate piecewise polynomial systems

We present a subdivision based algorithm to compute the solution of an under-constrained piecewise polynomial system of $n-2$ equations with $n$ unknowns, exploiting properties of B-spline basis functions. The solution of such systems is, typically, a two-manifold in ${\mathbb{R}}^n$. To guarantee the topology of the approximated solution in each sub-domain, we provide subdivision termination criteria, based on the (known) topology of the univariate solution on the domain’s boundary, and the existence of a one-to-one projection of the unknown solution on a two dimensional plane, in ${\mathbb{R}}^n$. We assume the equation solving problem is regular, while sub-domains containing points that violate the regularity assumption are detected, bounded, and returned as singular locations of small (subdivision tolerance) size. This work extends (and makes extensive use of) topological guarantee results for systems with zero and one dimensional solution sets. Test results in ${\mathbb{R}}^3$ and ${\mathbb{R}}^4$ are also demonstrated, using error-bounded piecewise linear approximations of the two-manifolds.     

Cooperative learning for radial basis function networks using particle swarm optimization

This paper presents a new evolutionary cooperative learning scheme, able to solve function approximation and classification problems with improved accuracy and generalization capabilities. The proposed method optimizes the construction of radial basis function (RBF) networks, based on a cooperative particle swarm optimization (CPSO) framework. It allows for using variable-width basis functions, which increase the flexibility of the produced models, while performing full network optimization by concurrently determining the rest of the RBF parameters, namely center locations, synaptic weights and network size. To avoid the excessive number of design variables, which hinders the optimization task, a compact representation scheme is introduced, using two distinct swarms. The first swarm applies the non-symmetric fuzzy means algorithm to calculate the network structure and RBF kernel center coordinates, while the second encodes the basis function widths by introducing a modified neighbor coverage heuristic. The two swarms work together in a cooperative way, by exchanging information towards discovering improved RBF network configurations, whereas a suitably tailored reset operation is incorporated to help avoid stagnation. The superiority of the proposed scheme is illustrated through implementation in a wide range of benchmark problems, and comparison with alternative approaches.     

Modelling of a magneto-rheological damper by evolving radial basis function networks

This paper presents an approach to approximate the forward and inverse dynamic behaviours of a magneto-rheological (MR) damper using evolving radial basis function (RBF) networks. Due to the highly nonlinear characteristics of MR dampers, modelling of MR dampers becomes a very important problem to their applications. In this paper, an alternative representation of the MR damper in terms of evolving RBF networks, which have a structure of four input neurons and one output neuron to emulate the forward and inverse dynamic behaviours of an MR damper, respectively, is developed by combining the genetic algorithms (GAs) to search for the network centres with other standard learning algorithms. Training and validating of the evolving RBF network models are achieved by using the data generated from the numerical simulation of the nonlinear differential equations proposed for the MR damper. It is shown by the validation tests that the evolving RBF networks can represent both forward and inverse dynamic behaviours of the MR damper satisfactorily.     

Design of a radial basis function neural network with a radius-modification algorithm using response surface methodology

A radial basis function (RBF) neural network was designed for time series forecasting using both an adaptive learning algorithm and response surface methodology (RSM). To improve the traditional RBF networks forecasting capability, the generalized delta rule learning method was employed to modify the radius of the kernel function. Then RSM was utilized to explore the mean square error response surface so that the appropriate combination of network parameters, such as the number of hidden nodes and the initial learning rates, could be found. Extensive studies were performed on the effect of the initial values of connection weights on the accuracy of the backpropagation learning method that was employed in the training of the RBF artificial neural network. The effectiveness of the neural network with the proposed radius-modification technique and the RSM method was demonstrated with an example of forecasting intensity pulsations of a laser. It was found that, by utilizing the proposed techniques, the neural network provided a more accurate prediction of the response.     

Numerical solution of nonlinear differential equations using computer algebra

This paper describes a numerical method for finding periodic solutions to nonlinear ordinary differential equations. The solution is approximated by a trigonometric series. The series is substituted into the differential equation using the FORMAC computer algebra system for the resulting lengthy algebraic manipulations. This lead to a set of nonlinear algebraic equations for the series coefficients. Modern search methods are used to solve for the coefficients. The method is illustrated by application to Duffing’ equation.