Metamodeling using extended radial basis functions: a comparative approach

The process of constructing computationally benign approximations of expensive computer simulation codes, or metamodeling, is a critical component of several large-scale multidisciplinary design optimization (MDO) approaches. Such applications typically involve complex models, such as finite elements, computational fluid dynamics, or chemical processes. The decision regarding the most appropriate metamodeling approach usually depends on the type of application. However, several newly proposed kernel-based metamodeling approaches can provide consistently accurate performance for a wide variety of applications. The authors recently proposed one such novel and effective metamodeling approach—the extended radial basis function (E-RBF) approach—and reported highly promising results. To further understand the advantages and limitations of this new approach, we compare its performance to that of the typical RBF approach, and another closely related method—kriging. Several test functions with varying problem dimensions and degrees of nonlinearity are used to compare the accuracies of the metamodels using these metamodeling approaches. We consider several performance criteria such as metamodel accuracy, effect of sampling technique, effect of sample size, effect of problem dimension, and computational complexity. The results suggest that the E-RBF approach is a potentially powerful metamodeling approach for MDO-based applications, as well as other classes of computationally intensive applications.     

Robustness of radial basis functions

Neural networks are intended to be used in future nanoelectronic technology since these architectures seem to be robust to malfunctioning elements and noise in its inputs and parameters. In this work, the robustness of radial basis function networks is analyzed in order to operate in noisy and unreliable environment. Furthermore, upper bounds on the mean square error under noise contaminated parameters and inputs are determined if the network parameters are constrained. To achieve robuster neural network architectures fundamental methods are introduced to identify sensitive parameters and neurons.     

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.     

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.     

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.     

Approximate implicitization of parametric surfaces by using compactly supported radial basis functions

In this paper, a new approach is proposed to solve the approximate implicitization of parametric surfaces. It is primarily based on multivariate interpolation of scattered data by using compactly supported radial basis functions. Experimental results are provided to illustrate the proposed method is flexible and effective.     

A hybrid radial basis function and data envelopment analysis neural network for classification

We propose a hybrid radial basis function network-data envelopment analysis (RBFN-DEA) neural network for classification problems. The procedure uses the radial basis function to map low dimensional input data from input space ℜ to a high dimensional ℜ⁺ feature space where DEA can be used to learn the classification function. Using simulated datasets for a non-linearly separable binary classification problem, we illustrate how the RBFN-DEA neural network can be used to solve it. We also show how asymmetric misclassification costs can be incorporated in the hybrid RBFN-DEA model. Our preliminary experiments comparing the RBFN-DEA with feed forward and probabilistic neural networks show that the RBFN-DEA fares very well.     

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     

Compactly supported radial basis functions for adaptive process control

An adaptive nonlinear control strategy based on networks of compactly supported radial basis functions is proposed. The local influence of the basis functions allows efficient on-line adaptation that is performed using a gradient law, and new basis functions are added to the network only when new regions in state space are encountered and the prediction error exceeds a pre-specified tolerance. The approximate model is used to construct an input-output linearizing control law. The adaptive control strategy is applied to a nonlinear chemical reactor model.     

Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions

This paper presents a new algorithm for derivative-free optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. The proposed algorithm, called ConstrLMSRBF, uses radial basis function (RBF) surrogate models and is an extension of the Local Metric Stochastic RBF (LMSRBF) algorithm by Regis and Shoemaker (2007a) [1] that can handle black-box inequality constraints. Previous algorithms for the optimization of expensive functions using surrogate models have mostly dealt with bound constrained problems where only the objective function is expensive, and so, the surrogate models are used to approximate the objective function only. In contrast, ConstrLMSRBF builds RBF surrogate models for the objective function and also for all the constraint functions in each iteration, and uses these RBF models to guide the selection of the next point where the objective and constraint functions will be evaluated. Computational results indicate that ConstrLMSRBF is better than alternative methods on 9 out of 14 test problems and on the MOPTA08 problem from the automotive industry (Jones, 2008 [2]). The MOPTA08 problem has 124 decision variables and 68 inequality constraints and is considered a large-scale problem in the area of expensive black-box optimization. The alternative methods include a Mesh Adaptive Direct Search (MADS) algorithm (Abramson and Audet, 2006 [3]; Audet and Dennis, 2006 [4]) that uses a kriging-based surrogate model, the Multistart LMSRBF algorithm by Regis and Shoemaker (2007a) [1] modified to handle black-box constraints via a penalty approach, a genetic algorithm, a pattern search algorithm, a sequential quadratic programming algorithm, and COBYLA (Powell, 1994 [5]), which is a derivative-free trust-region algorithm. Based on the results of this study, the results in Jones (2008) [2] and other approaches presented at the ISMP 2009 conference, ConstrLMSRBF appears to be among the best, if not the best, known algorithm for the MOPTA08 problem in the sense of providing the most improvement from an initial feasible solution within a very limited number of objective and constraint function evaluations.     

Interactive 3D medical image segmentation with energy-minimizing implicit functions

We present an interactive segmentation method for 3D medical images that reconstructs the surface of an object using energy-minimizing, smooth, implicit functions. This reconstruction problem is called variational interpolation. For an intuitive segmentation of medical images, variational interpolation can be based on a set of user-drawn, planar contours that can be arbitrarily oriented in 3D space. This also allows an easy integration of the algorithm into the common manual segmentation workflow, where objects are segmented by drawing contours around them on each slice of a 3D image.Because variational interpolation is computationally expensive, we show how to speed up the algorithm to achieve almost real-time calculation times while preserving the overall segmentation quality. Moreover, we show how to improve the robustness of the algorithm by transforming it from an interpolation to an approximation problem and we discuss a local interpolation scheme.A first evaluation of our algorithm by two experienced radiology technicians on 15 liver metastases and 1 liver has shown that the segmentation times can be reduced by a factor of about 2 compared to a slice-wise manual segmentation and only about one fourth of the contours are necessary compared to the number of contours necessary for a manual segmentation.     

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.     

A review of genetic algorithms applied to training radial basis function networks

The problems associated with training feedforward artificial neural networks (ANNs) such as the multilayer perceptron (MLP) network and radial basis function (RBF) network have been well documented. The solutions to these problems have inspired a considerable amount of research, one particular area being the application of evolutionary search algorithms such as the genetic algorithm (GA). To date, the vast majority of GA solutions have been aimed at the MLP network. This paper begins with a brief overview of feedforward ANNs and GAs followed by a review of the current state of research in applying evolutionary techniques to training RBF networks.     

Grey-box radial basis function modelling

A fundamental principle in data modelling is to incorporate available a priori information regarding the underlying data generating mechanism into the modelling process. We adopt this principle and consider grey-box radial basis function (RBF) modelling capable of incorporating prior knowledge. Specifically, we show how to explicitly incorporate the two types of prior knowledge: (i) the underlying data generating mechanism exhibits known symmetric property, and (ii) the underlying process obeys a set of given boundary value constraints. The class of efficient orthogonal least squares regression algorithms can readily be applied without any modification to construct parsimonious grey-box RBF models with enhanced generalisation capability.     

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.     

The improved radial basis function neural network and its application

Compared with other feed-forward neural networks, radial basis function neural networks (RBFNN) have many advantages which make them more suitable for nonlinear system modeling, and they have recently received considerable attention. In this paper, a RBFNN is employed to model strongly nonlinear systems. First, the problems of nonlinear system modeling are analyzed, and then the structure of the RBFNN as well as the training algorithm are improved to solve these problems. Finally, an industrial high-purity distillation column, which is a strongly nonlinear system, is successfully modeled with the improved RBFNN. Owing to the complexities of a nonlinear system, it is necessary to use a real-time model correction method to modify the parameters of the RBFNN model in real time. One efficient method is proposed in this paper. The idea is to employ the Givens transformation to modify the parameters of the RBFNN-based model. This work was presented, in part, at the International Symposium on Artificial Life and Robotics, Oita, Japan, February 18–20, 1996     

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.     

Effect of different basis functions on a radial basis function network in prediction of drill flank wear from motor current signals

The Gaussian kernel has almost exclusively been used as the basis function of the cluster centers (hidden layer nodes) of a radial basis function network (RBFN) in most of its applications, especially in tool condition monitoring (TCM) problems. This study explores the possible usage of a set of five other basis functions in addition to the standard Gaussian function, in one such important TCM problem, i.e., prediction of drill flank wear. The analysis focuses on a comparative study of the wear prediction capabilities of the RBFN employing these six different basis functions for a wide range of the basis width parameter (wherever applicable) and changing the number of cluster centers in the hidden layer. This analysis is carried out following a series of experiments employing high speed steel (HSS) drills for drilling holes on mild steel workpieces, under different sets of cutting conditions (spindle speed, feed-rate and drill diameter) and noting the root mean square (RMS) value of spindle motor current as well as the average flank wear in each case. The results show that other basis functions can also match the performance of the Gaussian kernel, and depending upon the nature of application at hand and the requirements of time and space, the use of basis functions other than the Gaussian kernel may just prove advantageous.     

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.