Radial Basis Function Type Research Articles

Decentralized Adaptive Control of Large-Scale Nonaffine Nonlinear Systems Using Radial Basis Function Neural Networks

In this paper, a novel decentralized adaptive neural network controller is proposed for a class of large-scale nonlinear systems with unknown nonlinear, nonaffine subsystems and unknown nonlinear interconnections. The stability of the closed loop system is guaranteed by introducing a robust adaptive bound based on Lyapunov stability analysis. A radial-basis function type neural network is used in the paper. To show the effectiveness of the proposed method, we performed some simulation studies. The results of simulation become very promising.

Locality properties of radial basis function expansion coefficients for equispaced interpolation

Many types of radial basis functions (RBFs) are global in terms of having large magnitude across the entire domain. Yet, in contrast, e.g. with expansions in orthogonal polynomials, RBF expansions exhibit a strong property of locality with regard to their coefficients. That is, changing a single data value mainly affects the coefficients of the RBFs which are centred in the immediate vicinity of that data location. This locality feature can be advantageous in the development of fast and well-conditioned iterative RBF algorithms. With this motivation, we employ here both analytical and numerical techniques to derive the decay rates of the expansion coefficients for cardinal data, in both 1D and 2D. Furthermore, we explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs.

Analysis of thick composite laminates using a higher-order shear and normal deformable plate theory (HOSNDPT) and a meshless method

The meshless local Petrov–Galerkin (MLPG) method with radial basis functions (RBFs), and the higher order shear and normal deformable plate theory (HOSNDPT) are used to analyze static infinitesimal deformations of thick laminated composite elastic plates under different boundary conditions. Two types of RBFs, namely, multiquadrics (MQ) and thin plate splines (TPS), are employed for constructing trial functions while a fourth order spline function is used as the test function. Computed results for different lamination schemes are found to match well with those obtained by other researchers. A benefit of using RBFs over those generated by the moving least squares approximation is that no special treatment is needed to impose essential boundary conditions, which substantially reduces the computational cost. Furthermore, the MLPG method does not require nodal connectivity which reduces the time required to prepare the input data.

Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization

Gradient-based aerodynamic shape optimization using computational fluid dynamics (CFD), and time dependent problems in aeroelasticity, that is, coupled calculations between computational structural mechanics (CSM) and CFD, require repeated deformations of the CFD mesh. An interpolation scheme, based on radial basis functions (RBF), is devised in order to propagate the deformations from the boundaries to the interior of the CFD mesh. This method can lower the computational costs due to the deformation of the mesh, in comparison with the usual Laplace smoothing. Moreover, the algorithm is independent of the mesh connectivities. Therefore, structured and unstructured meshes are equally treated as well as hybrid meshes. The application of this interpolation scheme in problems of aerodynamic shape optimization is also carefully investigated. When the optimization is executed by a gradient-based algorithm the cost function is differentiated with respect to the design parameters in order to obtain the gradient. The gradient is most efficiently and accurately calculated by solving a certain adjoint equation derived from the discretized flow equations. The calculation of the gradient, which is detailed in this presentation, involves the Jacobian matrix of the mesh deformation. Finally, we present the results of an optimization of the ONERA M6 wing at transonic speed using the interpolation algorithm. The results are used for comparison with another technique of mesh deformation. The quality of the mesh obtained by the new algorithm, and the interpolation error, are analyzed with respect to the parameters of the interpolation scheme: the type of RBF, the RBF’s shape parameter, and the sets of control points.

Analysis of thick plates by using a higher-order shear and normal deformable plate theory and MLPG method with radial basis functions

Infinitesimal deformations of a homogeneous and isotropic thick elastic plate have been analyzed by using a meshless local Petrov–Galerkin (MLPG) method and a higher-order shear and normal deformable plate theory (HONSDPT). Radial basis functions (RBF) are employed for constructing trial solutions, while a spline function is used as the weight function over a local subdomain. The present method uses a number of randomly distributed nodes in the domain and is truly meshless. Two types of RBFs, i.e. multiquadrics (MQ) and thin plate splines (TPS), are employed and effects of their shape parameters on the quality of the computed solution are examined for deformations of thick plates under different boundary conditions. It is found that the present MLPG formulations give results very close to those obtained by other researchers. A benefit of using RBFs is that no special treatment is needed to impose the essential boundary conditions, which substantially reduces the computational cost.

Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions

Infinitesimal deformations of a thick functionally graded elastic plate have been analyzed by using the meshless local Petrov-Galerkin (MLPG) method and the higher-order shear and normal deformable plate theory (HOSNDPT). Radial basis functions (RBF) are employed for constructing trial functions, while a spline function is used as the weighting function over a local subdomain. The present method employs a number of randomly located nodes in the domain. It does not require a mesh for either interpolation of the field variables or integration of the weak form and hence is truly meshless. Two types of RBFs, i.e. Multiquadrics (MQ) and Thin Plate Splines (TPS), are used. Effective material moduli of the plate, made of two isotropic constituents with volume contents varying only in the thickness direction, are computed using the Mori-Tanaka homogenization technique. Computed results for a simply supported plate are found to agree well with their analytical solutions. The through-the-thickness distributions of the deflection and stresses in the plate are also presented for three types of boundary conditions: simply supported, clamped and free.

RBF network optimization of complex microwave systems represented by small FDTD modeling data sets

This paper outlines an original algorithm of neural optimization backed by three-dimensional full-wave finite-difference time-domain (FDTD) simulation and suitable for viable computer-aided design of complex microwave (MW) systems. The frequency response of an S-parameter is optimized with a decomposed radial basis function (RBF) network capable of dealing with various MW devices. The key feature of the optimization is the dynamic generation of as much FDTD data as the network needs to find a solution satisfying the constraints or the stopping criteria. Other functions contributing to the reduction of computational cost include a choice of an RBF type, radius optimization of the Gaussian RBF, optimization of the regularization parameter, etc. Performance of the algorithm is illustrated by its application to the systems, which can be adequately described only with the full-wave numerical analysis: a double waveguide window, a loaded MW oven, and a patch antenna with two long slits. In all these examples, the network demonstrates excellent generalizing capabilities with the use of relatively small data sets, and the optimized solutions are obtained within fairly reasonable time. The algorithm is shown to be advantageous over conventional gradient and nongradient local-optimization techniques because it is independent of the starting point and having the potential to find the "best" local optimum in the specified domain. Finally, parameters of FDTD simulations and the network operations influencing the computational cost of the optimization are thoroughly discussed.

Artificial neural networks in their simplest forms for analysis and synthesis of RF/microwave planar transmission lines

In this work, the multiplayer perceptron (MLP) and radial basis function (RBF) neural networks in their simplest forms are employed in function approximation for highly nonlinear and complex analysis and synthesis of the most commonly used planar RF/microwave transmission lines, that is, microstrip lines, coupled microstrip lines, and basic and shielded coplanar-waveguides. Since the analysis and synthesis processes for these systems have “one-to-one mapping” relations with each other, a forward model is defined for the analysis process for all these types of the planar transmission lines; on the other hand, a reverse model is also considered for the synthesis of the same lines. This reverse model is realized by swapping some of the inputs/outputs in the analysis model, and training the neural networks accordingly. Both MLP and RBF types of neural models are applied to the four widely used anisotropic and isotropic dielectric materials: PTFE/microstrip glass, RT/Duroid 6006, alumina and gallium arsenide. The results are shown to agree very well with the targets. A low-pass filter with 30-dB attenuation frequency at 3.5 GHz on an alumina substrate is designed by the use of a neural-network synthesis and its resulting performance agrees well with the one using analytical formulas for the synthesis. © 2005 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2005.

Investigation of the use of radial basis functions in local collocation method for solving diffusion problems

Multidimensional diffusion problems can be solved by a local collocation method using radial basis functions. This method is computationally efficient because it is meshless and yields a sparse system of algebraic equations. Three types of radial basis functions—the generalized multiquadric function, the Gaussian function, and the generalized thin-plate spline function—are considered as interpolation functions in this method. Analysis of truncation error indicates that the use of the generalized multiquadric function or the Gaussian function produces satisfactory results, but the generalized thin-plate spline function is not a good interpolation function.

Revised GMDH-type Neural Networks with Radial Basis Functions and their Application to Medical Image Recognition of Stomach

In this paper, the revised Group Method of Data Handling (GMDH)-type neural networks with radial basis functions are proposed. This algorithm has been developed based on the basic GMDH-type neural networks which have both characteristics of the GMDH, the conventional multi-layered neural networks and the multi-layered radial basis function networks. The GMDH-type neural networks are automatically organized by using the heuristic self-organization method proposed by A.G. Ivakhnenko. In the GMDH-type neural networks, many types of neurons such as the sigmoid function type, the radial basis function type, the high order Polynomial type and the linear function type, are used and these neurons are automatically selected so as to fit the complexity of the nonlinear system by using an error criterion defined as Akaike's Information Criterion (AIC). The revised GMDH-type neural networks with radial basis functions proposed in this paper can identify a complex nonlinear system more accurately compared with the conventional radial basis function networks. Revised GMDH-type neural networks with radial basis functions are applied to the medical image recognition of the stomach and it is shown that this algorithm is very useful method for the medical image recognition.

Interpolation in the limit of increasingly flat radial basis functions

Many types of radial basis functions, such as multiquadrics, contain a free parameter. In the limit where the basis functions become increasingly flat, the linear system to solve becomes highly ill-conditioned, and the expansion coefficients diverge. Nevertheless, we find in this study that limiting interpolants often exist and take the form of polynomials. In the 1-D case, we prove that with simple conditions on the basis function, the interpolants converge to the Lagrange interpolating polynomial. Hence, differentiation of this limit is equivalent to the standard finite difference method. We also summarize some preliminary observations regarding the limit in 2-D.

A Radial Basis Function Method for Global Optimization

We introduce a method that aims to find the global minimum of a continuous nonconvex function on a compact subset of \dRd. It is assumed that function evaluations are expensive and that no additional information is available. Radial basis function interpolation is used to define a utility function. The maximizer of this function is the next point where the objective function is evaluated. We show that, for most types of radial basis functions that are considered in this paper, convergence can be achieved without further assumptions on the objective function. Besides, it turns out that our method is closely related to a statistical global optimization method, the P-algorithm. A general framework for both methods is presented. Finally, a few numerical examples show that on the set of Dixon-Szego test functions our method yields favourable results in comparison to other global optimization methods.

Approximation of nonlinear systems with radial basis function neural networks

A technique for approximating a continuous function of n variables with a radial basis function (RBF) neural network is presented. The method uses an n-dimensional raised-cosine type of RBF that is smooth, yet has compact support. The RBF network coefficients are low-order polynomial functions of the input. A simple computational procedure is presented which significantly reduces the network training and evaluation time. Storage space is also reduced by allowing for a nonuniform grid of points about which the RBFs are centered. The network output is shown to be continuous and have a continuous first derivative. When the network is used to approximate a nonlinear dynamic system, the resulting system is bounded-input bounded-output stable. For the special case of a linear system, the RBF network representation is exact on the domain over which it is defined, and it is optimal in terms of the number of distinct storage parameters required. Several examples are presented which illustrate the effectiveness of this technique.

Runoff Forecasting Using RBF Networks with OLS Algorithm

This paper illustrates the application of the radial basis function type of artificial neural networks (ANNs) using the orthogonal least-squares (OLS) algorithm to model the rainfall runoff process. Models using this approach differ from the more commonly used ANN models that adopt the back propagation (BP) algorithm, in that the former are linear in the parameters. The OLS algorithm is also capable of synthesizing the suitable network architecture, relieving the user of a time-consuming trial-and-error procedure. The proposed method is then applied to forecast runoff in a small catchment. One-hour predictions using the model are compared with those predicted by an ANN model that uses the BP algorithm, an ARMAX model, and with observed values. Results indicate that the OLS algorithm-based approach produces forecasts of comparable accuracy to those based on the BP algorithm. This approach has the added advantage of requiring less time for model development, and is also readily usable by the hydrologist with little or no background knowledge of ANNs.

Use of Radial Basis Function Type Artificial Neural Networks for Runoff Simulation

In this paper, the applicability of the radial basis function (RBF) type artificial neural networks (ANNs) approach for modeling a hydrologic system is investigated. The method differs from the more widely used multilayer perceptron (MLP) approach in that the nonlinearity of the model is embedded only in the hidden layer of the network. Search for optimal model parameters is carried out in two steps, each of which can be made to be more efficient and much faster than in MLP. This approach is applied to simulate runoff discharges in a small catchment. The results show that the models based on RBF networks can predict runoff with accuracy comparable with that with the MLP approach. An added advantage of RBF network‐based models is that they can be developed with relative ease and with much less time compared with their MLP counterparts.

A neural-network learning theory and a polynomial time RBF algorithm

This paper presents a new learning theory (a set of principles for brain-like learning) and a corresponding algorithm for the neural-network field. The learning theory defines computational characteristics that are much more brain-like than that of classical connectionist learning. Robust and reliable learning algorithms would result if these learning principles are followed rigorously when developing neural-network algorithms. This paper also presents a new algorithm for generating radial basis function (RBF) nets for function approximation. The design of the algorithm is based on the proposed set of learning principles. The net generated by this algorithm is not a typical RBF net, but a combination of "truncated" RBF and other types of hidden units. The algorithm uses random clustering and linear programming (LP) to design and train this "mixed" RBF net. Polynomial time complexity of the algorithm is proven and computational results are provided for the well known Mackey-Glass chaotic time series problem, the logistic map prediction problem, various neuro-control problems, and several time series forecasting problems. The algorithm can also be implemented as an online adaptive algorithm.

Cooperative-competitive genetic evolution of radial basis function centers and widths for time series prediction

In a radial basis function (RBF) network, the RBF centers and widths can be evolved by a cooperative-competitive genetic algorithm. The set of genetic strings in one generation of the algorithm represents one REP network, not a population of competing networks. This leads to moderate computation times for the algorithm as a whole. Selection operates on individual RBFs rather than on whole networks. Selection therefore requires a genetic fitness function that promotes competition among RBFs which are doing nearly the same job while at the same time promoting cooperation among RBFs which cover different parts of the domain of the function to be approximated. Niche creation resulting from a fitness function of the form |w(i)|(beta)/E(|w(i')|(beta)), 1<beta<2 can facilitate the desired cooperative-competitive behavior. The feasibility of the resulting algorithm to evolve networks of Gaussian, inverse multiquadric, and thin-plate spline RBFs is demonstrated by predicting the Mackey-Glass time series. For each type of RBF, and for networks of 25, 50, 75, 100, 125, and 150 RBF units, prediction errors for the evolved Gaussian RBF networks are 50-70% lower than RBF networks obtained by k-means clustering.

Radial Basis Function Approximation: from Gridded Centres to Scattered Centres

Abstract : The paper studies L-infinity (IR d)-norm approximations from a space spanned by a discrete set of translates of a basis function theta. Attention here is restricted to functions theta whose Fourier transform is smooth on IRd/ 0, and has a singularity at the origin. Examples of such basis functions are the thin-plate splines and the multiquadrics, as well as other types of radial basis functions that are employed in Approximation Theory. The above approximation problem is well-understood in case the set of points used for translating theta forms a lattice in IR d, and many optimal and quasi-optimal approximation schemes can already be found in the literature. In contrast, only few, mostly specific, results are known for a set of scattered points. The main objective of this paper is to provide a general tool for extending approximation schemes that use integer translates of a basis function to the non-uniform case. We introduce a single, relatively simple, conversion method that preserves the approximation orders provided by a large number of schemes presently in the literature (more precisely, to almost all stationary schemes ). In anticipation of future introduction of new schemes for uniform grids, an effort is made to impose only a few mild conditions on the function theta, which still allow for a unified error analysis to hold. In the course of the discussion here, the recent results of BuDL on scattered center approximation are reproduced and improved upon.