A Radial Basis Function Method with Improved Accuracy for Fourth Order Boundary Value Problems

Accurately approximating higher order derivatives is an inherently difficult problem. It is shown that a random variable shape parameter strategy can improve the accuracy of approximating higher order derivatives with Radial Basis Function methods. The method is used to solve fourth order boundary value problems. The use and location of ghost points are examined in order to enforce the extra boundary conditions that are necessary to make a fourth-order problem well posed. The use of ghost points versus solving an overdetermined linear system via least squares is studied. For a general fourth-order boundary value problem, the recommended approach is to either use one of two novel sets of ghost centers introduced here or else to use a least squares approach. When using either ghost centers or least squares, the random variable shape parameter strategy results in significantly better accuracy than when a constant shape parameter is used.

Numerical Simulation of Dam-Break Flows Using Radial Basis Functions: Application to Urban Flood Inundation

Dam-break flows pose significant threats to urban areas due to their potential for causing rapid and extensive flooding. Traditional numerical methods for simulating these events struggle with complex urban landscapes. This paper presents an alternative approach using Radial Basis Functions to simulate dam-break flows and their impact on urban flood inundation. The proposed method adapts a new strategy based on Particle Swarm Optimization for variable shape parameter selection on meshfree formulation to enhance the numerical stability and convergence of the simulation. The method’s accuracy and efficiency are demonstrated through numerical experiments, including well-known partial and circular dam-break problems and an idealized city with a single building, highlighting its potential as a valuable tool for urban flood risk management.

Crack Fault Diagnosis and Location Method for a Dual-Disk Hollow Shaft Rotor System Based on the Radial Basis Function Network and Pattern Recognition Neural Network

The crack fault is one of the most common faults in the rotor system,and researchers have paid close attention to its fault diagnosis.However,most studies focus on discussing the dynamic response characteristics caused by the crack rather than estimating the crack depth and position based on the obtained vibration signals.In this paper,a novel crack fault diagnosis and location method for a dual-disk hollow shaft rotor system based on the Radial basis function(RBF)network and Pattern recognition neural network(PRNN)is presented.Firstly,a rotor system model with a breathing crack suitable for a short-thick hollow shaft rotor is established based on the finite element method,where the crack's periodic opening and closing pattern and different degrees of crack depth are considered.Then,the dynamic response is obtained by the harmonic balance method.By adjusting the crack parameters,the dynamic characteristics related to the crack depth and position are analyzed through the amplitude-frequency responses and waterfall plots.The analysis results show that the first critical speed,first subcritical speed,first critical speed amplitude,and super-harmonic resonance peak at the first subcritical speed can be utilized for the crack fault diagnosis.Based on this,the RBF network and PRNN are adopted to determine the depth and approximate location of the crack respectively by taking the above dynamic characteristics as input.Test results show that the proposed method has high fault diagnosis accuracy.This research proposes a crack detection method adequate for the hollow shaft rotor system,where the crack depth and position are both unknown.

Evolution Performance of Symbolic Radial Basis Function Neural Network by Using Evolutionary Algorithms

Radial Basis Function Neural Network(RBFNN)ensembles have long suffered from non-efficient training,where incorrect parameter settings can be computationally disastrous.This paper examines different evolutionary algorithms for training the Symbolic Radial Basis Function Neural Network(SRBFNN)through the behavior’s integration of satisfiability programming.Inspired by evolutionary algorithms,which can iteratively find the nearoptimal solution,different Evolutionary Algorithms(EAs)were designed to optimize the producer output weight of the SRBFNN that corresponds to the embedded logic programming 2Satisfiability representation(SRBFNN-2SAT).The SRBFNN’s objective function that corresponds to Satisfiability logic programming can be minimized by different algorithms,including Genetic Algorithm(GA),Evolution Strategy Algorithm(ES),Differential Evolution Algorithm(DE),and Evolutionary Programming Algorithm(EP).Each of these methods is presented in the steps in the flowchart form which can be used for its straightforward implementation in any programming language.With the use of SRBFNN-2SAT,a training method based on these algorithms has been presented,then training has been compared among algorithms,which were applied in Microsoft Visual C++software using multiple metrics of performance,including Mean Absolute Relative Error(MARE),Root Mean Square Error(RMSE),Mean Absolute Percentage Error(MAPE),Mean Bias Error(MBE),Systematic Error(SD),Schwarz Bayesian Criterion(SBC),and Central Process Unit time(CPU time).Based on the results,the EP algorithm achieved a higher training rate and simple structure compared with the rest of the algorithms.It has been confirmed that the EP algorithm is quite effective in training and obtaining the best output weight,accompanied by the slightest iteration error,which minimizes the objective function of SRBFNN-2SAT.

A Novel Radial Basis Function Neural Network Approach for ECG Signal Classification

ions in the ECG signal.The cardiologist and medical specialistfind numerous difficulties in the process of traditional approaches.The specified restrictions are eliminated in the proposed classifier.The fundamental aim of this work is tofind the R-R interval.To analyze the blockage,different approaches are implemented,which make the computation as facile with high accuracy.The information are recovered from the MIT-BIH dataset.The retrieved data contain normal and pathological ECG signals.To obtain a noiseless signal,Gaborfilter is employed and to compute the amplitude of the signal,DCT-DOST(Discrete cosine based Discrete orthogonal stock well transform)is implemented.The amplitude is computed to detect the cardiac abnormality.The R peak of the underlying ECG signal is noted and the segment length of the ECG cycle is identified.The Genetic algorithm(GA)retrieves the primary highlights and the classifier integrates the data with the chosen attributes to optimize the identification.In addition,the GA helps in performing hereditary calculations to reduce the problem of multi-target enhancement.Finally,the RBFNN(Radial basis function neural network)is applied,which diminishes the local minima present in the signal.It shows enhancement in characterizing the ordinary and anomalous ECG signals.

Local Radial Basis Function Methods: Comparison, Improvements, and Implementation

Radial Basis Function methods for scattered data interpolation and for the numerical solution of PDEs were originally implemented in a global manner. Subsequently, it was realized that the methods could be implemented more efficiently in a local manner and that the local approaches could match or even surpass the accuracy of the global implementations. In this work, three localization approaches are compared: a local RBF method, a partition of unity method, and a recently introduced modified partition of unity method. A simple shape parameter selection method is introduced and the application of artificial viscosity to stabilize each of the local methods when approximating time-dependent PDEs is reviewed. Additionally, a new type of quasi-random center is introduced which may be better choices than other quasi-random points that are commonly used with RBF methods. All the results within the manuscript are reproducible as they are included as examples in the freely available Python Radial Basis Function Toolbox.

A Numerical Method for Solving Ill-Conditioned Equation Systems Arising from Radial Basis Functions

Continuously differentiable radial basis functions (C∞-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C∞-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C∞-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.

Analysis of radial basis function interpolation approach

The radial basis function （RBF） interpolation approach proposed by Freedman is used to solve inverse problems encountered in well-logging and other petrophysical issues. The approach is to predict petrophysical properties in the laboratory on the basis of physical rock datasets, which include the formation factor, viscosity, permeability, and molecular composition. However, this approach does not consider the effect of spatial distribution of the calibration data on the interpolation result. This study proposes a new RBF interpolation approach based on the Freedman＇s RBF interpolation approach, by which the unit basis functions are uniformly populated in the space domain. The inverse results of the two approaches are comparatively analyzed by using our datasets. We determine that although the interpolation effects of the two approaches are equivalent, the new approach is more flexible and beneficial for reducing the number of basis functions when the database is large, resulting in simplification of the interpolation function expression. However, the predicted results of the central data are not sufficiently satisfied when the data clusters are far apart.

An Evolutionary Programming Based on Hidden Neuron Modifiable Radial Basis Function Networks

In this paper, an improved radial basis function networks named hidden neuron modifiable radial basis function (HNMRBF) networks is proposed for target classification, and evolutionary programming (EP) is used as a learning algorithm to determine and modify the hidden neuron of HNMRBF nets. The result of passive sonar target classification shows that HNMRBF nets can effectively solve the problem of traditional neural networks, i. e. learning new target patterns on line will cause forgetting of the old patterns.

Application of Radial Basis Function Network in Sensor Failure Detection

Aim To detect sensor failure in control system using a single sensor signal. Methods A neural predictor was designed based on a radial basis function network(RBFN), and the neural predictor learned the sensor signal on line with a hybrid algorithm composed of n means clustering and Kalman filter and then gave the estimation of the sensor signal at the next step. If the difference between the estimation and the actural values of the sensor signal exceeded a threshold, the sensor could be declared to have a failure. The choice of the failure detection threshold depends on the noise variance and the possible prediction error of neural predictor. Results and Conclusion\ The computer simulation results show the proposed method can detect sensor failure correctly for a gyro in an automotive engine.

Comparison Between Radial Basis Function Neural Network and Regression Model for Estimation of Rice Biophysical Parameters Using Remote Sensing

The radial basis function （RBF） emerged as a variant of artificial neural network. Generalized regression neural network （GRNN） is one type of RBF, and its principal advantages are that it can quickly learn and rapidly converge to the optimal regression surface with large number of data sets. Hyperspectral reflectance （350 to 2500 nm） data were recorded at two different rice sites in two experiment fields with two cultivars, three nitrogen treatments and one plant density （45 plants m^-2）. Stepwise multivariable regression model （SMR） and RBF were used to compare their predictability for the leaf area index （LAI） and green leaf chlorophyll density （GLCD） of rice based on reflectance （R） and its three different transformations, the first derivative reflectance （D1）, the second derivative reflectance （D2） and the log-transformed reflectance （LOG）. GRNN based on D1 was the best model for the prediction of rice LAI and CLCD. The relationships between different transformations of reflectance and rice parameters could be further improved when RBF was employed. Owing to its strong capacity for nonlinear mapping and good robustness, GRNN could maximize the sensitivity to chlorophyll content using D1. It is concluded that RBF may provide a useful exploratory and predictive tool for the estimation of rice biophysical parameters.

Application of the optimal Latin hypercube design and radial basis function network to collaborative optimization

Improving the efficiency of ship optimization is crucial for modem ship design. Compared with traditional methods, multidisciplinary design optimization （MDO） is a more promising approach. For this reason, Collaborative Optimization （CO） is discussed and analyzed in this paper. As one of the most frequently applied MDO methods, CO promotes autonomy of disciplines while providing a coordinating mechanism guaranteeing progress toward an optimum and maintaining interdisciplinary compatibility. However, there are some difficulties in applying the conventional CO method, such as difficulties in choosing an initial point and tremendous computational requirements. For the purpose of overcoming these problems, optimal Latin hypercube design and Radial basis function network were applied to CO. Optimal Latin hypercube design is a modified Latin Hypercube design. Radial basis function network approximates the optimization model, and is updated during the optimization process to improve accuracy. It is shown by examples that the computing efficiency and robustness of this CO method are higher than with the conventional CO method.

SOME PROBLEMS WITH THE METHOD OF FUNDAMENTAL SOLUTION USING RADIAL BASIS FUNCTIONS

The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.

Application of Near Infrared Diffuse Reflectance Spectroscopy with Radial Basis Function Neural Network to Determination of Rifampincin Isoniazid and Pyrazinamide Tablets

Partial least squares（PLS）,back-propagation neural network（BPNN）and radial basis function neural network（RBFNN）were respectively used for estalishing quantative analysis models with near infrared（NIR）diffuse reflectance spectra for determining the contents of rifampincin（RMP）,isoniazid（INH）and pyrazinamide（PZA）in rifampicin isoniazid and pyrazinamide tablets.Savitzky-Golay smoothing,first derivative,second derivative,fast Fourier transform（FFT）and standard normal variate（SNV）transformation methods were applied to pretreating raw NIR diffuse reflectance spectra.The raw and pretreated spectra were divided into several regions,depending on the average spectrum and RSD spectrum.Principal component analysis（PCA）method was used for analyzing the raw and pretreated spectra in different regions in order to reduce the dimensions of input data.The optimum spectral regions and the models＇ parameters were chosen by comparing the root mean square error of cross-validation（RMSECV）values which were obtained by leave-one-out cross-validation method.The RMSECV values of the RBFNN models for determining the contents of RMP,INH and PZA were 0.00288,0.00226 and 0.00341,respectively.Using these models for predicting the contents of INH,RMP and PZA in prediction set,the RMSEP values were 0.00266,0.00227 and 0.00411,respectively.These results are better than those obtained from PLS models and BPNN models.With additional advantages of fast calculation speed and less dependence on the initial conditions,RBFNN is a suitable tool to model complex systems.

MESHLESS METHOD BASED ON COLLOCATION WITH CONSISTENT COMPACTLY SUPPORTED RADIAL BASIS FUNCTIONS

Based on our previous study,the accuracy of derivatives of interpolating functions are usually very poor near the boundary of domain when Compactly Supported Radial Basis Functions (CSRBFs)are used,so that it could result in significant error in solving partial differential equations with Neumann boundary conditions.To overcome this drawback,the Consistent Compactly Supported Radial Basis Functions(CCSRBFs)are developed,which satisfy the predetermined consistency con- ditions.Meshless method based on point collocation with CCSRBFs is developed for solving partial differential equations.Numerical studies show that the proposed method improves the accuracy of approximation significantly.

Band structures of transverse waves in nanoscale multilayered phononic crystals with nonlocal interface imperfections by using the radial basis function method

A radial basis function collocation method based on the nonlocal elastic continuum theory is developed to compute the band structures of nanoscale multilayered phononic crystals. The effects of nonlocal imperfect interfaces on band structures of transverse waves propagating obliquely or vertically in the system are studied. The correctness of the present method is verified by comparing the numerical results with those obtained by applying the transfer matrix method in the case of nonlocal perfect interface. Furthermore, the influences of the nanoscale size, the impedance ratio and the incident angle on the cut-off frequency and band structures are investigated and discussed in detail. Numerical results show that the nonlocal interface imperfections have significant effects on the band structures in the macroscopic and microscopic scale.

HERMITE—BIRKHOFF INTERPOLATION OF SCATTERED DATA BY RADIAL BASIS FUNCTIONS

For Hermite-Birkhoff interpolation of scattered multidumensional data by radial basis function (?),existence and characterization theorems and a variational principle are proved. Examples include (?)(r)=r^b,Duchon's thin-plate splines,Hardy's multiquadrics,and inverse multiquadrics.

Estimation of vegetation biophysical parameters by remote sensing using radial basis function neural network

Hyperspectral reflectance (350~2500 nm) data were recorded at two different sites of rice in two experiment fields including two cultivars, and three levels of nitrogen (N) application. Twenty-five Vegetation Indices (VIs) were used to predict the rice agronomic parameters including Leaf Area Index (LAI, m2 green leaf/m2 soil) and Green Leaf Chlorophyll Density (GLCD, mg chlorophyll/m2 soil) by the traditional regression models and Radial Basis Function Neural Network (RBF). RBF emerged as a variant of Artificial Neural Networks (ANNs) in the late 1980’s. A large variety of training algorithms has been tested for training RBF networks. In this study, Original RBF (ORBF), Gradient Descent RBF (GDRBF), and Generalized Regression Neural Network (GRNN) were employed. Results showed that green waveband Normalized Difference Vegetation Index (NDVIgreen) and TCARI/OSAVI have the best prediction power for LAI by exponent model and ORBF respectively, and that TCARI/OSAVI has the best prediction power for GLCD by exponent model and GDRBF. The best performances of RBF are compared with the traditional models, showing that the relationship between VIs and agronomic variables are further improved when RBF is used. Compared with the best traditional models, ORBF using TCARI/OSAVI improves the prediction power for LAI by lowering the Root Mean Square Error (RMSE) for 0.1119, and GDRBF using TCARI/OSAVI improves the prediction power for GLCD by lowering the RMSE for 26.7853. It is concluded that RBF provides a useful exploratory and predictive tool when applied to the sensitive VIs.

Synchronization of chaos using radial basis functions neural networks

The Radial Basis Functions Neural Network （RBFNN） is used to establish the model of a response system through the input and output data of the system. The synchronization between a drive system and the response system can be implemented by employing the RBFNN model and state feedback control. In this case, the exact mathematical model, which is the precondition for the conventional method, is unnecessary for implementing synchronization. The effect of the model error is investigated and a corresponding theorem is developed. The effect of the parameter perturbations and the measurement noise is investigated through simulations. The simulation results under different conditions show the effectiveness of the method.

A numerical method based on boundary integral equations and radial basis functions for plane anisotropic thermoelastostatic equations with general variable coefficients

A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coe?cients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coe?cients as long as all the requirements of the laws of physics are satis?ed. To check the validity and accuracy of the proposed numerical method, some speci?c test problems with known solutions are solved.