Radial Basis Function Method Research Articles

Meshless symplectic and multi-symplectic algorithm for Klein–Gordon–Schrödinger system with local RBF collocation

This work proposes a novel meshless symplectic and multi-symplectic approximation for Klein–Gordon–Schrödinger system. The main features of this method are: (I) the application of local radial basis function (RBF) collocation method (LRBFCM) in spatial discretization; (II) the application of symplectic integrator in time discretization. LRBFCM method can deal with the ill-conditioned problem and shape-parameter sensitivity of global RBF method, and is simple and effective. The conservatism of this method is discussed and its accuracy is evaluated. Numerical simulations are given for one dimension (1D)/2D with regular and irregular domains to verify the effectiveness of our new method.

Numerical analysis of time-fractional Sobolev equation for fluid-driven processes in impermeable rocks

This paper proposes a local meshless radial basis function (RBF) method to obtain the solution of the two-dimensional time-fractional Sobolev equation. The model is formulated with the Caputo fractional derivative. The method uses the RBF to approximate the spatial operator, and a finite-difference algorithm as the time-stepping approach for the solution in time. The stability of the technique is examined by using the matrix method. Finally, two numerical examples are given to verify the numerical performance and efficiency of the method.

Assessment of Three-Dimensional Interpolation Method in Hydrologic Analysis in the East China Sea

The water mass in the East China Sea (ECS) shelf has a complicated three-dimensional (3D) hydrologic structure. However, previous studies mostly concentrated on the sea surface based on the sparse in situ and incomplete satellite-derived observations. Therefore, the 3D interpolation technology was introduced for the reconstruction of hydrologic structure in the ECS shelf using in situ temperature and salinity observations in the summer and autumn of 2010 to 2011. Considering the high accuracy and good fitness of the radial basis function (RBF) methods, we applied the RBF methods to the in situ observations to completely reconstruct the 3D hydrologic fields. Other 3D interpolation methods and 2D methods were also tested for a comparison. The cubic and thin plate spline RBFs were recommended because their mean absolute error (MAE) in the 10-fold cross-validation experiments maintained the order of ~10−2. The 3D RBF reconstructions showed a reasonable 3D hydrologic structure and extra details of the water masses in the ECS shelf. It also helps evaluate regional satellite-derived sea surface temperature (SST). Comparisons between the interpolated and satellite-derived SST indicates that the large bias of satellite-derived SST in the daytime corresponds to weak mixing during low-speed wind and shows seasonal variation.

Laminar natural convection in a 2-D pentagonal house with sloping roof heated by constant heat flux

Natural convection in a pentagonal house with a protruding cooler for summer is numerically investigated. The sloping roof is subjected to constant heat flux, and the cooler maintains at a uniform cold temperature. The remaining walls of the house are insulated. Local multi-quadratic radial basis function (LMQRBF) method is applied in solution of Navier–Stokes equations in stream function-vorticity form. The working fluid in the house is air with Pr = 0.71. The effects of Rayleigh number (103 ≤ Ra ≤ 106), heat flux of the roof (0.1 ≤ q ≤ 2.0), and the cooler’s vertical position (0.65 ≤ H/L ≤ 0.85) are explored. Streamlines, isotherms, velocity profiles, and the distribution of Nusselt number are displayed. Numerical results indicate that for the top and right wall of the cooler, the heat transfer is stronger than that for the bottom wall. It reveals that all the governing parameters efficiently impact the generation and evolution of the secondary and tertiary vortex in the gable roof. It is also observed that the variation of average Nusselt number on the right wall with Rayleigh number is under the combined influence of the heat flux and the cooler’s vertical position.

A meshless multi-symplectic local radial basis function collocation scheme for the “good” Boussinesq equation

A novel meshless multi-symplectic scheme is proposed for the “good” Boussinesq equation. The scheme consists of a local radial basis function (RBF) collocation method (LRBFCS) in space and a symplectic integrator in time. The LRBFCS is simple and efficient, since only a sparse banded linear system has to be solved. Moreover, it can avoid the ill-conditioned problem and shape-parameter-sensitivity of the global RBF method. The multi-symplectic LRBFCS (MLRBFCS) is more accurate than traditional methods. Numerical experiments with uniform knots and random knots are designed to verify the effectiveness of the method.

Improved phenomenological nuclear charge radius formulae with kernel ridge regression * *Supported by National Natural Science Foundation of China (11875027, 11775112, 11775026, 11775099, 11975096), Fundamental Research Funds for the Central Universities (2021MS046)

The kernel ridge regression (KRR) method with a Gaussian kernel is used to improve the description of the nuclear charge radius by several phenomenological formulae. The widely used , and formulae, and their improved versions including isospin dependence, are adopted as examples. The parameters in these six formulae are refitted using the Levenberg–Marquardt method, which give better results than the previous versions. The radius for each nucleus is predicted with the KRR network, which is trained with the deviations between experimental and calculated nuclear charge radii. For each formula, the resultant root-mean-square deviations of 884 nuclei with proton number and neutron number can be reduced to about 0.017 fm after considering the modification by the KRR method. The extrapolation ability of the KRR method for the neutron-rich region is examined carefully and compared with the radial basis function method. It is found that the improved nuclear charge radius formulae using the KRR method can avoid the risk of overfitting, and have a good extrapolation ability. The influence of the ridge penalty term on the extrapolation ability of the KRR method is also discussed. Finally, the nuclear charge radii of several recently observed K and Ca isotopes are analyzed.

Backpropagation and Radial Basis Function Methods for Predicting Rainfall in Sukabumi City Using Artificial Neural Networks: A Comparative Analysis

The weather has a substantial impact on the ability to live organisms to carry out everyday activities, particularly outside activities. Weather data is helpful in various fields, including marine, aviation, and agriculture. The maritime domain is beneficial for establishing the optimal navigation time for a fisherman, the aviation domain helps reduce climate-related mishaps, and the agriculture sector uses weather information to develop harvest season models for agricultural products. Indonesia is a tropical nation with heavy precipitation. Utilized for various objectives, rainfall forecasting models seek the utmost precision, particularly in specialized areas such as flood control. This study is based on two techniques: the Radial Basis Function Neural Network (RBFNN) and Backpropagation Neural Network (BPNN) techniques using multiple training functions. The RBFNN approach yields less accurate results for predicting precipitation, but the multi-practice BPNN method yields more accurate results.

Bidirectional Deep Learning of Polarization Transfer in Liquid Crystals with Application to Quantum State Preparation

Accurate control of light polarization represents a core building block in polarization metrology, imaging, and optical and quantum communications. Voltage-controlled liquid crystals offer an efficient way of polarization transformation. However, common twisted nematic liquid crystals are notorious for lacking an accurate theoretical model linking control voltages and output polarization. An inverse model, which would predict control voltages required to prepare a target polarization, is even more challenging. Here we report both the direct and inverse models based on deep neural networks, radial basis functions, and linear interpolation. We present an inverse-direct compound model solving the problem of control voltages ambiguity. We demonstrate one order of magnitude improvement in accuracy using deep learning compared to the radial basis function method and two orders of magnitude improvement compared to the linear interpolation. Errors of the deep neural network model also decrease faster than the other methods with an increasing number of training data. The best direct and inverse models reach the average infidelities of $4 \times 10^{-4}$ and $2 \times 10^{-4}$, respectively, which is the accuracy level not reported yet. Furthermore, we demonstrate local and remote preparation of an arbitrary single-photon polarization state using the deep learning models. The results will impact the application of twisted-nematic liquid crystals, increasing their control accuracy across the board. The presented bidirectional learning can be used for optimal classical control of complex photonic devices and quantum circuits beyond interpolation.

Improvement of the five-hole probe calibration using artificial neural networks

In the present study, the artificial neural networks (ANNs) technique was implemented to link non-dimensional pressure coefficients and flow characteristics to calibrate a five-hole probe. The experimental data of this work were obtained from a subsonic open-circuit wind tunnel at the velocity of 10 m/s. Here, the efficiency of ANNs was compared with two conventional data reduction methods, including linear interpolation technique and 5th-order polynomial surface fit algorithm. Based on the statistical parameters of calibration data, it was concluded that the radial basis function (RBF) algorithm was more accurate and had more flexibility compared to the multi-layer perceptron (MLP) regression algorithm, the linear interpolation and 5th-order polynomial methods. In the RBF method, the mean absolute errors of 0.11, 0.64, 0.02 and 0.03 were achieved for α, β, Cpt and Cps , respectively. Furthermore, the effects of training data reduction and data selection on the performance of RBF were studied. The accuracy of the proposed RBF method was analyzed at different α angles and for random test data. Finally, the influence of increasing number of test data on the efficiency of calculated RBF method was evaluated.

Analytical assessment of the time‐space fractional bioheat transfer equation by the radial basis function method for living tissues

AbstractIn recent years, thermal treatment has proven to be helpful, notably in oncology. In fact, ablating the superfluous mass and eliminating the malignant tumor using a different modality such as warmth or cooling is a therapeutic method. In this study, the radial basis function approach was used to answer the time‐space fractional heat transfer equations of human body tissue during thermal therapy. To validate the radial basis function technique, it was also compared to the fourth‐order Runge–Kutta numerical method. The findings demonstrated that this methodology is extremely accurate and efficient, with an error rate of less than 1%. The effect of time and spatial fractional parameters, blood perfusion coefficient, metabolic coefficient, and internal source coefficient on the temperature profile inside human living tissues has been studied and depicted well.

A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems

In this article, we propose a simplified radial basis function (RBF) method with exterior fictitious sources for solving elliptic boundary value problems (BVPs). Three simplified RBFs, including Gaussian, multiquadric (MQ), and inverse multiquadric (IMQ) without the shape parameter, are adopted in this study. With the consideration of many exterior fictitious sources outside the domain, the radial distance of the RBF is always greater than zero, such that we can remove the shape parameter from RBFs. Additionally, simplified Gaussian, MQ, and IMQ RBFs and their derivatives in the governing equation are always smooth and nonsingular. Comparative analysis is conducted for three different collocation types, including conventional uniform centers, randomly fictitious centers, and exterior fictitious sources. Numerical examples of elliptic BVPs in two and three dimensions are carried out. The results demonstrate that the proposed simplified RBFs with exterior fictitious sources can significantly improve the accuracy, especially for the Laplace equation. Furthermore, the proposed simplified RBFs exhibit the simplicity of solving elliptic BVPs without finding the optimum shape parameter.

Mathematical modeling of fractional derivatives for magnetohydrodynamic fluid flow between two parallel plates by the radial basis function method

Investigations into the magnetohydrodynamics of viscous fluids have become more important in recent years, owing to their practical significance and numerous applications in astro-physical and geo-physical phenomena. In this paper, the radial base function was utilized to answer fractional equation associated with fluid flow passing through two parallel flat plates with a magnetic field. The magnetohydrodynamics coupled stress fluid flows between two parallel plates, with the bottom plate being stationary and the top plate moving at a persistent velocity. We compared the radial basis function approach to the numerical method (fourth-order Range-Kutta) in order to verify its validity. The findings demonstrated that the discrepancy between these two techniques is quite negligible, indicating that this method is very reliable. The impact of the magnetic field parameter and Reynolds number on the velocity distribution perpendicular to the fluid flow direction is illustrated. Eventually, the velocity parameter is compared for diverse conditions α, Reynolds and position (y), the maximum of which occurs at α = 0.4. Also, the maximum velocity values occur in α=0.4 and Re=1000 and the concavity of the graph is less for α=0.8.

Solving Inverse Problems of Stationary Convection–Diffusion Equation Using the Radial Basis Function Method with Polyharmonic Polynomials

In this article, the radial basis function method with polyharmonic polynomials for solving inverse problems of the stationary convection–diffusion equation is presented. We investigated the inverse problems in groundwater pollution problems for the multiply-connected domains containing a finite number of cavities. Using the given data on the part of the boundary with noises, we aim to recover the missing boundary observations, such as concentration on the remaining boundary or those of the cavities. Numerical solutions are approximated using polyharmonic polynomials instead of using the certain order of the polyharmonic radial basis function in the conventional polyharmonic spline at each source point. Additionally, highly accurate solutions can be obtained with the increase in the terms of the polyharmonic polynomials. Since the polyharmonic polynomials include only the radial functions. The proposed polyharmonic polynomials have the advantages of a simple mathematical expression, high precision, and easy implementation. The results depict that the proposed method could recover highly accurate solutions for inverse problems with cavities even with 5% noisy data. Moreover, the proposed method is meshless and collocation only such that we can solve the inverse problems with cavities with ease and efficiency.

Proper orthogonal decomposition and physical field reconstruction with artificial neural networks (ANN) for supercritical flow problems

The development of mathematical models, and the associated numerical simulations, are challenging in higher-dimensional systems featuring flows of supercritical fluids in various applications. In this paper, a data-driven methodology is presented to achieve system order reduction, and to identify important physical information within the principal flow features. Firstly, a new hybrid neural network based on radial basis function (RBF) and multi-layer perceptron (MLP) methods, namely RBF-MLP, is tested to achieve a highly nonlinear approximation. When provided with 1000 nonlinear test samples, this model provides an excellent prediction accuracy with a maximum regression coefficient (R) of 0.99 and a minimum root mean square error (RMSE) below 1%. Furthermore, the model is also proven to be flexible enough to capture accurately the turbulent fluctuation characteristics, even at significant nonlinear buoyancy conditions. Secondly, the high-dimensional buoyancy data is collected and integrated into a matrix database. Subsequently, a proper orthogonal decomposition (POD) approach is employed to reduce the high-dimensional database, and to obtain a set of low-dimensional POD basis-spanned space, which defines a reduced-order system with low-dimensional basis vectors. The results reveal that the first five order modes contain dominant flow features, accounting for 93% of the total mode energy, which can be selected to reconstruct the physical flow field. Thirdly, a new data-driven POD-ANN model is established to construct the nonlinear mapping between the full-field buoyancy data and decomposed basis vectors. It is also demonstrated that the POD-ANN model reconstructs the principal flow features accurately and reliably. This POD-ANN model can be used to provide new insights for reduced-order modelling and for reconstructing physical fields of higher-dimensional nonlinear flow cases.

Numerical investigation of the resonance modes for the dielectric optical micro-spheres by using radial basis function method

In this paper, we present a numerical meshless method, which is based on the radial basis functions (RBF), to solve the morphology-dependent resonance (MDR) modes of a dielectric micro-sphere. To check the accuracy of the considered RBF method, we calculated the transverse electric (TE) and transverse magnetic (TM) modes of a perfect sphere that has an analytical characteristic equation. Besides, we employed the traditional mesh-dependent finite difference (FD) method to study the same problem. When we compared the results obtained by the three methods, we observe that the RBF method’s results are in very good agreement with the analytical and FD method’s results. We conclude that the presented meshless method may also be applicable to prolate and oblate spheroidal geometries, which do not have an analytical equation to calculate the modes.

Nonintrusive reduced‐order modeling approach for parametrized unsteady flow and heat transfer problems

AbstractA novel nonintrusive reduced‐order modeling approach is proposed for parametrized unsteady flow and heat transfer problems. A set of reduced basis functions are extracted via a proper orthogonal decomposition (POD) method from a collection of high‐fidelity numerical solutions (snapshots of spatial distribution at a series of time steps) that are computed for properly chosen parameters (e.g., material property, initial/boundary conditions) using a full‐order model (finite‐volume/finite‐element methods). Here, the time dimension is treated separately from other parameters, reflecting the dynamic features of the flow problems. Dynamic mode decomposition is used to decompose the time‐resolved data (POD coefficients) into dynamics modes and reconstruct/predict the dynamic evolution of the flow systems. The POD coefficients at every time step under the condition of a set of parameters are approximated through interpolation with the radial basis function method from those obtained from the snapshots of the chosen parameter space. Hence, the POD coefficients at a set of given time and parameters can be approximated, and the spatial distribution of the solution data can be reconstructed. The current reduced‐order model has been applied to the problems governed by a set of parametrized unsteady Navier–Stokes and heat conduction equations. The model capability has been illustrated numerically by three numerical test cases: flow past a cylinder, melt solidification, and dam break. The model prediction capabilities have been evaluated by varying the material properties (viscosity and density) and the heat transfer conditions on some of the boundaries. An error analysis is also carried out to show the model prediction accuracy.

Estimating Soil Water Retention Curve by Extreme Learning Machine, Radial Basis Function, M5 Tree and Modified Group Method of Data Handling Approaches

AbstractThis study was conducted to assess the applicability of novel types of neural network methods (extreme learning machine (ELM), radial basis function (RBF), modified group method of data handling (M‐GMDH)) and M5 tree methods for the prediction of the soil water retention curve (SWRC) and compare their performance with that of derived methods and pedotransfer functions (PTFs) of other studies for soils in Iran. Then, 15 PTFs were developed. Predictions were evaluated by the integral root mean square error (IRMSE), integral mean error (IME), Akaike's information criterion (AIC), and coefficient of determination (R2). The RBF‐based PTFs were better than the M5 tree, M‐GMDH, and ELM‐based PTFs in terms of the IRMSE criterion in the testing step. Also, PTFs and methods developed in the present study were more reliable than other derived PTFs and methods by different researchers. The values of the IRMSE and R2 in the best PTFs (with inputs of sand, clay, total porosity and the moisture content at field capacity and permanent wilting point) of the testing data set of the RBF method were 0.037 cm3cm−3 and 0.988, respectively. For the testing data set, the average values of the IRMSE criterion for all the PTFs of the RBF, ELM, M‐GMDH, and M5 tree methods were 0.051, 0.062, 0.055, and 0.054 cm3cm−3, respectively. Therefore, the differences were considerable only between the ELM and other methods. The IRMSE criterion results of the testing data set showed the suitability of the RBF method in the development of PTFs for the prediction of the SWRC.

A Novel Radial Basis Function Approach for Infiltration-Induced Landslides in Unsaturated Soils

In this article, the modeling of infiltration--induced landslides, in unsaturated soils using the radial basis function (RBF) method, is presented. A novel approach based on the RBF method is proposed to deal with the nonlinear hydrological process in the unsaturated zone. The RBF is first adopted for curve fitting to build the representation of the soil water characteristic curve (SWCC) that corresponds to the best estimate of the relationship between volumetric water content and matric suction. The meshless method with the RBF is then applied to solve the nonlinear Richards equation with the infiltration boundary conditions. Additionally, the fictitious time integration method is adopted in the meshless method with the RBF for tackling the nonlinearity. To model the stability of the landslide, the stability analysis of infinite slope coupled with the nonlinear Richards equation considering the fluctuation of transient pore water pressure is developed. The validation of the proposed approach is accomplished by comparing with exact solutions. The comparative analysis of the factor of safety using the Gardner model, the van Genuchten model and the proposed RBF model is provided. Results illustrate that the RBF is advantageous for reconstructing the SWCC with better estimation of the relationship than conventional parametric Gardner and van Genuchten models. We also found that the computed safety factors significantly depend on the representation of the SWCC. Finally, the stability of landslides is highly affected by matric potential in unsaturated soils during the infiltration process.

Anti-aliasing of gray-scale/color/outline images: Looking through the lens of numerical approaches for PDE-based models

Down-sampling in the process of saving and transferring images often leads to the appearance of jaggies or step like edges in the objects of images where they should be smooth to seem natural. The application of blurring filters for the purpose of anti-aliasing images seems inappropriate since it leaves a blurring effect that destroys the edges in a way that they become indistinguishable. In 2012 Ziou and Horé proposed a novel partial differential equation (PDE) in which anti-aliased images efficiently. However, solving this equation as it is severely nonlinear and defined in a large domain due to the size of images is a challenging problem. In this paper, we propose a new numerical method that is based on a combination of iterative operator splitting (IOS) technique with Gaussian radial basis functions (RBFs) method for solving this equation in an efficient way. The proposed method converts the problem to several one-dimensional linear problems that are solved in smaller sizes to overcome the large domain of the problem. The Crank-Nicolson method is applied for the discretization of the time dimension. The proposed method is tested on different gray-scale, color, and outline images to illustrate the efficiency of the proposed method visually and numerically in the section of numerical results.

Identification of Flexural Modulus and Poisson’s Ratio of Fresh Femoral Bone Based on a Finite Element Model

Finite element analysis (FEA) is increasingly applied to medicine because it could increase accuracy and rapid outcomes. However, there is a lack of the method to determine Young’s modulus and Poisson’s ratio for fresh femoral bone and the mathematical principle’s optimization for calculating nonuniform configuration. This study aimed to investigate the surrogate model for the optimization method to determine Young’s modulus and Poisson’s ratio of the fresh femoral bone. Young’s modulus and Poisson’s ratio obtained 20 ranked pairs by the Latin hypercube sampling method. The values ​​were calculated in the finite element for root mean square error (RMSE) and were then used for solutions by a quadratic function, radial basis function (RBF), and Kriging (KG). The lowest RMSE value was 0.1518 for the RBF method, with the young’s modulus at 304.4756 and the Poisson’s ratio at 0.3334. The current study identified the RBF technique to determine the properties of the femoral bone. Moreover, the RBF procedure might apply to other long bones because of the comparable nonuniform configuration.