Generalized-Multiquadric Radial Basis Function Neural Networks (RBFNs) with Variable Shape Parameters for Function Recovery

Abstract

After being introduced to approximate two-dimensional geographical surfaces in 1971, the multivariate radial basis functions (RBFs) have been receiving a great amount of attention from scientists and engineers. In 1987 the idea was extended into the construction of neural networks corresponding to the beginning of the era of artificial intelligence, forming what is now called ‘Radial Basis Function Neural Networks (RBFNs)’. Ever since, RBFNs have been developed and applied to a wide variety of problems; approximation, interpolation, classification, prediction, in nowadays science, engineering, and medicine. This also includes numerically solving partial differential equations (PDEs), another essential branch of RBFNs under the name of the ‘Meshfree/Meshless’ method. Amongst many, the so-called ‘Multiquadric (MQ)’ is known as one of the mostly-used forms of RBFs and yet only a couple of its versions have been extensively studied. This study aims to extend the idea toward more general forms of MQ. At the same time, the key factor playing a very crucial role for MQ called ‘shape parameter’ (where selecting a reliable one remains an open problem until now) is also under investigation. The scheme was applied to tackle the problem of function recovery as well as an approximation of its derivatives using six forms of MQ with two choices of the variable shape parameter. The numerical results obtained in this study shall provide useful information on selecting both a suitable form of MQ and a reliable choice of MQ-shape for further applications in general.

ON A LOCAL $k$-POINT MULTIQUADRIC NEURAL NETWORK IN FUNCTION RECOVERY APPLICATIONS

“…In this work, apart from the popular choice with β = 1/2, β = 5/2 is also included. This is recommended from our previous experiment [24].…”

“…To achieve this, the multiquadric itself needs to be differentiated, as expressed (in x-direction) below: On the choice of shape parameter, the following shape finding strategy (equation 21) was proposed by Xiang et.al. (2012) [25] and has recently been tested to be best amongst other choices, documented in [24], so it is now being employed for the sake of comparison in this work (referred hereafter as STG-2):…”

Numerical Comparison of Shapeless Radial Basis Function Networks in燩attern Recognition

“…A year later, Zheng et al [16] used the approximation error of the least-squares to measure the shape parameter of the radial basis function and proposed a corresponding optimization problem to obtain the optimal shape parameter. Later in 2021, Kaennakham et al [17] compared two interesting forms of shape (one is in exponential base and the other in trigonometric) in the context of recovering functions and their derivatives, and also in this year, Cavoretto et al [18] proposed finding an optimal value of the shape parameter by using leaveone-out cross-validation (LOOCV) technique combined with univariate global optimization methods. Despite the broad range of research under this path, the topic is still highly problem-dependent, i.e., one reasonable good shape for a certain task might be useless for others when problem configuration changes.…”

Generalized Multiquadric Neural Networks in Image Reconstruction

“…The study in this path is named as 'Generalized MQ or GMQ) and was firstly approached by Maggie E. Chenoweth [8] in 2009. Over the years, not much numerical work has been done until one of our preliminary works done in 2021 [9].…”

“…The focus of this work is paid to the numerical application of GMQ under the context of image reconstruction investigated under the structure of a neural network (with space limitation, more details are found in [9]). For this, ten forms of GMQ were numerically investigated and their performances were monitored and recorded.…”