A new variable shape parameter strategy for RBF approximation using neural networks

TL;DR

This paper introduces a neural network-based method to adaptively select shape parameters for RBFs, improving approximation accuracy and stability in interpolation and finite difference applications.

Contribution

It presents a novel neural network approach for predicting optimal RBF shape parameters, enhancing the flexibility and performance of RBF methods.

Findings

Neural network effectively predicts shape parameters for RBFs.

Improved accuracy in RBF interpolation tasks.

Enhanced stability in RBF-finite difference methods.

Abstract

The choice of the shape parameter highly effects the behaviour of radial basis function (RBF) approximations, as it needs to be selected to balance between ill-condition of the interpolation matrix and high accuracy. In this paper, we demonstrate how to use neural networks to determine the shape parameters in RBFs. In particular, we construct a multilayer perceptron trained using an unsupervised learning strategy, and use it to predict shape parameters for inverse multiquadric and Gaussian kernels. We test the neural network approach in RBF interpolation tasks and in a RBF-finite difference method in one and two-space dimensions, demonstrating promising results.