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

TL;DR

The paper tackles the critical issue of selecting RBF shape parameters by introducing a neural network that predicts adaptive shape parameters for inverse multiquadric and Gaussian kernels. It applies the approach to RBF interpolation and RBF-FD-based PDE solvers in 1D and 2D settings, reporting improved accuracy and conditioning with negligible runtime overhead. The method enables data-driven, locally adaptive parameter selection that enhances robustness on smooth problems and shows potential for extension to more general geometries. Overall, it contributes a practical framework for integrating neural-network-based shape parameter prediction into RBF-based numerical 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.