Finite differences provided the first numerical approach that permitted large-scale
simulations in many applications areas, such as geophysical fluid dynamics. As accuracy …

This book is focused on a powerful numerical methodology for solving PDEs to high
accuracy in any number of dimensions: Radial Basis Functions (RBFs). During the past …

When radial basis functions (RBFs) are made increasingly flat, the interpolation error
typically decreases steadily until some point when Runge-type oscillations either halt or …

Radial Basis Function (RBF) methods have become the primary tool for interpolating
multidimensional scattered data. RBF methods also have become important tools for solving …

Radial basis functions (RBFs) are receiving much attention as a tool for solving PDEs
because of their ability to achieve spectral accuracy also with unstructured node layouts …

Physics-informed neural networks (PINNs) are neural networks trained by using physical
laws in the form of partial differential equations (PDEs) as soft constraints. We present a new …

The current paper establishes the computational efficiency and accuracy of the RBF-FD
method for large-scale geoscience modeling with comparisons to state-of-the-art methods …

Radial basis function (RBF) approximation has the potential to provide spectrally accurate
function approximations for data given at scattered node locations. For smooth solutions, the …

In this paper, we present a method based on radial basis function (RBF)-generated finite
differences (FD) for numerically solving diffusion and reaction–diffusion equations (PDEs) …

In this paper we present a high-order kernel method for numerically solving diffusion and
reaction-diffusion partial differential equations (PDEs) on smooth, closed surfaces …