Oberseminar Mathematical Fluid Dynamics

The choice of the shape parameter highly affects 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. In the end of this talk, I will also talk about work in progress towards self-improving codes,
in which a machine learning algorithm is integrated within a numerical solver, and improves over time by using
local error estimators and online learning.
This is joint work with Fatemeh Nassajian Mojarrad, Jan S. Hesthaven and Philipp Öffner