Learning a robust shape parameter for RBF approximation

• Shape parameter tuning is an open problem that appears in interpolation, kernel density estimation and meshless methods. • New optimization problem to find the shape parameter ε while controlling the condition number of the interpolation matrix A = . • Creation of a dataset to train a neural network that predicts the shape parameter ε given any set of points x ‾ ⊂ R n of size N . • Novel fallback procedure to guarantee that the proposed shape parameter generates a well-posed interpolation matrix. • Method is evaluated for 1-dimensional and 2-dimensional problems in interpolation and numerical solution of PDEs. Radial basis functions (RBFs) play an important role in function interpolation, in particular, when considering an arbitrary set of interpolation nodes. The accuracy of the interpolation depends on a parameter called the shape parameter . There are many approaches in the literature on how to appropriately choose it to increase the accuracy of interpolation while avoiding stability issues. However, finding the optimal shape parameter remains a challenge in general. We introduce a new method for determining the shape parameter in RBFs. First, we construct an optimization problem to obtain a shape parameter that leads to a bounded condition number for the interpolation matrix, then, we introduce a data-driven method that controls the condition number of the interpolation matrix to avoid numerically unstable interpolations, while keeping good accuracy. In addition, a fallback procedure is proposed to enforce a strict upper bound on the condition number, as well as a learning strategy to improve the performance of the data-driven method by learning from previously run simulations. Several numerical results are presented to demonstrate the robustness of our strategy in both 1- and 2-dimensional spaces.