A modified radial basis function network for multiscale scattered data approximation in euclidean space and on the sphere

Radial basis function (RBF) networks are a powerful tool for constructing high-order accurate approximations of scattered data in arbitrary dimension and on manifolds. We develop an algorithm that uses a cross-validated elastic net model together with a modified RBF network to create such approximations in Euclidean space and on the sphere. The elastic net model determines a global background profile. The modified RBF network captures localized features, and has information about the manifold on which the data lies encoded into it. We study the performance of the algorithm in generating approximants with relatively few degrees of freedom. Our results demonstrate the ability of the algorithm to recreate known underlying functions in the plane and capture multiscale features of scattered data sets on the sphere, and compare its accuracy to spectral data approximation. The median root-mean-square relative errors of our approximations of simulation data are on the order of 10 -1 to 10 -4 with approximation ratios on the order of 10 -4 . The accuracy of these approximations is better than spectral approximations with similar approximation ratios, reducing error by a factor of approximately 4.7 to 6.3.