any continuous function on a compact interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number of radial basis functions is used. The approximant is differentiable with respect to the weights . The weights could thus be learned using any of the standard iterative methods for neural networks. Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly. RBFs for PDEs Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the Kansa method and was used to solve the elliptic Poisson equation and the linear advection-diffusion equation. The function values at points in the domain are approximated by the linear combination of RBFs: The derivatives are approximated as such: where are the number of points in the discretized domain, the dimension of the domain and the scalar coefficients that are unchanged by the differential operator. Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method, the RBF-QR method and the RBF-PUM method. See also Matérn covariance function Radial basis function interpolation Kansa method References Further reading Sirayanone, S., 1988, Comparative studies of kriging, multiquadric-biharmonic, and other methods for solving mineral resource problems, PhD. Dissertation, Dept. of Earth Sciences, Iowa State University, Ames, Iowa. Artificial neural networks Interpolation Numerical analysis