Radial Basis Function Surrogates
Radial basis function interpolants build a surrogate from weighted bumps centered on training points, giving smooth fits with simple linear algebra.
The interpolant
A radial basis function (RBF) surrogate represents the response as f_hat(x) = sum of w_i * phi(||x - x_i||), a weighted sum of basis functions that depend only on the distance from each training center x_i. Because each basis is radially symmetric about its center, the method extends cleanly to any number of input dimensions.
Choosing the basis
- Gaussian: phi(r) = exp(-(epsilon r)^2), compact and smooth
- Multiquadric: phi(r) = sqrt(1 + (epsilon r)^2), globally supported
- Inverse multiquadric: phi(r) = 1/sqrt(1 + (epsilon r)^2)
- Thin-plate spline: phi(r) = r^2 log r, parameter-free and popular for smooth surfaces
Solving for the weights
Requiring the interpolant to pass through every training point gives a linear system Phi w = y, where Phi is the matrix of basis values between all pairs of centers. Solving this system once yields the weights; evaluation afterward is a fast sum. For conditionally positive-definite bases like multiquadrics, a low-order polynomial term is added with side constraints to guarantee a unique solution.
The shape parameter
Gaussian and multiquadric bases carry a shape parameter epsilon that sets how wide each bump is. Small epsilon (flat bumps) gives high accuracy but an ill-conditioned matrix; large epsilon is stable but less accurate. Choosing epsilon by cross-validation is a routine but delicate step; the trade-off is sometimes called the uncertainty principle of RBF interpolation.
Strengths and weaknesses
RBFs are meshfree, handle scattered data in high dimensions, and are cheap to evaluate once fit. Unlike Gaussian processes they do not natively provide error bars, though a probabilistic reinterpretation recovers them. Dense global bases produce a full linear system that scales poorly to very large datasets unless compactly supported bases are used.
Use in practice
RBF surrogates are a workhorse for surrogate-based optimization and for reconstructing smooth engineering fields from a handful of expensive runs. In Kronos scans they provide fast interpolants where a full probabilistic treatment is unnecessary and only a smooth response map is needed.