  Surrogate modeling based on Gaussian processes (GPs) has received increasing
attention in the analysis of complex problems in science and engineering.
Despite extensive studies on GP modeling, the developments for functional
inputs are scarce. Motivated by an inverse scattering problem in which
functional inputs representing the support and material properties of the
scatterer are involved in the partial differential equations, a new class of
kernel functions for functional inputs is introduced for GPs. Based on the
proposed GP models, the asymptotic convergence properties of the resulting mean
squared prediction errors are derived and the finite sample performance is
demonstrated by numerical examples. In the application to inverse scattering, a
surrogate model is constructed with functional inputs, which is crucial to
recover the reflective index of an inhomogeneous isotropic scattering region of
interest for a given far-field pattern.
