Functional-Input Gaussian Processes with Applications to Inverse Scattering Problems

TL;DR

This paper introduces a new class of Gaussian process models with functional inputs, providing theoretical convergence analysis and demonstrating their effectiveness in inverse scattering problems for recovering material properties.

Contribution

The paper develops novel kernel functions for Gaussian processes with functional inputs and analyzes their convergence, addressing a gap in surrogate modeling for complex functional data.

Findings

Derived asymptotic convergence properties of the proposed GP models.

Numerical examples show good finite sample performance.

Applied the models to inverse scattering, successfully recovering material properties.

Abstract

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.