A covariance-based reduced-order framework for solving acoustic scattering problems

Abstract

This paper presents a physics-aware reduced-order method (ROM) for the efficient computation of wave-scattering problems. Standard model order reduction techniques, typically treating scattering as generic parameterized systems, frequently overlook the underlying physical structure, limiting their effectiveness in practice. To address this limitation, we propose an algorithmic framework that utilizes the intrinsic low-rank structure of the induced contrast source density. By modeling the incident wave as a random variable governed by a specified prior probability measure, we formulate the contrast source as a spatial random field whose covariance function captures essential spatial correlation and physical interactions. The reduced-order basis is then constructed via the Karhunen-Lo\`eve (KL) expansion, effectively extracting the dominant features from the scattering process to resolve multiple scattering scenarios. A central algorithmic contribution is the efficient reconstruction of the covariance matrix for arbitrary scatterer geometries and specified incident wave priors. To circumvent the prohibitive computational cost of assembling high-fidelity covariance matrices, we introduce a non-intrusive, physics-informed graph neural network (GNN) surrogate to learn the complex mapping from scatterer geometry to the source correlation kernel, enabling a highly efficient offline-online computational paradigm suitable for large-scale scattering configurations. Extensive numerical experiments demonstrate that the proposed framework achieves robust computational acceleration over full-order models without sacrificing accuracy.