Transcending Sparse Measurement Limits: Operator-Learning-Driven Data Super-Resolution for Inverse Source Problem

TL;DR

This paper introduces a novel operator-learning framework that significantly enhances inverse source localization accuracy from extremely sparse measurements by combining deep operator networks with classical inversion methods.

Contribution

The work extends uniqueness theorems for inverse source problems and develops a modular DeepONet-based super-resolution approach to improve localization from sparse data.

Findings

DeepONet interpolation reduces localization error by about an order of magnitude.

The framework achieves grid-level precision with sparse data for single sources.

Effective localization is maintained even with apertures as small as π/4.

Abstract

Inverse source localization from Helmholtz boundary data collected over a narrow aperture is highly ill-posed and severely undersampled, undermining classical solvers (e.g., the Direct Sampling Method). We present a modular framework that significantly improves multi-source localization from extremely sparse single-frequency measurements. First, we extend a uniqueness theorem for the inverse source problem, proving that a unique solution is guaranteed under limited viewing apertures. Second, we employ a Deep Operator Network (DeepONet) with a branch-trunk architecture to interpolate the sparse measurements, lifting six to ten samples within the narrow aperture to a sufficiently dense synthetic aperture. Third, the super-resolved field is fed into the Direct Sampling Method (DSM). For a single source, we derive an error estimate showing that sparse data alone can achieve grid-level precision. In two- and three-source trials, localization from raw sparse measurements is unreliable, whereas DeepONet-reconstructed data reduce localization error by about an order of magnitude and remain effective with apertures as small as $\pi/4$. By decoupling interpolation from inversion, the framework allows the interpolation and inversion modules to be swapped with neural operators and classical algorithms, respectively, providing a practical and flexible design that improves localization accuracy compared with standard baselines.