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

TL;DR

The paper addresses inverse source localization from boundary data gathered over a narrow aperture, a highly ill-posed problem under sparse measurements. It proposes a modular framework that decouples interpolation from inversion by using a DeepONet-based neural operator to densify sparse boundary data, followed by the Direct Sampling Method (DSM) for localization. Key theoretical contributions include a finite-aperture uniqueness theorem for Dirac sources and DSM error estimates in the single-source case, together with an empirical demonstration that operator-driven interpolation reduces localization errors by about an order of magnitude for 2- and 3-source configurations, even when the aperture is as small as $\pi/4$. The approach yields a plug-and-play design that preserves the interpretability of classical solvers while boosting accuracy, with broad potential for limited-aperture imaging in acoustics, electromagnetism, and biomedical contexts.

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 $π/4$. 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.

Figures (14)

• Figure 1: DSM indicator for a single source (Example \ref{['singleDiracradiator']}). Maps of $I_{DSM}(x)$ for apertures (a) $S_1 = [-\pi/2,\pi/2]$, (b) $S_2 = [-\pi/3,\pi/3]$ and (c) $S_3 = [-\pi/4,\pi/4]$ at $k=4$. Red crosses: true source; blue circles: DSM peaks.
• Figure 2: Three-source DSM with sparse angles (Example \ref{['exa:10_measurements']}, $M=10$). Indicator maps for wavenumber sets (a) $\mathcal{A}_1$ to (f) $\mathcal{A}_6$. Red crosses: truth; blue circles: detected peaks. As the number of wavenumbers decreases, peaks drift away from the true locations.
• Figure 3: Three-source DSM with dense angles (Example \ref{['exa:128_measurements']}, $M=128$). Indicator maps for wavenumber sets (a) $\mathcal{A}_1$ to (f) $\mathcal{A}_6$. Red crosses: truth; blue circles: detected peaks. Dense angular sampling brings peaks closer to the ground truth and suppresses spurious extrema.
• Figure 4: Measured field along the measurement curve. (a-c): real part; (d-f): imaginary part at $k=4,6,8$. Black dots: sparse ($M=10$) measurements; dashed curves: dense ($M=128$) measurements. Sparse measurements under-resolve oscillations, while dense measurements capture the waveform and supplie richer information to DSM.
• Figure 5: Operator-learning-driven framework for sparse partial apertures. Sparse boundary measurements are interpolated to a dense, self-consistent trace by a pre-trained neural operator; the dense data are then consumed by a classical inverse solver. The reconstruction and inversion modules are decoupled and can be swapped with alternative neural operators and classical solvers.

Theorems & Definitions (13)

• Theorem 1
• proof
• Theorem 2
• proof
• Remark 1
• Example 1
• Example 2
• Example 3
• Remark 2
• Remark 3