Multi-UUV radiated-noise spectral prediction in shallow water using neural field priors and graph residual learning

Predicting the received radiated-noise power spectral density (PSD) of multiple unmanned underwater vehicles (UUVs) in shallow-water waveguides is challenging. The measured spectrum is jointly affected by incoherent multi-source power superposition, propeller rotation speed, source–receiver geometry, and multipath propagation caused by boundary reflections. Traditional numerical methods typically require detailed structural parameters, excitation inputs, and environmental information. In contrast, purely data-driven models lack mechanisms to preserve single-UUV spectral characteristics and account for the permutation-invariant nature of multi-source configurations. To address this, we propose the multi-UUV neural radiated-noise field (MNRNF). MNRNF decomposes the process of predicting the received PSD of the radiated noise from multiple UUVs into single-UUV prior spectrum prediction and graph neural network (GNN)-based residual correction. For each UUV, the model first predicts the single-source spectrum under the corresponding source–receiver conditions and aggregates the predictions in the power domain to construct the prior spectrum. A permutation-invariant GNN residual learner then estimates frequency-dependent corrections based on the source-set geometry, receiver positions, and prior-spectrum information. Experiments involving three simultaneously operating homogeneous UUVs were conducted in a shallow-water lake under three propeller-rotation speeds, with the same MNRNF architecture trained and evaluated independently for each condition. Across the three speed conditions, MNRNF achieved the lowest prediction errors among the compared methods, with average mean-absolute-error and root-mean-square-error values of 3.18 and 4.09 dB, respectively. Residual analyses, ablation studies, and evaluations across different spatial configurations further demonstrate that combining spectral priors with residual learning provides an effective approach for predicting the received spectra of multiple UUVs. The proposed framework also establishes a practical basis for assessing the radiated noise from UUV formations.