A parabolic equation-based physics-informed machine learning method for underwater sound propagation modeling

Underwater sound propagation modeling is crucial for ocean environmental monitoring, underwater communication, and target localization. Traditional underwater acoustics models are limited by high computational costs and restricted adaptability, while data-driven machine learning methods lack physical constraints, leading to poor generalization and reliance on large datasets. Although Physics-Informed Neural Networks have recently emerged to integrate physical priors, they still face challenges in achieving accurate long-range extrapolation. To address this limitation, we propose U-PARANET, a physics-informed machine learning method that incorporates the parabolic equation as a hard constraint directly into its architecture. The model leverages the parabolic equation's recursive, range-stepping structure within a neural network framework, enhancing stability and mitigating error accumulation over long-range propagation. Validation on both simulated and experimental data shows that U-PARANET accurately predicts transmission loss and phase structures, with good agreement in spatial field patterns. Specifically, the mean absolute error for transmission loss prediction is 1.40 dB in an ideal shallow-water environment, 1.06 dB in a simulation using SWellEx-96 environmental parameters, and 2.87 dB on SWellEx-96 experimental data. In conclusion, the proposed method exhibits excellent long-range modeling capabilities, demonstrating robust extrapolation in challenging, realistic environments.