An accelerated coupled normal-mode model based on physics-informed neural networksa)

The coupled normal-mode model is a fundamental tool for simulating underwater sound propagation in range-dependent environments, but its cost is high because the modal Sturm-Liouville problem must be solved repeatedly for every segment. This paper proposes an accelerated coupled normal-mode model based on physics-informed neural networks. Empirical orthogonal function analysis compresses the input sound-speed profiles, and a dual-branch physics-informed neural network predicts the horizontal wavenumbers and modal depth functions. The training loss combines a data-driven term with physics constraints from the modal equation and boundary conditions. On the 2015 Shallow Water Sound Fluctuation experiment dataset, the surrogate reaches a mode-averaged relative error of 1.52 × 10-3% for the wavenumbers and 6.7% for the depth functions, one orders of magnitude smaller than first-order modal perturbation theory, and cuts the online modal-solving time by 85% relative to a finite difference solver. The reconstructed field is evaluated against KRAKENC for idealized internal solitary wave environments, measured 2015 Shallow Water Sound Fluctuation experiment sound-speed sections, and piecewise-linear sloping bathymetry. Transmission loss errors stay within 3 dB and the complex pressure normalized mean squared error remains on the order of 10-5, with comparable accuracy in the measured and sloping environments. The framework offers practical accuracy and computational efficiency for shallow water environments with combined sound-speed and bathymetric range dependence.