Range-only physics-informed neural network for underwater acoustic field reconstruction with application to source depth estimation

Physics-informed neural networks (PINNs) have been applied in underwater acoustics mainly to reconstruct pressure fields by solving the Helmholtz equation. When measurements are collected by a single hydrophone with a moving source, the data provide limited depth diversity, so the range dependence of the field is the most directly constrained component. We derive a one-dimensional governing equation that describes the complex pressure field as a function of range, enabling physics-constrained reconstruction from single depth measurements. The reconstructed field is used for source depth estimation through synthetic aperture beamforming. By suppressing range-induced phase fluctuations and yielding a smooth, physically consistent range dependence through the range-only governing equation, the proposed PINN reduces spectral broadening and sidelobe levels in the wavenumber spectrum. Numerical simulations and SWellEx-96 experimental data show that the proposed PINN can reconstruct the field under incomplete sampling and range perturbations. The method is also robust to sound speed profile mismatch of a certain level, due to the combination of physical constraint with the data-fidelity loss term.