A PI-DeepONet-Based Rapid and Accurate Wide-Area ELF Computation Method for Smart Ocean Sensing with Theoretical and Experimental Validations

Rapid three-dimensional electromagnetic field simulation in stratified marine environments is essential for underwater target sensing system design. This paper presents a Physics-Informed Deep Operator Network (PI-DeepONet) that integrates analytical Sommerfeld integral solutions with seafloor experimental measurements to establish a validated, mesh-free forward modeling framework for extremely low-frequency (ELF) electromagnetic propagation. The architecture uses Fourier feature encoding to resolve multiscale dipole fields and incorporates Maxwell’s divergence constraint through automatic differentiation. The model is evaluated using a tiered validation strategy that combines analytical benchmarks, controlled seafloor experiments, and comparison with a purely data-driven DeepONet. The results show close agreement across stratified marine scenarios, improved accuracy and physical consistency from the embedded constraint, and substantially faster pointwise inference than conventional finite element solvers. Analysis of near-field discrepancies further identifies seabed anisotropy and environmental uncertainty as important sources of model–experiment mismatch, thereby clarifying the framework’s applicability and limitations for marine sensing-system design.