Hybrid neural-symbolic construction of localized, periodic, and interacting internal waves in stratified ocean media

Nonlinear internal ocean waves are important in stratified marine environments because they are linked to energy transport, momentum transfer, and mixing processes. In this study, the (3 + 1)-dimensional potential Yu–Toda–Sasa–Fukuyama (pYTSF) equation is used as an idealized mathematical model to study wave propagation, energy localization, and nonlinear interactions in multidimensional dispersive media. A multivariate bilinear neural network method (mBNNM), which serves as a hybrid neural-symbolic framework, is applied to obtain exact solutions of the governing equation. The developed [4 2 2 1] neural architecture is used to derive lump wave (LW), one-wave (OW), new lump wave (NLW), and triangular periodic wave (TPW) solutions. The [4 3 2 1] architecture is then used to obtain lump-kink wave (LKW), lump-periodic-kink interaction wave (INT3-W), and lump-periodic interaction wave (INT2-W) solutions. These solutions show localization, nonlinear deformation, periodic behavior, and multidimensional wave interactions. Their main features are illustrated using three-dimensional surface and contour plots. The results provide exact analytical benchmarks for multidimensional nonlinear-wave studies and for validating numerical and physics-informed models. The derived solutions provide qualitative benchmark structures that may be incorporated into future studies of ocean mixing, climate variability, sea-surface roughness, and electromagnetic scattering when combined with realistic physical data and coupled models.