Hybrid localized structures of the (3+1)-dimensional gBKP equation via bilinear neural networks with PINN verification

This study investigates hybrid localized structures of the (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili (gBKP) equation using a bilinear neural network method (BNNM). The gBKP equation governs weakly dispersive and nonlinear wave propagation in shallow water, nonlinear optics, and plasma physics. An activation function pool is established that includes Jacobi elliptic functions, hyperbolic tangent, polynomials, and exponentials. A nonlinear coupled neural network architecture with one or two hidden layers is then constructed. Four families of hybrid localized structures are obtained from this architecture: algebraic-elliptic composite waves, topological-periodic composite waves, exponential-elliptic composite waves, and elliptic-modulated topological waves. From an analysis of the non-trivial solution families, a phenomenon termed wavefield degeneracy is identified. Multiple parameter configurations reduce to identical functional forms, indicating that waveform morphology is governed by the integrable structure rather than specific dispersion parameters. To validate the physical realizability of these solutions, physics-informed neural networks (PINNs) are employed to simulate the anisotropic localized wave constructed by the BNNM. The relative $ L_2 $ error of the PINN approximation converges to $ 2.66\% $, with the partial differential equation (PDE) residual decreasing to $ \mathcal{O}(10^{-6}) $, confirming the numerical reproducibility of the BNNM-derived solution. The framework combines symbolic construction of exact solutions via neural networks with PINN-based numerical reproduction for the study of higher-dimensional integrable wave dynamics. The numerical verification is demonstrated on the most structurally complex solution family (Type Ⅱ), with the validation scope limited due to computational constraints.