Solving forced KdV equation using physics-informed neural networks

Abstract Physics-informed neural networks (PINNs) are applied to solve forced-KdV (fKdV) equations and to assess their capability for nonlinear wave dynamics under external forcing. We first validate the solver in analytically tractable forced-KdV benchmarks with periodic and exponential time-dependent forcing, demonstrating accurate reproduction of two-soliton interactions. Robustness is sensitive to forcing characteristics and to the coupled choice of loss weights and sampling density: more rapidly varying forcing and imbalanced weighting increase across-seed variability in the attained relative $$\:{L}^{2}$$ error, consistent with known PINN optimization pathologies. We then demonstrate an event-based ocean application for internal solitary wave (ISW) variability near Dongsha Atoll using a reduced-order, one-dimensional fKdV representation. A spatiotemporal forcing proxy $$\:F\left(x,\:t\right)$$ is constructed from TPXO $$\:{M}_{2}$$ barotropic currents and the along-transect bathymetric slope; a ridge-based partitioning provides a consistent ridge-to-site linkage across events. A forced–free ablation of the propagation-stage forcing shows that West-Ridge corridor forcing primarily modulates site waveform morphology—systematically reducing peak magnitude while broadening the main pulse—without qualitatively altering the westward propagation pathway. These results suggest that PINNs provide a practical framework for investigating fKdV dynamics in both controlled benchmarks and mechanism-oriented ocean case studies.