Deep wavelet neural network with lateral connections for solving fractional differential equations arising in oil pollution dynamics

Deep learning has proven to be a powerful approach for solving complex problems in physical and environmental systems. Inspired by its success, we propose a novel framework called the Wavelet Integrated Lateral Neural Network (WLNN) for approximating solutions of time-fractional differential equations that describe the dynamics of oil pollution. The proposed architecture incorporates lateral connections between hidden layers to enhance spatiotemporal learning and integrates wavelet transforms to efficiently capture multi-scale and non-local features inherent in fractional dynamics. To ensure theoretical soundness, the existence and uniqueness of the solution to the time-fractional differential equation are rigorously established. The framework is validated using a time-fractional Allen–Cahn equation, a representative model for oil dispersion in aquatic environments. Comparative analyses with Physics-Informed Neural Networks (PINNs) and gradient-enhanced PINNs (gPINNs) demonstrate that the proposed WLNN achieves higher accuracy in solving time-fractional equations related to oil pollution.