Victory Obieke

Physics-Informed Neural Networks (PINNs) offer a flexible framework for solving nonlinear partial differential equations (PDEs), yet conventional implementations often fail to preserve key physical invariants during long-term integration. This paper introduces a \emph{structure-preserving PINN} framework for the nonlinear Korteweg--de Vries (KdV) equation, a prototypical model for nonlinear and dispersive wave propagation. The proposed method embeds the conservation of mass and Hamiltonian energy directly into the loss function, ensuring physically consistent and energy-stable evolution throughout training and prediction. Unlike standard \texttt{tanh}-based PINNs~\cite{raissi2019pinn,wang2022modifiedpinn}, our approach employs sinusoidal activation functions that enhance spectral expressiveness and accurately capture the oscillatory and dispersive nature of KdV solitons. Through representative case studies -- including single-soliton propagation (shape-preserving translation), two-soliton interaction (elastic collision with phase shift), and cosine-pulse initialization (nonlinear dispersive breakup) -- the model successfully reproduces hallmark behaviors of KdV dynamics while maintaining conserved invariants. Ablation studies demonstrate that combining invariant-constrained optimization with sinusoidal feature mappings accelerates convergence, improves long-term stability, and mitigates drift without multi-stage pretraining. These results highlight that computationally efficient, invariant-aware regularization coupled with sinusoidal representations yields robust, energy-consistent PINNs for Hamiltonian partial differential equations such as the KdV equation.

Seasonal rainfall forecasting is crucial for agricultural planning and water resource management in Delta State, Nigeria, as the region's economy is highly dependent on climate. This study investigates the trend and appropriate models for forecasting seasonal rainfall patterns in the region. We employed a range of methods, including traditional time series techniques like Holt's Winter exponential smoothing and the Seasonal Autoregressive Integrated Moving Average (SARIMA), alongside more advanced machine learning and deep learning models. Critical data properties such as stationarity and normality of error terms were first assessed. Model performance was then evaluated using standard metrics, including root mean square error, mean absolute error, mean absolute percentage error, and mean square error. The data was found to have a stationary pattern, and among the models explored, the Holt's Winter exponential smoothing model was identified as the best performing.