Victory Obieke

Victory Obieke

We propose a structure-preserving framework for learning compact finite-difference derivative stencils for the one-dimensional Maxwell system. The stencil coefficients are obtained from derivative data by solving a constrained least-squares problem. A skew-adjointness constraint enforces antisymmetry of the learned stencil and guarantees conservation of the semi-discrete electromagnetic energy, while centered moment constraints impose the desired formal order of accuracy. The resulting constrained optimization problem is solved using ADMM, with an equality-constrained stencil update and a projection step for box constraints. For data generated by the classical centered Maxwell derivative operator, the method recovers the corresponding standard finite-difference stencils and attains the expected formal orders of accuracy. Numerical tests with several initial conditions show that the learned stencils reproduce the standard finite-difference solutions to roundoff accuracy while preserving discrete energy to machine precision. We also consider noisy data generated by a hidden nonstandard skew-adjoint operator. In this setting, the learned stencil adapts to the data while retaining energy conservation and gives smaller errors than standard finite differences for broadband initial data.

Victory Obieke, Emmanuel Oguadimma

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.