{"ID":2847190,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2511.00418","arxiv_id":"2511.00418","title":"Structure-Preserving Physics-Informed Neural Network for the Korteweg--de Vries (KdV) Equation","abstract":"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.","short_abstract":"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 nonlin...","url_abs":"https://arxiv.org/abs/2511.00418","url_pdf":"https://arxiv.org/pdf/2511.00418v1","authors":"[\"Victory Obieke\",\"Emmanuel Oguadimma\"]","published":"2025-11-01T06:07:24Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math-ph\",\"nlin.PS\",\"physics.flu-dyn\"]","methods":"[\"Large Language Model\"]","has_code":false}