Structure-Preserving Physics-Informed Neural Network for the Korteweg--de Vries (KdV) Equation

TL;DR

This paper develops a structure-preserving PINN framework for the KdV equation that embeds conservation laws into the training process, employs sinusoidal activations for better spectral accuracy, and demonstrates improved stability and physical fidelity in simulating nonlinear dispersive waves.

Contribution

It introduces a novel invariant-preserving PINN approach with sinusoidal activations for the KdV equation, ensuring energy conservation and accurate wave dynamics.

Findings

Successfully models soliton propagation, interaction, and dispersive breakup.

Enhances long-term stability and reduces invariant drift.

Accelerates convergence with invariant-constrained optimization and sinusoidal features.

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.