Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Victory Obieke, Emmanuel Oguadimma

TL;DR

This work develops a structure-preserving PINN (SP-PINN) for the KdV equation, embedding mass conservation $\mathcal{M}(t)$ and Hamiltonian energy $\mathcal{E}(t)$ directly into a single-stage loss to ensure physically consistent, energy-stable dynamics. It combines sinusoidal activations for spectral expressiveness with dynamic gradient-based weighting and L-BFGS optimization, achieving accurate long-time predictions across soliton propagation, soliton interactions, and dispersive breakup while maintaining invariants. Across three benchmark scenarios, SP-PINN outperforms conventional PINNs in invariant preservation and stability, demonstrating robustness without multi-stage pretraining. The approach offers a computationally efficient framework for Hamiltonian PDEs and can be extended to higher-order invariants and other nonlinear dispersive systems.

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.

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

TL;DR

This work develops a structure-preserving PINN (SP-PINN) for the KdV equation, embedding mass conservation and Hamiltonian energy directly into a single-stage loss to ensure physically consistent, energy-stable dynamics. It combines sinusoidal activations for spectral expressiveness with dynamic gradient-based weighting and L-BFGS optimization, achieving accurate long-time predictions across soliton propagation, soliton interactions, and dispersive breakup while maintaining invariants. Across three benchmark scenarios, SP-PINN outperforms conventional PINNs in invariant preservation and stability, demonstrating robustness without multi-stage pretraining. The approach offers a computationally efficient framework for Hamiltonian PDEs and can be extended to higher-order invariants and other nonlinear dispersive systems.

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.

Paper Structure

This paper contains 16 sections, 17 equations, 16 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Abstract.
  2. Introduction
  3. Hamiltonian System
  4. KdV in 1D
  5. Physical Properties of KDV
  6. Methodology
  7. Neural Network Architecture
  8. Loss Function Components
  9. Dynamic Weighting of Invariant Loss Terms
  10. Training Procedure
  11. Post-Training Evaluation
  12. Algorithm
  13. Results
  14. Ablation Study: Impact of Conservation Laws on PINN Performance
  15. Numerical Simulations
  16. ...and 1 more sections

Figures (16)

  • Figure 1: Structure-Preserving PINN Framework for KdV
  • Figure 2: Comparison of predicted vs. exact soliton profiles at various time snapshots. The dashed lines denote PINN predictions; solid lines denote the true solution.
  • Figure 3: Mass (left) and energy (right) conservation over time for the PINN solution for one soliton profile.
  • Figure 4: Contour plots of the exact solution $u(t,x)$ (left), PINN-predicted solution (middle), and absolute error (right).
  • Figure 5: Collision of Two-Soliton Interaction
  • ...and 11 more figures