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<p>We propose a data-driven framework for learning and predicting solutions of the discrete sine-Gordon equation by combining universal-basis expansions with neural ODE architectures. In this approach, polynomial and trigonometric basis functions are embedded into the network’s representation of the Hamiltonian, enabling efficient approximation of non-polynomial interactions in the lattice. We further incorporate <i>symmetry-informed</i> penalties, which enforce invariants such as reflection symmetry and energy conservation, thereby enhancing stability and long-horizon accuracy. Numerical experiments on both sine-wave and breather soliton initializations demonstrate that our <i>universal-basis neural ordinary differential equations</i> (UB-NODEs) yield accurate particle trajectories and maintain essential soliton properties over extended times. Moreover, empirical comparisons reveal the advantages of adding symmetry-based constraints, including faster convergence and reduced overfitting. This methodology is broadly applicable to other Hamiltonian lattice systems and paves the way for deeper machine-learning investigations of complex nonlinear dynamics.</p>

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Universal-basis neural ODE modeling of the discrete sine-gordon system

  • 期刊论文

  • 发表日期：

    2025年8月6日

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  • Mingkun Li,
  • Zaijiu Shang,
  • Peng Wang,
  • Hong-Kun Zhang,
  • Junjie Fan

摘要

We propose a data-driven framework for learning and predicting solutions of the discrete sine-Gordon equation by combining universal-basis expansions with neural ODE architectures. In this approach, polynomial and trigonometric basis functions are embedded into the network’s representation of the Hamiltonian, enabling efficient approximation of non-polynomial interactions in the lattice. We further incorporate symmetry-informed penalties, which enforce invariants such as reflection symmetry and energy conservation, thereby enhancing stability and long-horizon accuracy. Numerical experiments on both sine-wave and breather soliton initializations demonstrate that our universal-basis neural ordinary differential equations (UB-NODEs) yield accurate particle trajectories and maintain essential soliton properties over extended times. Moreover, empirical comparisons reveal the advantages of adding symmetry-based constraints, including faster convergence and reduced overfitting. This methodology is broadly applicable to other Hamiltonian lattice systems and paves the way for deeper machine-learning investigations of complex nonlinear dynamics.