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

Mingkun Li, zaijiu shang, Peng Wang, Hong-Kun Zhang, Junjie Fan

Abstract:

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 \emph{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 \emph{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.