Universal-Basis Neural ODE Modeling of the Discrete Sine-Gordon System

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.