A symmetric difference data enhancement physics-informed neural network for the solving of discrete nonlinear lattice equations

Abstract In this paper, we propose a symmetric difference data enhancement physics-informed neural network (SDE-PINN) to study soliton solutions for discrete nonlinear lattice equations (NLEs). By considering known and unknown symmetric points, numerical simulations are conducted to one-soliton and two-soliton solutions of a discrete KdV equation, as well as a one-soliton solution of a discrete Toda lattice equation. Compared with the existing discrete deep learning approach, the numerical results reveal that within the specified spatiotemporal domain, the prediction accuracy by SDE-PINN is excellent regardless of the interior or extrapolation prediction, with a significant reduction in training time. The proposed data enhancement technique and symmetric structure development provides a new perspective for the deep learning approach to solve discrete NLEs. The newly proposed SDE-PINN can also be applied to solve continuous nonlinear equations and other discrete NLEs numerically.