Prediction of discrete rogue wave solutions and model parameters of Ablowitz–Ladik equation via symmetric difference data enhancement physics-informed neural networks

We introduce an innovative discrete deep learning approach, termed symmetric difference data enhancement physics-informed neural networks (SDE-PINNs), designed to address both forward and inverse problems associated with discrete rogue waves in nonlinear lattice equations. The methodology is characterized by three main contributions: (1) The development of structure-preserving discrete difference operators that maintain intrinsic symmetry within lattice systems, thereby overcoming the limitations of traditional continuous PINNs when applied to discrete environments; (2) To the best of our knowledge, this is the first application of deep learning techniques to inverse problems in discrete systems, achieving coefficient inversion under sparse, low-noise data conditions. Additionally, it was observed that increasing data density enhances robustness against high-noise environments; (3) A comprehensive statistical non-parametric test analysis elucidates the crucial relationship between initialization strategies and model robustness, notably identifying the optimal initialization point as the midpoint between true symmetry and domain boundaries. Through comprehensive numerical experiments and parameter inversion applied to first- and second-order discrete rogue waves within the Ablowitz–Ladik (AL) equation, we demonstrate the method’s ability to attain accuracy levels of $10^{-4}$ in forward modeling and a parameter recognition rate exceeding 99% in inverse problems. This approach uniquely integrates deep learning with discrete soliton theory, establishing novel connections between the two fields.