Lax pair and symmetric difference constrained PINNs for forward and inverse problems of soliton solutions in the discrete complex mKdV equation

We propose Lax pair and symmetric difference constrained physics-informed neural networks (LSD-PINNs), designed to study forward and inverse problems for soliton solutions of discrete nonlinear integrable equations (DNIEs). We take the discrete complex modified Korteweg-de Vries (mKdV) equation as an example for numerical simulations and parameter inversion. The results demonstrate that, in the forward problem, the newly proposed method can not only achieve accuracy levels of \(10^-4\) for both internal and external predictions, but also successfully obtain the corresponding spectral functions and spectral parameters. Furthermore, under identical conditions, LSD-PINNs exhibits the highest simulation accuracy and stability compared to classical discrete PINNs, symmetry-enhanced deep learning (SDE)-PINNs, and Lax-pair-enhanced (LD)-PINNs. For the inverse problem, LSD-PINNs achieves inversion errors on the order of \(10^-4\) for unknown parameters. In addition, under conditions of sparse data, LSD-PINNs is capable of reducing inversion errors by two orders of magnitude compared to classical discrete PINNs. This study offers a novel framework for analyzing discrete soliton solutions and spectral problems within discrete nonlinear integrable equations (DNIEs), providing new insights into integrable phenomena and contributing to the advancement of discrete integrable deep learning.