Data-driven discovery of soliton solutions and parameter inversion for the discrete mKdV equation via the traveling-wave constrained PINNs with soft and hard constraints

Abstract This paper proposes a discrete deep learning method, termed the traveling-wave constrained physics-informed neural networks (TC-PINNs), for addressing both the forward and inverse problems of discrete nonlinear lattice equations (NLEs). The discrete mKdV equation is used as a representative example to validate the effectiveness of the proposed approach. We respectively carry out numerical simulations and parameter inversion for one- and two-soliton solutions of the discrete mKdV equation by employing the traveling-wave transformation and propagation properties through soft constraint and hard constraint schemes, and further extend this approach to rational soliton solutions. Numerical experiments show that the TC-PINNs consistently outperform traditional PINNs and symmetric difference data enhancement physics-informed neural networks (SDE-PINNs) in both the interior and extrapolation predictions of soliton solutions as well as parameter inversion. Compared with soft constraints, TC-PINNs under hard constraints achieve a higher prediction accuracy. This method can also be applied to solve other discrete NLEs and perform parameter inversion.