Data-Driven Rogue Waves in Nonlocal \(\cal{PT}\)-Symmetric Schrödinger Equation via Mix-Training PINN

In this paper, by modifying loss function MSE (adding the mean square error of the complex conjugate term to the loss function) and training area of the physics-informed neural network (PINN), the authors proposed two neural network models: Mix-training PINN and prior information mix-training PINN. The authors demonstrated the advantages of these models by simulating rogue waves in the nonlocal \(\cal{PT}\) $$\mathcal{P}\mathcal{T}$$ -symmetric Schrödinger equation. Numerical experiments showed that the proposed models not only simulate first-order rogue waves, but also significantly improve the simulation capability. Compared with original PINN, the prediction accuracy of the first-order rouge waves are improved by one to three orders of magnitude. By testing the inverse problem of first-order rogue waves, it is also proved that these models have good performance.