A physics-informed neural network framework for modeling rogue and breather solutions in conformable fractional nonlinear wave systems

This study presents a robust and generalized Physics-Informed Neural Network (PINN) framework for solving a class of conformable fractional nonlinear wave equations (CFNWEs). These equations are widely used in modeling complex wave dynamics in physical systems exhibiting memory and hereditary effects. By embedding the conformable fractional operator directly into the neural network architecture, the proposed model accurately captures localized nonlinear structures, including rogue waves and breather-type solutions. The framework employs a composite loss function integrating initial, boundary, and PDE residual constraints, optimized through a hybrid training strategy combining Adam and L-BFGS optimizers. Extensive numerical experiments validate the accuracy, physical consistency, and reproducibility of the proposed approach, demonstrating close agreement with analytical solutions. Moreover, the model exhibits robust performance under sparse and noisy data conditions, highlighting its potential for broad applications in wave dynamics, viscoelastic media, and nonlinear signal forecasting.