Data-driven solutions and parameters discovery of the Mukherjee–Kundu equation via physics-informed neural networks with adaptive sampling

In this work, data-driven solutions and parameter discovery for the Mukherjee–Kundu equation are considered using Physics-Informed Neural Networks (PINNs) with Residual-based Adaptive Distribution (RAD) sampling. For data-driven solutions, the equation’s soliton, accelerated soliton, rogue wave, and Ma breather solutions are all simulated using PINNs. Experimental results show that RAD-PINN outperforms traditional PINN in spatiotemporal dynamic simulations, verifying the effectiveness of adaptive sampling in improving the accuracy of complex, nonlinear solutions. For parameter discovery, the PINN and RAD-PINN are compared in inverting the key parameter λ under three scenarios (no noise, 1% noise, and 3% noise), respectively. Results demonstrate RAD-PINN’s stronger noise robustness and higher inversion accuracy, providing a feasible path for nonlinear equation parameter identification in noisy environments.