Advanced physics-informed neural networks for nonlinear partial differential equations: machine learning, gradient-enhanced and extended approach

Nonlinear partial differential equations (NLPDEs) are fundamental in describing various physical and engineering processes. In this study, two advanced deep learning frameworks the gradient-enhanced physics-informed neural network (gPINN) (Yu et al. in Comput. Methods Appl. Mech. Eng. 393:114823, 2022) and the extended physics-informed neural network (XPINN) (Jagtap and Karniadakis in Commun. Comput. Phys. 28:2002–2041, 2020) are systematically employed to solve and compare the performance of two nonlinear models: the Fisher equation and the dispersive dissipative (DD) equation. The gPINN approach incorporates both the PDE residual and its gradient information within the loss function, accelerating convergence and improving local accuracy, while XPINN divides the computational domain into sub-domains and enforces interface continuity for efficient parallel training and localized error control. Neural networks with 4–10 hidden layers and 30–70 neurons per layer are implemented to approximate the solutions under well-defined initial and boundary conditions. Quantitative evaluations show that the predicted solutions closely match the analytical ones, achieving mean square errors (MSEs) between 10−2 and 10−6. Moreover, the XPINN framework reduces training time by approximately 35 percent and CPU memory usage by 25 percent compared to gPINN, confirming its superior computational efficiency and scalability. To the best of our knowledge and as supported by prior studies on PINN (Raissi, Perdikaris and Karniadakis in J. Comput. Phys. 378:686–707, 2019), gPINN (Yu et al. in Comput. Methods Appl. Mech. Eng. 393:114823, 2022), and XPINN (Jagtap and Karniadakis in Commun. Comput. Phys. 28:2002–2041, 2020; Shukla, Jagtap and Karniadakis in J. Comput. Phys. 447:110683, 2021) this work presents the first integrated comparative analysis combining gPINN and XPINN for these nonlinear PDEs under identical conditions. The findings highlight the potential of hybrid neural frameworks to achieve fast, accurate, and physics-consistent approximations for complex nonlinear systems.