The improved backward compatible physics-informed neural networks for reducing error accumulation and applications in...

Area of Science:

• Computational Physics
• Applied Mathematics
• Machine Learning for PDEs

Background:

• Rogue waves exhibit dynamic characteristics like instantaneity and steepness, necessitating advanced simulation techniques.
• Backward compatible physics-informed neural networks (bc-PINN) offer a temporally sequential approach to solve partial differential equations (PDEs).
• Error propagation in sequential PDE solvers can impact overall solution accuracy and stability.

Purpose of the Study:

• To improve the backward compatible physics-informed neural network (bc-PINN) algorithm for enhanced simulation of rogue waves.
• To address error propagation issues inherent in sequential domain decomposition methods for PDEs.
• To demonstrate the effectiveness of the improved method in solving complex nonlinear systems.

Main Methods:

• Modification of the bc-PINN loss term to ensure backward compatibility using the earliest learned solution as a pseudo-reference.
• Concatenation of solutions from individual subnetworks to form the final predicted solution in the improved bc-PINN (Ibc-PINN).
• Application of Ibc-PINN to study data-driven higher-order rogue waves for the nonlinear Schrödinger (NLS) equation and the AB system.
• Utilization of transfer learning and initial condition guided learning (ICGL) to accelerate training.

Main Results:

• The improved Ibc-PINN demonstrates superior accuracy and stability compared to the original bc-PINN in simulating rogue waves.
• Error analysis reveals a slower error accumulation speed in Ibc-PINN, leading to greater accuracy.
• Numerical results confirm that Ibc-PINN outperforms bc-PINN without sacrificing computational efficiency.

Conclusions:

• The proposed Ibc-PINN effectively enhances the simulation of rogue waves by improving backward compatibility and managing error propagation.
• The Ibc-PINN offers significant advantages in accuracy and stability for solving PDEs, particularly for complex phenomena like higher-order rogue waves.
• The method provides a robust and efficient approach for data-driven discovery of nonlinear wave phenomena.