Modified PINN Approach Integrating Conservation Laws for Efficient Multi-Stage Training of Coupled Nonlinear Systems

In this work, we propose an innovative conservation multi-stage algorithm (CM) based on Physics-Informed Neural Networks (PINNs) to investigate the dynamics of vector solitons governed by Zakharov equation. By ingeniously integrating conservation laws into the multi-stage training algorithm, we enhanced the method’s constraint and physical consistency in solving nonlinear systems. Numerical simulations of the Zakharov and NLS equations demonstrated that our method showed obvious improvements in approximation accuracy, convergence speed, and training efficiency, compared to traditional PINN methods. Moreover, when the parameter representing the speed of sound is sufficiently large, our method efficiently simulates the approximation from the Zakharov equation to the Nonlinear Schrödinger (NLS) equation. The approximative simulation not only confirms the conservation multi-stage algorithm’s applicability in various physical environments but also highlights its potential in controlling sound wave propagation characteristics to simulate NLS equation behavior.