Abstract: The use of physics-informed neural networks (PINNs) to solve the nonlinear Schrödinger equation (NLSE) has garnered significant attention. However, the majority of current studies use non-adaptive fixed sampling methods, which typically involve a large amount of data, and the contribution of obtained samples to the training of PINNs varies considerably. Additionally, when dealing with higherorder problems with increased complexity, PINNs exhibit notable deficiencies in computational performance. To address these two shortcomings, this study focuses on the first-order and second-order rogue wave problems of the NLSE. On the one hand, we integrate active learning algorithms into PINNs to reduce data usage and effectively enhance training efficiency. On the other hand, we incorporate an adaptive time-marching strategy into PINNs to compensate for the computational performance limitations of a single PINN, resulting in an improvement in computational accuracy by 1 – 2 orders of magnitude. This work provides insights and pathways for optimizing the performance of PINNs in solving partial differential equations and also demonstrates the promising application of PINNs and corresponding optimization algorithms in nonlinear physical problems.

Key words: nonlinear science, physics-informed neural networks, active learning, adaptive timemarching strategy