Physics-informed neural networks for nonlinear Schrödinger localized wave problems with gradient-residual adaptive collocation sampling
摘要
The nonlinear Schrödinger (NLS) equation admits a wide range of localized wave structures, including bright and dark solitons, breathers, and rogue waves. Such localized structures are challenging for physics-informed neural networks (PINNs) to resolve accurately because the relevant solution structure often occupies only a small fraction of the spacetime domain, making collocation-point placement a key factor in training quality. To address this issue, we propose GRAVIS (Gradient-Residual Adaptive Variance-based Importance Sampling), a dual-criterion adaptive collocation framework that combines residual magnitude with the output-layer gradient norm of the per-point residual loss, using the latter as a proxy for parameter-space sensitivity. The resulting score is converted into a smooth sampling distribution via a tempered Gaussian mixture model (GMM), supplemented by scheduled exploration and importance-corrected training. We evaluate the method on seven NLS-type benchmarks, including bright and dark solitons, a breather, first- and second-order rogue waves, and two (2+1)-dimensional extensions, under matched network architectures and training budgets. Under the matched architectures and training budgets used in this study, GRAVIS attains the lowest relative L2 error among the evaluated sampling strategies on all seven benchmarks, with improvement factors of 1.56