Enhancing Physics-Informed Neural Networks for the Nonlinear Schrödinger Equation via Residual-Based Refinement and Conservation Laws

Abstract

Physics-informed neural networks (PINNs) offer a powerful alternative for solving partial differential equations, yet their efficacy is highly sensitive to collocation point distribution and the formulation of the loss function. This research evaluates the performance of the PINNsTF2-DeepXDE framework against enhanced configurations to solve the focusing cubic Nonlinear Schrödinger Equation (NLSE). Specifically, it investigates three methods deployed to counteract limitations of original PINNs, namely, Residual-based Adaptive Refinement (RAR), an L2 energy conservation constraint, and the weighting of the PDE residual loss (wf). Across all tested configurations, RAR emerged as the most effective catalyst of error reduction. The model that employed RAR with wf = 4 achieved the best overall accuracy, with relative ℓ2 error for the solution h at 7.566 × 10⁻⁴ and the lowest validation errors for both components u and v. The L2 conservation law showed a mixed effect on aggregate error but integrated physically consistent solutions. Results also show that no single wf value is universally optimal, as the best weight depends on whether refinement and/or conservation constraints are active. The hybrid configuration (RAR + L2) provides a practical compromise between numerical accuracy and physical guidance, though it does not surpass the best RAR-only setting in this study (relative ℓ2 error for h reaching 8.689 × 10⁻⁴). Overall, the findings indicate that robust PINN performance for nonlinear wave dynamics requires coordinated tuning of adaptive collocation and loss composition.