Data Set for Research on Solving the Single Soliton Solution of the First-Order Nonlinear Schrödinger Equation and Its Nonlinear Coefficient InversionBasedon Physics-Informed Neural Networks

List of Figures and Their Data DescriptionsThesis FigureProgram Output File(s)Dataset DescriptionFig. 2-1(conceptual, not generated by script)Schematic of the PINN framework; no numerical data.Fig. 3-1{run_tag}_scatter_obs_points.pngSparse noisy training data: randomly chosen 20 000 points from the full grid, with amplitude |u| computed from the exact solution multiplied by (1+ε), where ε ~ N(0, σ²).

Visualises the scarcity and noise of observations used in the inverse problem.Fig. 4-1(conceptual, not generated by script)Two‑stage training strategy illustration; no numerical data.Fig. 5-1{run_tag}_amplitude_lines_t0.png, _tmid.png, _tend.pngCross‑section of the amplitude |u| at three time instants: initial, middle, and final. Each plot compares the PINN prediction with the exact analytic solution along the spatial axis.Fig. 5-2{run_tag}_final_amplitude_comparison.png, _final_real_comparison.png, _final_imag_comparison.pngLine plots at the final time t=5 showing |u|, Re(u), and Im(u).

Predicted and exact fields are overlaid to assess local accuracy.Fig. 5-3{run_tag}_heatmap_exact_amplitude.pngFull spacetime colour map of the exact soliton amplitude on a 256×120 grid.Fig. 5-4{run_tag}_heatmap_obs_amplitude.pngFull spacetime colour map of the noisy observed amplitude (multiplicative Gaussian noise), generated from the exact solution.Fig. 5-5{run_tag}_heatmap_pred_amplitude.pngPINN‑reconstructed amplitude spacetime map; directly comparable with Fig. 5-3.Fig. 5-6{run_tag}_heatmap_exact_phase.png, _heatmap_pred_phase.pngSpacetime maps of the phase φ = atan2(Im(u), Re(u)) for both the exact solution and the PINN prediction.Fig. 5-7{run_tag}_hist_pde_residual.pngHistogram of the PDE residual magnitude |f(x,t)| over all grid points, shown on a logarithmic scale.

Reflects how well the PINN output satisfies the nonlinear Schrödinger equation.Fig. 5-8{run_tag}_heatmap_pde_residual.pngSpacetime heatmap of the PDE residual magnitude; highlights where equation violation is largest (often near the soliton core and boundaries).Fig. 5-9{run_tag}_residual_rms_over_time.pngRoot‑mean‑square of the residual computed at each time step; demonstrates the temporal evolution of equation misfit.Fig. 5-10{run_tag}_beta_trajectory.pngEvolution of the trainable nonlinear coefficient β during optimisation, recorded at every iteration.

The dashed line marks the true value β=1.0.Fig. 5-11{run_tag}_training_losses.pngTotal loss and its four components (PDE, data, boundary, regularisation) versus training step, plotted on a log scale.Fig. 5-12 (loss weights){run_tag}_dynamic_weights.pngAdaptive loss weights λ_pde, λ_data, λ_bc, λ_reg over the course of training, illustrating the dynamic balancing strategy.Fig. 5-12 (error metric){run_tag}_relative_L2_over_time.pngRelative L² error between the PINN prediction and the exact solution, evaluated per time slice.

Shows error accumulation as time progresses.Fig. 5-13{run_tag}_heatmap_error_amplitude.pngSpacetime map of the absolute amplitude error ||u_pred| - |u_exact||.Fig. 5-14{run_tag}_mass_conservation.pngTime evolution of the mass integral ∫|u|² dx for both the exact (constant) and the PINN‑predicted solution.Monte‑Carlo (Sect. 5.2.3)mc_beta_hist_sigma*.pngDistributions of estimated β from five independent runs at each noise level (0, 0.02, 0.05, 0.10).Monte‑Carlomc_e_beta_vs_sigma.pngMean relative parameter error with 95% confidence intervals versus noise level.Monte‑Carlomc_relL2_vs_sigma.pngMean global relative L² error with confidence intervals against noise level.Monte‑Carlomc_residual_rms_vs_sigma.pngMean residual RMS with confidence intervals versus noise.Monte‑Carlomc_beta_boxplot.pngBoxplots summarising the spread of β estimates across all noise levels and repetitions.