Robust Physics Discovery from Highly Corrupted Data: A PINN Framework Applied to the Nonlinear Schrödinger Equation

TL;DR

The paper tackles the inverse problem of identifying the nonlinear coefficient $\beta$ in the 1D NLSE from highly noisy and sparse data. It introduces a Physics-Informed Neural Network framework that embeds the NLSE via PDE residuals and treats $\beta$ as a trainable parameter, leveraging automatic differentiation for accurate gradient computation. The approach achieves sub-1% relative error in $\beta$ under 20% Gaussian noise with only 500 data points and generalizes across $\beta$ values in $[0.5, 2.0]$ and data densities $N_u$ in $[100,1000]$, outperforming traditional finite-difference methods. The method runs on modest cloud hardware, demonstrating practical data efficiency and robustness, with code publicly available to support reproducibility and adoption in experimental settings.

Abstract

We demonstrate a deep learning framework capable of recovering physical parameters from the Nonlinear Schrodinger Equation (NLSE) under severe noise conditions. By integrating Physics-Informed Neural Networks (PINNs) with automatic differentiation, we achieve reconstruction of the nonlinear coefficient beta with less than 0.2 percent relative error using only 500 sparse, randomly sampled data points corrupted by 20 percent additive Gaussian noise, a regime where traditional finite difference methods typically fail due to noise amplification in numerical derivatives. We validate the method's generalization capabilities across different physical regimes (beta between 0.5 and 2.0) and varying data availability (between 100 and 1000 training points), demonstrating consistent sub-1 percent accuracy. Statistical analysis over multiple independent runs confirms robustness (standard deviation less than 0.15 percent for beta equals 1.0). The complete pipeline executes in approximately 80 minutes on modest cloud GPU resources (NVIDIA Tesla T4), making the approach accessible for widespread adoption. Our results indicate that physics-based regularization acts as an effective filter against high measurement uncertainty, positioning PINNs as a viable alternative to traditional optimization methods for inverse problems in spatiotemporal dynamics where experimental data is scarce and noisy. All code is made publicly available to facilitate reproducibility.

Figures (4)

• Figure 1: Parameter Discovery Evolution. The learned nonlinear coefficient $\beta$ (blue solid line) converges rapidly to the true value of 1.0 (red dashed line) from an unbiased initialization of 0.0. Training conducted for 5,000 epochs with Adam optimizer.
• Figure 2: Spatial-Temporal Error Analysis. Left: Ground truth amplitude. Middle: PINN reconstruction. Right: Absolute error heatmap. The error remains uniformly small across the domain, with maximum values below 2% of peak amplitude.
• Figure 3: Loss History. The Physics Loss (orange) drops significantly, confirming the model obeys the equation. The Data Loss (green) plateaus, avoiding overfitting to the 20% input noise. Training was conducted for 5,000 epochs with the Adam optimizer.
• Figure 4: Visual Reconstruction. Comparison between the Exact Solution (black dashed) and PINN Prediction (red solid) at three different time snapshots. The overlap indicates highly accurate recovery of the soliton wave dynamics across the spatial domain.