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

Source: arXiv - 2601.04176v1

Overview

A new deep‑learning pipeline shows that Physics‑Informed Neural Networks (PINNs) can accurately recover hidden parameters of the Nonlinear Schrödinger Equation (NLSE) even when the measurement data are extremely noisy and sparse. By treating the governing PDE as a regularizer, the framework sidesteps the noise‑amplification problems that cripple classic finite‑difference inverse methods.

Key Contributions

• Robust parameter inference: Recovers the NLSE’s nonlinear coefficient β with < 0.2 % relative error from only 500 randomly sampled points corrupted by 20 % Gaussian noise.
• Data‑efficiency: Demonstrates consistent sub‑1 % accuracy across a wide range of training set sizes (100–1000 points).
• Generalization across regimes: Works for β values from 0.5 to 2.0 without retraining the network architecture.
• Statistical reliability: Multiple independent runs yield a standard deviation < 0.15 % for β = 1.0, confirming repeatability.
• Practical runtime: End‑to‑end training and inference complete in ~80 minutes on a single NVIDIA Tesla T4 GPU—well within the budget of most cloud‑based ML workloads.
• Open‑source release: Full code, data‑generation scripts, and training notebooks are publicly available, enabling immediate replication and extension.

Methodology

1. Data generation – Synthetic NLSE solutions are sampled on a 2‑D space‑time grid. From these, 100–1000 points are randomly selected and corrupted with additive Gaussian noise (σ ≈ 0.2 × signal amplitude).
2. PINN architecture – A fully‑connected feed‑forward network takes space‑time coordinates (x, t) as input and outputs the complex field ψ(x,t). The network is trained to minimize a composite loss:
   • Data loss: Mean‑squared error between network predictions and the noisy observations.
   • Physics loss: Squared residual of the NLSE (computed via automatic differentiation) evaluated at a larger set of collocation points (unobserved).
3. Parameter embedding – The unknown coefficient β is treated as a trainable scalar variable, jointly optimized with the network weights.
4. Optimization – Adam optimizer (learning rate 1e‑3) for 10 k iterations, followed by L‑BFGS refinement to fine‑tune β.
5. Evaluation – Relative error on β, statistical spread over 30 random seeds, and runtime profiling.

The key idea is that the physics loss forces the network to honor the underlying PDE, effectively filtering out noise that would otherwise dominate the data loss.

Results & Findings

• Noise tolerance: Traditional finite‑difference inverse methods diverge beyond ~5 % noise, while the PINN maintains < 1 % error up to 20 % noise.
• Scalability: Increasing the number of collocation points (used only for the physics loss) improves stability without extra measurement data.
• Runtime: 80 min on a Tesla T4 translates to ~0.16 GPU‑hours per inference—compatible with batch processing pipelines.

Statistical analysis across 30 random seeds shows a mean relative error of 0.21 % and a standard deviation of 0.13 % for the canonical case (β = 1.0, 500 points).

Practical Implications

• Experimental physics & optics: Labs that measure wave‑packet dynamics (e.g., fiber optics, Bose‑Einstein condensates) can extract nonlinear coefficients from a handful of noisy sensor readings, reducing the need for high‑precision instrumentation.
• Model‑based control: Real‑time system identification for adaptive optics or quantum simulators becomes feasible; the PINN can be retrained on streaming data to keep β up‑to‑date.
• Edge deployment: The modest GPU budget means the pipeline can run on cloud‑based inference services or even on high‑end edge devices (e.g., NVIDIA Jetson) for on‑site diagnostics.
• Cross‑domain inverse problems: The same physics‑regularized framework can be transplanted to other PDE‑governed systems (fluid dynamics, electromagnetics), offering a template for robust parameter discovery when data are scarce.

Developers can integrate the provided PyTorch implementation into existing data‑processing pipelines, leveraging automatic differentiation and standard optimizers without needing custom PDE solvers.

Limitations & Future Work

• Synthetic data focus: The study uses numerically generated NLSE solutions; real experimental noise (e.g., systematic bias, outliers) may pose additional challenges.
• Single‑parameter inference: Only the nonlinear coefficient β is treated as unknown; extending to multi‑parameter or functional‑parameter identification will increase complexity.
• Scalability to higher dimensions: While the 1‑D NLSE is a common benchmark, applying the same approach to 2‑D or 3‑D wave equations will demand more sophisticated network architectures and memory management.
• Training time: Although acceptable for offline analysis, real‑time applications would benefit from further acceleration (e.g., physics‑aware network pruning or meta‑learning).

Future research directions highlighted by the authors include testing on laboratory measurements, incorporating Bayesian uncertainty quantification, and exploring hybrid PINN‑classical solvers for even faster convergence.

Authors

• Pietro de Oliveira Esteves

Paper Information

• arXiv ID: 2601.04176v1
• Categories: cs.LG, physics.comp-ph
• Published: January 7, 2026
• PDF: Download PDF