Advances Physics from Corrupted Data: PINN Delivers Sub-0.15% Nonlinearity Reconstruction

Recovering accurate physical parameters from noisy data remains a significant challenge in many scientific disciplines. Now, Pietro de Oliveira Esteves of the Federal University of Ceará, alongside co-authors, present a new deep learning framework that successfully addresses this problem, specifically for the Nonlinear Schrödinger Equation. Their research demonstrates the power of Physics-Informed Neural Networks (PINNs) to reconstruct crucial parameters, such as the nonlinear coefficient beta, even when data is heavily corrupted by noise , a scenario where conventional methods often struggle. This innovative approach achieves remarkable accuracy with limited and imperfect data, offering a robust and accessible solution for inverse problems in fields like optics and fluid dynamics, and paving the way for more reliable analysis of complex spatiotemporal phenomena. The team’s publicly available code further encourages reproducibility and wider application of this promising technique.

The research successfully determines the coefficient β with less than 0.2% relative error, utilising only 500 sparse, randomly sampled data points corrupted by 20% additive Gaussian noise. This represents a significant advancement as traditional finite difference methods often struggle in such conditions due to noise amplification during numerical derivative calculations. Validation of the method’s generalisation ability was conducted across a range of physical regimes (β ∈ [0.5, 2.0]) and varying data availability (Nu ∈ [100, 1000]). To facilitate this, scientists decomposed the complex field ψ(x, t) into real and imaginary components, u(x, t) and v(x, t), transforming the NLSE into a system of two coupled partial differential equations. These residuals, denoted as fu and fv, were then incorporated as physical constraints during the training process of the neural network. The core of the methodology is a fully connected Deep Neural Network (DNN) with an input layer of two neurons representing spatial and temporal coordinates, followed by four hidden layers each containing 50 neurons, and culminating in a two-neuron output layer representing the predicted real and imaginary components.

The approach enables robust parameter estimation across a range of physical regimes, specifically with beta values between 0.5 and 2.0, and varying data availability from 100 to 1000 training points. Statistical analysis, conducted over multiple independent runs, further confirmed robustness, revealing a standard deviation of less than 0.15 percent for beta equal to 1.0. The complete computational pipeline executes in approximately 80 minutes on NVIDIA Tesla T4 cloud GPU resources, making this technique accessible for broader implementation. This study pioneered the use of physics-based regularization as an effective filter against high measurement uncertainty, positioning PINNs as a viable alternative to traditional optimisation methods. The team harnessed the power of PINNs to effectively separate signal from stochastic noise, achieving high precision in reconstructing spatiotemporal dynamics. Experiments revealed consistent sub-1 percent accuracy in reconstructing beta across a range of physical regimes, specifically when beta values fell between 0.5 and 2.0. The study further validated the method’s performance with varying data availability, utilising between 100 and 1000 training points to demonstrate robust and reliable results. The complete computational pipeline, encompassing data processing and model training, executes in approximately 80 minutes utilising modest cloud GPU resources, an NVIDIA Tesla T4.

Data shows that the physics-based regularization inherent in the Physics-Informed Neural Network (PINN) acts as an effective filter, mitigating the impact of high measurement uncertainty. Results indicate that PINNs offer a viable alternative to traditional optimization methods for solving inverse problems in spatiotemporal dynamics, particularly when dealing with scarce and noisy experimental data. The team employed a fully connected Deep Neural Network with four hidden layers, each containing 50 neurons, and utilized the Adam optimizer with a learning rate of 10−3. Collocation points, numbering 20,000, were sampled using Latin Hypercube Sampling to ensure comprehensive coverage of the spatiotemporal domain. The researchers successfully recovered the nonlinear coefficient, beta, with a relative error of less than 0.2 percent even when training data was significantly corrupted by 20 percent Gaussian noise. This level of precision was achieved using a relatively small dataset of 500 randomly sampled points, highlighting the effectiveness of physics-informed regularization in mitigating the effects of noise. Validation across a range of physical regimes and data densities confirmed the model’s generalizability and consistent sub-1 percent accuracy.

The complete computational pipeline requires approximately 80 minutes on standard cloud GPU resources, making this approach accessible for a wide range of researchers. The authors acknowledge limitations including the current focus on one-dimensional systems and the assumption of additive Gaussian noise, suggesting future work should explore extensions to higher dimensions and validation against more complex, realistic noise profiles. Further research could also investigate the simultaneous recovery of multiple parameters and explore adaptive loss weighting strategies to optimise performance.