[Paper Review] RAR-PINN algorithm for the data-driven vector-soliton solutions and parameter discovery of coupled nonlinear equations

Shu-Mei Qin, Min Li+2 more|arXiv (Cornell University)|Apr 29, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes the RAR-PINN algorithm, a physics-informed neural network enhanced with residual-based adaptive refinement, to accurately predict vector-soliton solutions and discover unknown parameters in coupled nonlinear equations such as the CGNLS equation. The method achieves faster convergence and higher accuracy than traditional PINN, especially for solutions with steep gradients and shape-changing interactions, and successfully identifies parameters even under 12% noise corruption with high robustness.

ABSTRACT

This work aims to provide an effective deep learning framework to predict the vector-soliton solutions of the coupled nonlinear equations and their interactions. The method we propose here is a physics-informed neural network (PINN) combining with the residual-based adaptive refinement (RAR-PINN) algorithm. Different from the traditional PINN algorithm which takes points randomly, the RAR-PINN algorithm uses an adaptive point-fetching approach to improve the training efficiency for the solutions with steep gradients. A series of experiment comparisons between the RAR-PINN and traditional PINN algorithms are implemented to a coupled generalized nonlinear Schrödinger (CGNLS) equation as an example. The results indicate that the RAR-PINN algorithm has faster convergence rate and better approximation ability, especially in modeling the shape-changing vector-soliton interactions in the coupled systems. Finally, the RAR-PINN method is applied to perform the data-driven discovery of the CGNLS equation, which shows the dispersion and nonlinear coefficients can be well approximated.

Motivation & Objective

• To develop a deep learning framework capable of accurately predicting complex vector-soliton solutions in coupled nonlinear systems, particularly those with steep gradients and shape-changing interactions.
• To address the limitations of traditional PINN methods in training efficiency and convergence for solutions with sharp features by introducing adaptive point selection.
• To enable data-driven discovery of unknown physical parameters—such as dispersion and nonlinear coefficients—in the coupled generalized nonlinear Schrödinger (CGNLS) equation.
• To evaluate the robustness of the method under varying levels of data noise and sparse training data.
• To demonstrate the superiority of RAR-PINN over traditional PINN in approximating multisoliton solutions and minimizing residual errors.

Proposed method

• The RAR-PINN method integrates physics-informed neural networks (PINN) with a residual-based adaptive refinement (RAR) strategy that dynamically selects training points based on local residual magnitude.
• Instead of random sampling, the algorithm prioritizes regions with high residual error—indicating steep gradients or complex solution features—thereby improving training efficiency and convergence.
• The method uses automatic differentiation to enforce the PDE constraints (e.g., the CGNLS equation) within the neural network loss function, ensuring physical consistency.
• The RAR mechanism iteratively refines the training dataset by adding new points in high-error regions, reducing the overall loss and improving solution accuracy over epochs.
• The framework is applied to the CGNLS equation with complex coefficients, enabling simultaneous prediction of vector-soliton solutions and discovery of unknown parameters.
• The method is evaluated using loss curves, absolute and relative error metrics, and parameter identification accuracy under varying noise levels and data sparsity.

Experimental results

Research questions

• RQ1 Can the RAR-PINN method achieve faster convergence and better approximation accuracy than traditional PINN for vector-soliton solutions of coupled nonlinear equations?
• RQ2 How does the RAR-PINN method perform in modeling complex vector-soliton interactions, particularly shape-changing collisions with steep gradients?
• RQ3 To what extent can the RAR-PINN framework discover unknown physical parameters (e.g., dispersion and nonlinear coefficients) in the CGNLS equation from noisy or sparse data?
• RQ4 How robust is the RAR-PINN method to data corruption, such as uncorrelated Gaussian noise up to 12%?
• RQ5 Does increasing the number of training data points improve the accuracy of parameter discovery and solution approximation in the RAR-PINN framework?

Key findings

• The RAR-PINN method achieves significantly faster convergence and lower residual errors compared to traditional PINN, particularly for solutions with steep gradients and complex interactions.
• For the CGNLS equation, RAR-PINN accurately predicts one-, two-, and three-soliton vector solutions, including shape-changing interactions, with relative errors reduced by up to 90% compared to traditional PINN.
• The method successfully identifies the unknown parameters of the CGNLS equation with high accuracy: for clean data, the identified coefficients deviate from the true values by less than 0.003% in magnitude.
• Even with 1% and 3% Gaussian noise, the parameter identification error remains below 0.012%, demonstrating strong robustness to data corruption.
• As the number of training data points increases from 1,015 to 5,010, the parameter identification error decreases monotonically, with the smallest error (0.0001152) observed at 5,010 points.
• The method maintains stable performance under up to 12% noise, with parameter errors remaining within 0.012, indicating strong resilience to data quality degradation.