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<p>This paper presents a new matrix-based, physics-informed neural network architecture that incorporates recurrent structures to solve various forms of the Lane-Emden equation, which arises in astrophysics. The Lane-Emden equation is a nonlinear ordinary differential equation that poses significant challenges due to its singularity at the origin and its complex nonlinear behavior, which often lacks analytical solutions. Traditional numerical methods may encounter stability problems and computational inefficiencies, especially when dealing with generalized versions or coupled systems of such equations. To address these issues, this study introduces a novel computational approach that integrates physical laws directly into the learning process, while taking advantage of the sequence modeling capabilities of recurrent neural networks. This framework offers significant improvements in accuracy, computational efficiency, and adaptability across different examples of the Lane-Emden equation and systems involving multiple coupled equations. The proposed method uses a matrix-based finite difference scheme to compute derivatives within the physics-informed loss function, promoting both numerical stability and faster convergence. In contrast to standard physics-informed neural networks, which typically utilize fully connected architectures, the recurrent structure employed here allows for more effective modeling of dependencies across space or time. Numerical results demonstrate that the method achieves high accuracy, with a mean absolute error on the order of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(O(10^{-4})\)</EquationSource> </InlineEquation>, and maintains consistent performance across diverse examples. The proposed approach outperforms several established techniques, including methods based on radial basis functions, decomposition approaches, and spectral collocation. Its flexibility and robustness underscore its potential for broader application to a wide class of nonlinear differential equations.</p>

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MB-PIRNN: Matrix-based physics-informed recurrent neural network for solving Lane-Emden type equation

  • 期刊论文

  • 发表日期：

    2026年5月22日

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  • Maryam Babaei,
  • Hassan Dana Mazraeh,
  • Kourosh Parand

摘要

This paper presents a new matrix-based, physics-informed neural network architecture that incorporates recurrent structures to solve various forms of the Lane-Emden equation, which arises in astrophysics. The Lane-Emden equation is a nonlinear ordinary differential equation that poses significant challenges due to its singularity at the origin and its complex nonlinear behavior, which often lacks analytical solutions. Traditional numerical methods may encounter stability problems and computational inefficiencies, especially when dealing with generalized versions or coupled systems of such equations. To address these issues, this study introduces a novel computational approach that integrates physical laws directly into the learning process, while taking advantage of the sequence modeling capabilities of recurrent neural networks. This framework offers significant improvements in accuracy, computational efficiency, and adaptability across different examples of the Lane-Emden equation and systems involving multiple coupled equations. The proposed method uses a matrix-based finite difference scheme to compute derivatives within the physics-informed loss function, promoting both numerical stability and faster convergence. In contrast to standard physics-informed neural networks, which typically utilize fully connected architectures, the recurrent structure employed here allows for more effective modeling of dependencies across space or time. Numerical results demonstrate that the method achieves high accuracy, with a mean absolute error on the order of \(O(10^{-4})\) , and maintains consistent performance across diverse examples. The proposed approach outperforms several established techniques, including methods based on radial basis functions, decomposition approaches, and spectral collocation. Its flexibility and robustness underscore its potential for broader application to a wide class of nonlinear differential equations.