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<p>The Lane-Emden equation, a nonlinear second-order ordinary differential equation, plays a fundamental role in theoretical physics and astrophysics, particularly in modeling the structure of stellar interiors. Also referred to as the polytropic differential equation, it describes the behavior of self-gravitating polytropic spheres. In this study, we present a novel approach to the solution of the eigenvalue problem which arises when considering the Lane-Emden equation for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n = 0, 1, 2, 3, 4\)</EquationSource> </InlineEquation> using Physics-Informed Neural Networks (PINNs). The novelty of this work is that, we not only solve the Lane-Emden equation via PINNs but we also determine the eigenvalue, <i>r</i>, which is the stellar radius. Hyperparameter tuning was conducted using Bayesian optimization in the Optuna framework to identify optimal values for the number of hidden layers, number of neurons, activation function, optimizer, and learning rate for each value of <i>n</i>. The results show that, for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(n = 0, 1\)</EquationSource> </InlineEquation>, PINNs achieve near-exact agreement with theoretical eigenvalues (errors &lt; <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(0.000806\%\)</EquationSource> </InlineEquation>). While for more nonlinear cases, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n = 2, 3\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(n=4\)</EquationSource> </InlineEquation>, PINNs yield errors below <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(0.0009\%\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(0.05\%\)</EquationSource> </InlineEquation> respectively, validating their robustness.</p>

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Solving Lane-Emden-Type Eigenvalue Problems with Physics-Informed Neural Networks

  • 期刊论文

  • 发表日期：

    2025年11月18日

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  • Luke Oluwaseye Joel,
  • Charis Harley,
  • Ebrahim Momoniat

摘要

The Lane-Emden equation, a nonlinear second-order ordinary differential equation, plays a fundamental role in theoretical physics and astrophysics, particularly in modeling the structure of stellar interiors. Also referred to as the polytropic differential equation, it describes the behavior of self-gravitating polytropic spheres. In this study, we present a novel approach to the solution of the eigenvalue problem which arises when considering the Lane-Emden equation for \(n = 0, 1, 2, 3, 4\) using Physics-Informed Neural Networks (PINNs). The novelty of this work is that, we not only solve the Lane-Emden equation via PINNs but we also determine the eigenvalue, r, which is the stellar radius. Hyperparameter tuning was conducted using Bayesian optimization in the Optuna framework to identify optimal values for the number of hidden layers, number of neurons, activation function, optimizer, and learning rate for each value of n. The results show that, for \(n = 0, 1\) , PINNs achieve near-exact agreement with theoretical eigenvalues (errors < \(0.000806\%\) ). While for more nonlinear cases, \(n = 2, 3\) and \(n=4\) , PINNs yield errors below \(0.0009\%\) and \(0.05\%\) respectively, validating their robustness.