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September 23, 2025Open Access

Single-layer Laguerre neural network model for solving Lane-Emden-Fowler type equations

Key Points

The Laguerre neural network effectively solves Lane-Emden-Fowler type equations, showing promise in computational methods.
Results obtained from the Laguerre neural network are benchmarked against other models, providing a framework for comparison.
An unsupervised backpropagation algorithm using Adam optimization is employed, ensuring efficient parameter adjustment.
Model comparisons reveal advantages of the Laguerre approach in accuracy and performance when dealing with nonlinear equations.

Abstract

In this study, a single-layer functional link artificial neural network (FLANN) model based on Laguerre polynomials, referred to as the Laguerre Neural Network (LgNN), is utilized to solve second-order linear and non-linear equations of Lane-Emden-Fowler type. Using this model, in which the hidden layers are replaced with Laguerre polynomials, we initially expand the input patterns corresponding to a given set of nonlinear Lane-Emden-Fowler type equations. Subsequently, the network parameters are adjusted through an unsupervised error backpropagation algorithm that employs Adam optimization. Consequently, we compare the LgNN results with those obtained by other FLANNbased models, namely the Chebyshev Neural Network (ChNN) and the Legendre Neural Network (LeNN), by solving some initial value problems of Lane-Emden–Fowler type equations.

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