Estudio de las redes neuronales ortogonales y su aplicación en la ecuación de Bagley-Torvik

Abstract

This monograph lies within the field of mathematical physics and explores the interplay between fractional calculus and orthogonal neural networks in modeling viscoelastic systems with memory, taking the fractional Bagley-Torvik equation as a representative case. The first part examines the theoretical framework of fractional calculus and its role in describing dissipation and temporal relaxation, deriving the Bagley-Torvik equation rigorously from Stokes’ problem for a Newtonian fluid. The second part reviews the mathematical foundations of orthogonal neural networks based on classical Hermite, Laguerre, and Legendre polynomials, emphasizing their consistency with distinct physical domains. Finally, a comparative analysis of three architectures (HerNN, LaNN, and LeNN) under various training schemes is conducted to approximate the equation’s numerical solution. Experimental results show a clear pattern: the Extreme Learning Machine (ELM) scheme achieves the best balance between accuracy and computational efficiency, with the Legendre-based network (LeNN) reaching a loss of 0.04035, RMSE of 0.20088, and correlation coefficient r = 0.9929 in less than 0.1 s of computation. Second-order iterative methods (BFGS and Newton-CG) also yielded competitive results, though with greater sensitivity to the network architecture and numerical conditions. The theoretical-numerical analysis further highlighted the impact of the convergence interval of orthogonal polynomials and the discretization sensitivity of the fractional operator on solution stability. Overall, the findings confirm that the Legendre network trained with ELM provides the most accurate and stable representation of the viscoelastic dynamics described by the Bagley-Torvik equation, establishing the fractional-orthogonal framework as a powerful tool for modeling physical systems with memory and nonlocal behavior.