A new algorithm for solving nonlinear parabolic equations using extreme learning machine method with parameter retention

• Published in: European physical journal plus 2024-02, Vol.139 (2), p.107, Article 107
• Main Authors: He, Jilong, Zheng, Zhoushun
• Format: Article
• Language: English
• Subjects: Accuracy; Algorithms; Applied and Technical Physics; Approximation; Artificial neural networks; Atomic; Bias; Complex Systems; Condensed Matter Physics; Finite difference method; Initial conditions; Iterative methods; Iterative solution; Machine learning; Mathematical analysis; Mathematical and Computational Physics; Molecular; Neural networks; Normal distribution; Operators (mathematics); Optical and Plasma Physics; Parameters; Partial differential equations; Physics; Physics and Astronomy; Regular Article; Supervised learning; Theoretical
• ISSN: 2190-5444; 2190-5444
• DOI: 10.1140/epjp/s13360-024-04897-7

This paper proposes a new iterative method using extreme learning machine (ELM) to solve nonlinear parabolic equations. Unlike feedforward neural networks, ELM does not require training a large number of parameters, but only needs to calculate the pseudo-inverse of the matrix, reducing a significant amount of computation time. The important step of the new iterative method is to first use supervised learning to obtain the initial conditions of the discretized partial differential equation and then continue to calculate the differential operator while retaining the parameters. Finally, the finite difference method is used for iterative solution in time. The key difference here is that the parameters of the ELM are retained throughout the entire process and do not need to be updated. The feasibility of our method is verified by applying it to two nonlinear parabolic equations.