Developing a Gudermannian neural network for solving the Painlevé model-II in the context of nonlinear optics

Journal article

A design of Gudermannian neural network (GNN) is executed for the numerical results of the Painlevé model-II in the context of nonlinear optics (PM-II-NO). One of the forms of artificial neural networks is GNN, which uses the Gudermannian function (GF) as a merit function. This function has a nonlinearity that performs a complicated association in inputs and outputs. The design of the merit function is performed based on the differential PM-II-NO and boundary conditions, which is optimized further using the hybrid of the global search particle swarm optimization (PSO) and local search sequential quadratic programming (SQP), i.e., PSO-SQP. The algorithm’s accuracy is perceived via matching of obtained and reference database solutions, and insignificant performance of absolute error. Furthermore, the statistical investigations based on multiple independent executions are performed in order to check the reliability of the scheme by various tests, e.g., mean square error, semi inter-quartile range, and Theil inequality coefficient in order to present the significance and reliability of the designed GNN-PSO-SQP for the PM-II-NO. The comparison of the proposed results taking 5, 15, and 45 numbers of neurons and literature results is also presented to perform the neuron analysis of this study. This designed neural network is mainly valuable to solve differential systems, such as PM-II-NO, which can competently estimate the solutions by applying the GF properties.

Authors: Faisal, S; Sabir, Z; Khan, SU; Aamir, M; Cyran, KA

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Nature Research
• Journal: Scientific Reports
• Volume: 16
• Issue: 1
• Article number: 16470
• Publication date: 2026-04-05
• Acceptance date: 2026-03-05
• DOI: 10.1038/s41598-026-43643-0
• EISSN: 2045-2322
• ISSN: 2045-2322
• Language: English
• Keywords: Gudermannian function; Sequential quadratic programming; Nonlinear optics; Neural networks; Particle swarm optimization; Painlevé model-II