aspg-40-4480-1792533367 20261020215607 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Pure Mathematics for Theoretical Computer Science PMTCS 2995-3162 2026 6 2 Qasim Tayyeh Department of Mechanical Techniques, Al-Nasiriya Technical Institute, Southern Technical University, Thi-Qar, Al-Nasiriya 64001, Iraq The purpose of this work was to benchmark three population-based metaheuristic optimizers—Particle Swarm Optimization, Differential Evolution, and Grey Wolf Optimizer—when used to solve nonlinear ordinary differential equations within the Neural Network Trial Solution methodology. Problems used for testing were the Riccati initial value problem, the nonlinear pendulum IVP, the Bratu boundary value problem, and the Lane-Emden equation with index five. All problems were implemented such that their boundary/initial conditions were satisfied exactly through analytical construction while their residuals at collocation points were minimized through unconstrained optimization. Thirty Monte Carlo runs of each algorithm were performed with same underlying settings to facilitate statistical comparisons between algorithms. Metrics used for comparisons were mean absolute error (MAE), root mean square error (RMSE), maximum error at any point, and rate of convergence. All significant testing was performed with the Wilcoxon signed-rank test. PSO is shown to consistently provide the smallest mean absolute error across three of the four problems, with an MAE as small as 2.78×10−5 on the Bratu BVP, while GWO was shown to stagnate prematurely when solving boundary value problems. 2026 01 10 4480 https://creativecommons.org/licenses/by/4.0/ 10.54216/PMTCS.060201 https://americaspg.com/journal/40/article/4480 https://americaspg.com/storage/articles/manuscripts/01M30K7BBKZ6QPBWNXV3XRJ19X.pdf