In this paper, we examine the effects of swarm intelligence techniques on the obtaining of numerical solutions of ordinary differential equations (ODEs) with Dirichlet boundary conditions (DBCs) via feed-forward neural networks. Population-based global optimization methods such as artificial bee colony (ABC), ant colony optimization (ACO), gravitational search algorithm (GSA) and particle swarm optimization (PSO) are utilized in order to solve ODEs with DBCs numerically. Furthermore, we hybridize ACO, ABC and GSA with PSO to improve the numerical solution of optimization algorithms. Unlike the approaches in the literature, we use the personal best and global best solutions of PSO for the updating position of employed and onlooker bees in ABC, in the hybrid method that combines PSO and ABC. We also prefer the adaptive number of swarm scheme on the hybrid system of PSO and ACO. In the experiments, we give some different Dirichlet Boundary Problems to indicate the efficiency of optimization methods. Also the results are compared with the well-known traditional methods as shooting method, finite difference method and Lobatto IIIa.

在本文中，我们研究了群智能技术对通过前馈神经网络获得具有狄利克雷边界条件 (DBC) 的常微分方程 (ODE) 数值解的影响。利用基于种群的全局优化方法，例如人工蜂群 (ABC)、蚁群优化 (ACO)、引力搜索算法 (GSA) 和粒子群优化 (PSO)，以数值方式求解具有 DBC 的 ODE。此外，我们将 ACO、ABC 和 GSA 与 PSO 混合以改进优化算法的数值解。与文献中的方法不同，我们在结合 PSO 和 ABC 的混合方法中，使用 PSO 的个人最佳和全局最佳解决方案来更新 ABC 中雇佣和旁观蜜蜂的位置。我们也更喜欢 PSO 和 ACO 混合系统上的自适应群数方案。在实验中，我们给出了一些不同的狄利克雷边界问题来表明优化方法的效率。并将结果与​​著名的传统方法如射击法、有限差分法和 Lobatto IIIa 进行了比较。