摘要
We present a neuro-heuristic computing platform for finding the solution for initial value problems （IVPs） of non- linear pantograph systems based on functional differential equations （P-FDEs） of different orders. In this scheme, the strengths of feed-forward artificial neural networks （ANNs）, the evolutionary computing technique mainly based on genetic algorithms （GAs）, and the interior-point technique （IPT） are exploited. Two types of mathematical models of the systems are constructed with the help of ANNs by defining an unsupervised error with and without exactly satisfying the initial conditions. The design parameters of ANN models are optimized with a hybrid approach GA-IPT, where GA is used as a tool for effective global search, and IPT is incorporated for rapid local convergence. The proposed scheme is tested on three different types oflVPs of P-FDE with orders 1-3 The correctness of the scheme is established by comparison with the existing exact solutions. The accuracy and convergence ofthc proposed scheme are further validated through a large number of numerical experiments by taking different numbers of neurons in ANN models.
本文提出了一种启发式神经网络计算平台,用于解决基于不同阶数泛函微分方程的非线性受电弓系统(Pantograph systems based on functional differential equations,P-FDEs)中的初值问题(Initial value problems,IVPs)。该方案利用了前馈人工神经网络(Artificial neural networks,ANNs)、基于遗传算法(Genetic algorithms,GAs)的进化计算技术,以及内点技术(Interior-point technique,IPT)。通过设定一个无监督学习误差,针对完全和不完全满足初始条件两种情况,利用ANNs创建了系统的两种数学模型。采用GA–IPT混合算法,对ANN模型的设计参数进行了优化。在GA–IPT中,GA是有效的全局搜索工具,IPT则用于快速的局部收敛。针对三种不同类型的1–3阶P-FDEs的IVPs对该方案进行了测试。通过对比现有的精确解,确认了该方案的正确性。通过采用不同数量神经元的ANN模型进行了大量的数值实验,进一步验证了该方案的准确性和收敛性。
