本文为印度ROURKELA国立技术研究所（作者：Pramod Kumar Parida）的硕士论文，共29页。
本文介绍了利用人工神经网络求解常微分方程的方法。在不直接使用优化技术的情况下，采用无监督型前馈神经网络对给定ODE进行逼近，使其精度达到要求。该问题是以这样一种方式表述的，即通过其构造来满足初始/边界条件。ODE的轨迹解为两项之和。第一项满足初始或边界条件，第二项是由n个输入和h个隐sigmoid单元产生的前馈神经网络输出。通过采用一般的学习方法，减小了误差梯度，得到了理想输出结果。本文对不同的问题进行了验证，并对任意点的人工神经网络输出收敛性进行了检验。可以注意到，通过这个过程也可以进行插值。当目标或精确结果未知或难以发现时，神经元处理器的优点是可以产生任意精度的输出。
In this investigation we introduced the method for solving Ordinary Differential Equations (ODEs) using artificial neural network. The feed forward neural network of the unsupervised type has been used to get the approximation of the given ODEs up to the required accuracy without direct use of the optimization techniques. The problem is formulated in such a manner that it satisfies the initial/boundary conditions by its construction. The trail solution of the ODE is the sum of two terms. The first term satisfies the initial or boundary conditions, while the second one is the feed forward neural output produced by n number of inputs and h number of hidden sigmoid units. The error gradient has been reduced by applying general learning method to get the desired output. The results have been verified for different problems and the convergence of Artificial Neural Network (ANN) output has been checked for arbitrary points. It may be noted that the interpolation is also possible through this process. The advantage of neuron processor is that the output can be produced to any arbitrary accuracy, while the targets or exact results are unknown or hard to find out.
1 引言
2 文献综述
3 ANN细节
4 采用ANN实现ODE求解的一般方法
5 示例与结果
6 讨论
7 结论与未来研究展望
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