Lu Yanfei, Zhang Shiqing, Weng Futian, Sun Hongli
School of Electronics and Information Engineering, Taizhou University, Zhejiang, Taizhou, China.
Data Mining Research Center, Xiamen University, Fujian, Xiamen, China.
Front Comput Neurosci. 2023 Mar 9;17:1120516. doi: 10.3389/fncom.2023.1120516. eCollection 2023.
In this study, we investigate a new neural network method to solve Volterra and Fredholm integral equations based on the sine-cosine basis function and extreme learning machine (ELM) algorithm. Considering the ELM algorithm, sine-cosine basis functions, and several classes of integral equations, the improved model is designed. The novel neural network model consists of an input layer, a hidden layer, and an output layer, in which the hidden layer is eliminated by utilizing the sine-cosine basis function. Meanwhile, by using the characteristics of the ELM algorithm that the hidden layer biases and the input weights of the input and hidden layers are fully automatically implemented without iterative tuning, we can greatly reduce the model complexity and improve the calculation speed. Furthermore, the problem of finding network parameters is converted into solving a set of linear equations. One advantage of this method is that not only we can obtain good numerical solutions for the first- and second-kind Volterra integral equations but also we can obtain acceptable solutions for the first- and second-kind Fredholm integral equations and Volterra-Fredholm integral equations. Another advantage is that the improved algorithm provides the approximate solution of several kinds of linear integral equations in closed form (i.e., continuous and differentiable). Thus, we can obtain the solution at any point. Several numerical experiments are performed to solve various types of integral equations for illustrating the reliability and efficiency of the proposed method. Experimental results verify that the proposed method can achieve a very high accuracy and strong generalization ability.
在本研究中，我们研究了一种基于正弦余弦基函数和极限学习机（ELM）算法来求解沃尔泰拉积分方程和弗雷德霍姆积分方程的新神经网络方法。考虑到ELM算法、正弦余弦基函数以及几类积分方程，设计了改进模型。该新型神经网络模型由输入层、隐藏层和输出层组成，其中利用正弦余弦基函数消除了隐藏层。同时，利用ELM算法的特点，即隐藏层偏差以及输入层与隐藏层的输入权重无需迭代调整即可完全自动实现，我们可以大大降低模型复杂度并提高计算速度。此外，寻找网络参数的问题被转化为求解一组线性方程。该方法的一个优点是，我们不仅可以获得第一类和第二类沃尔泰拉积分方程的良好数值解，还可以获得第一类和第二类弗雷德霍姆积分方程以及沃尔泰拉 - 弗雷德霍姆积分方程的可接受解。另一个优点是，改进算法以封闭形式（即连续且可微）提供了几种线性积分方程的近似解。因此，我们可以在任何点获得解。进行了几个数值实验来求解各种类型的积分方程，以说明所提出方法的可靠性和效率。实验结果验证了所提出的方法可以实现非常高的精度和强大的泛化能力。
