基于前馈神经网络的函数积分计算

Function integral computation based on feedforward neural networks

由于三层前馈神经网络可以逼近任何连续函数,因此可以利用三层前馈神经网络来逼近被积函数的原函数,并计算函数的积分。对于定积分、在矩形或长方体区域上的二重积分或三重积分的计算,首先构造一个三层前馈神经网络,通过训练网络使其在积分区域上对输入量的导数值、二阶混合偏导数值或三阶混合偏导数值等于相应被积函数值,训练好的网络就可逼近被积函数的原函数。对于非矩形或非长方体区域上的二重积分或三重积分,可通过换元法将积分区域转化为矩形或长方体区域。实例分析表明该方法理论简单、思路清晰、易于实现,同时精度也能得到满足。

Due to strong capability of mapping any continuous functions, three-layered feedforward neural networks can be employed to approximate primit/ve function of integrand, and compute definite integral. In order to compute definite integral, double integral in rectangular domain, triple integral in orthogonal box domain, a feedforward neural network with three layers was constructed, and could be the substitution of primitive function of integrand by training samples to satisfy first- order partial derivative, second-order mixed partial derivative, third-order mixed partial derivative of networks with respect to its input equal to integrand value in integral domain. If integral domain is not rectangular domain or orthogonal box domain, substitution method was employed to change domain shape. Case analysis shows this method has the properties of simplicity, easy operation, and high accuracy.