【摘要】 一阶神经网络的函数逼近理论虽已研究得相当深入,但对高阶神经网络在这方面的研究却报道甚少.本文给出了高阶神经网络的逼近理论.其具体内容是:如果网络的激励函数σ∈C~∞不是一个多项式,那么高阶神经网络能够以任意精度逼近任意定义在紧集上的连续函数.根据这个理论,可知稀疏或部分连接的高阶神经网络象全连接的网络一样能够逼近任意连续函数.基于此观点,我们设计了一个稀疏连接的高阶网络和另一个具有相同数目连接权的一阶神经网络,并把它们应用于太阳黑子序列进行逼近和预测.模拟实验结果表明,不论在收敛速度还是在推广能力方面,稀疏连接的高阶神经网络都优于一阶神经网络.更多还原

【Abstract】 The function approximation capabilities of first-order neural networks have been investigated rigorously, but few related works about higher-order neural networks have been reported. In this paper, we prove that higher-order neural networks can approximate any continuous function on a compact set with arbitrary degree of accuracy, and provided that the activation function belongs to C∞ and non-polynomial . According to this theory, we know that partially connected higher-order neural networks can approximate any continuous functions as fully connected neural networks can do. Based on this argument, we design a partially connected higher-order network to compare the first-order one with that of almost the same number of connections. The theory is validated by applying sunspot series, and the simulation results show that the partially connected higher-order network exhibits better convergence rate and generalisation ability compared to the first-order one.更多还原