神经网络具有一定意义上的万有逼近性, 因此有广泛的应用。在应用中,神经网络的复杂度, 即一个神经网络中神经元的个数, 很大程度上影响程序的执行效率和复杂度. 在[1] 中, 作者构造了具有一定神经元个数的三层前向反馈神经网络, 来任意逼近此多项式. 并且说明了对于任意Rd上的连续函数, 都可以找到一定复杂度的神经网络可以达到与 n 次多项式相同的逼近度. 本文在利用其对于多项式的方法进一步研究神经网络与样条的关系. 证明了存在具有一定神经元个数的神经网络, 可以任意逼近等距节点的某些样条函数. 进一步地可以得到, 对于 R 上的连续的周期函数, 能找到一定复杂度的神经网络, 使它具有与等距节点的 k 阶最小亏格的样条函数相同的逼近度.

Neural network is popular in many applications for its universal approximation property insome sense. In application, the complexity of a neural network (usually the number of neurons inthe neural network), influenced the complexity and efficiency of a program. So the research aboutthe relationship between the complexity and its approximate rate is useful. In paper[1], a threelayers forward neural network is established to approximate any polynomials in Rd, within anyapproximate error; and illustrated that for any continuous function in Rd, a neuron network withsomecomplexitycanbefoundtohaveatleastthesameapproximaterateasndegreespolynomials.In this paper, I worked on the basis of paper[1], and research on the relationship between neuronnetwork and spline. It is proved in this paper that there exist a neuron network with fixed numberof neurons, it can universally approximate a spline with fixed knots. And illustrated that for anyperiodic continuous function in R, there exist a neuron network with some complexity, it has thesome approximate rate with k degrees splines with m fixed knots.