Turchetti Claudio, Crippa Paolo, Pirani Massimiliano, Biagetti Giorgio
DEIT-Dipartimento di Elettronica, Intelligenza Artificiale e Telecomunicazioni, Università Politecnica delle Marche, I-60131 Ancona, Italy.
IEEE Trans Neural Netw. 2008 Jun;19(6):1033-60. doi: 10.1109/TNN.2007.2000055.
The learning capability of neural networks is equivalent to modeling physical events that occur in the real environment. Several early works have demonstrated that neural networks belonging to some classes are universal approximators of input-output deterministic functions. Recent works extend the ability of neural networks in approximating random functions using a class of networks named stochastic neural networks (SNN). In the language of system theory, the approximation of both deterministic and stochastic functions falls within the identification of nonlinear no-memory systems. However, all the results presented so far are restricted to the case of Gaussian stochastic processes (SPs) only, or to linear transformations that guarantee this property. This paper aims at investigating the ability of stochastic neural networks to approximate nonlinear input-output random transformations, thus widening the range of applicability of these networks to nonlinear systems with memory. In particular, this study shows that networks belonging to a class named non-Gaussian stochastic approximate identity neural networks (SAINNs) are capable of approximating the solutions of large classes of nonlinear random ordinary differential transformations. The effectiveness of this approach is demonstrated and discussed by some application examples.
神经网络的学习能力等同于对现实环境中发生的物理事件进行建模。一些早期的研究表明，某些类别的神经网络是输入-输出确定性函数的通用逼近器。近期的研究使用一类名为随机神经网络（SNN）的网络扩展了神经网络逼近随机函数的能力。用系统理论的语言来说，确定性和随机函数的逼近都属于非线性无记忆系统的辨识范畴。然而，迄今为止所呈现的所有结果都仅局限于高斯随机过程（SP）的情况，或者局限于保证该特性的线性变换。本文旨在研究随机神经网络逼近非线性输入-输出随机变换的能力，从而拓宽这些网络对具有记忆的非线性系统的适用范围。特别地，本研究表明，属于一类名为非高斯随机近似恒等神经网络（SAINN）的网络能够逼近一大类非线性随机常微分变换的解。通过一些应用示例展示并讨论了该方法的有效性。
