反馈神经网络对非线性动力系统的本质逼近能力研究

Nonlinear science is a rapidly developing research field. Much.attention has been paid all over the world to its promising. Neural networks, which is one of active branches in nonlinear science, has drawn great attention. In application, it involves various areas in natural and social science. Now it has become a powerful tool of exploring and solving many complicated problems in natural science and engineering. In this project, we make a systematic investigation into approximation ability of recurrent neural networks. . Neural networks represent a class of functions for the efficient identification and forecasting of dynamical systems. The method of recurrent neural networks approximation is used in nonlinear systems. So far , the study of approximation ability that recurrent neural network works on the nonlinear dynamical systems are basically concentrated on the dense problems，that is, the qualitative issues of the recurrent network approximation. However，from the application point of view, quantitative study of recurrent neural network approximation of dynamical systems and relative algorithms are particularly important. The quantitative study of recurrent neural network approximation, especially the study of reflecting the relationship between network's approximation speed and network topology structure, has begun to attract strong attention recently. Therefore, this project will have a deep study of the approximation capability of recurrent neural network over the nonlinear dynamical system. The specific tasks include: designing a new model of the core algorithm which has a high-precision approximation ability of recurrent neural network. By establishing both upper bound and lower bound estimations on approximation order, the essential approximation order of these networks is estimated and the theorem of saturation is proved. These results can precisely characterize the approximation ability of these networks and clarify the relationship among the rate of approximation, the number of hidden-layer unitsand the properties of dynamical systems. Depicting the limit behavior that the neural network we have constructed works on the approximation capability of the nonlinear dynamical system and also describing the dependencies between the quality of the information of the power system and the network topology. Qualitative analysis and quantitative analysis are combined in this project in order to provide a reference for the evaluation and improvement in this subject. . Moreover, the efficiently,credibly and intelligent optimization algorithms must be designed, which is rarely seen in the related research until now.

目前关于反馈神经网络对非线性动力系统逼近能力的研究基本上集中于稠密性问题，即反馈网络逼近的定性问题。但是，从应用角度来看，反馈神经网络逼近动力系统的定量研究和算法尤为重要。有关反馈神经网络逼近的定量研究, 特别是反映网络的逼近速度与网络拓扑结构之间关系的研究, 最近开始引起人们的强烈关注。本项目将深入研究反馈神经网络对非线性动力系统的本质逼近能力。具体任务包括：设计具有高精度逼近能力的反馈神经网络新模型和新算法；研究该类神经网络对非线性动力系统逼近速度的上、下界估计和本质逼近阶估计；刻画所构造的神经网络对非线性动力系统的本质逼近能力的极限行为与网络拓扑结构、动力系统演化规律的空间性质（元规则的空间性质、支持度及置信度等）之间的相依关系；构造具有较高逼近能力的，结构更简单化的反馈神经网络新模型。另外，设计求解这些模型的优化算法时，必须设计高效、可信的智能优化算法，这是目前相关研究所少有的。

本项目基本按照申请书原订计划，就反馈神经网络对非线性动力系统的本质逼近能力及其应用问题进行了较系统的研究。主要工作如下：1. 构造出了几类结构更简单化且具有较高逼近能力的反馈神经网络新模型和新算法。2. 给出了所构造的新型反馈神经网络逼近非线性动力系统精度的定性及定量研究。 研究网络的特性，采用多元函数逼近理论，数值逼近理论，以非线性动力系统的元规则空间性质、元规则的置信度等来刻画动力系统演化规律的空间性质，应用神经元个数，动力系统轨迹、分形状态及元规则的空间属性集的上、下界估计尺度等作为误差度量工具。给出逼近的上、下界估计和本质逼近阶及插值条件。3. 现实生活中归结的微分方程不满足解析解存在条件的情况比比皆是，根本无法讨论它的解析解。针对这一问题，项目组成员利用“分割、近似代替、求和、取极限”的思想与有限无重叠区域分解算法来处理拟线性奇异摄动问题并讨论了该算法一阶和二阶差分格式下的收敛性及误差的定性估计。4. 神经网络可用于近似代替，但其会遭遇较复杂的参数辨识问题。为了较好的解决这一问题，项目组将擅长全局搜索的蚁群-粒子群混合算法用于对系统参数进行最优化选取，再将此混合算法与神经网络结合，以充分发挥人工神经网络的“智能”性来更好的解决参数辨识问题。5. 非线性动态系统模型需要由模型构建元素来实现非线性映射和动力学行为。在我们的研究中状态模型和输入输出模型不仅仅用于建模，还应用于模型的灵敏度分析。6. 将所建立的几类神经网络模型与算法应用于宁东能源化工基地的大气与生态环境及黄河宁夏段水体环境的评价中，从而为宁东基地大气与生态环境及黄河宁夏段水污染防治和水资源保护提供了参考。研究结果在北大核心以上级别刊物上发表及录用论文18篇，其中被SCI、EI检索6篇，另有部分论文仍在修改、审稿或整理阶段。参加了4次神经网络及交叉学科方面的国际学术会议，以本项目内容做选题，指导8名硕士研究生完成了硕士学位论文。