瞬时混沌神经网络的混沌动力学

首先利用"不可分意味着混沌"从理论上证明了一维瞬时混沌神经网络在一定的条件下按Li-Yorke意义是混沌的;特别地,进一步推出了混沌神经网络按Li-Yorke意义是混沌的充分条件,而这将从理论上证明Aihara等人通过数值计算所得结论;最后,为说明前面的结论给出了一个例子及其数值计算的结果。     

Harmonic Analysis of Neural Networks

It is known that superpositions of ridge functions (single hidden-layer feedforward neural networks) may give good approximations to certain kinds of multivariate functions. It remains unclear, however, how to effectively obtain such approximations. In this paper, we use ideas from harmonic analysis to attack this question. We introduce a special admissibility condition for neural activation functions. The new condition is not satisfied by the sigmoid activation in current use by the neural networks community; instead, our condition requires that the neural activation function be oscillatory. Using an admissible neuron we construct linear transforms which represent quite general functionsfas a superposition of ridge functions. We develop

• • • a continuous transform which satisfies a Parseval-like relation;
• • • a discrete transform which satisfies frame bounds.

Both transforms representfin a stable and effective way. The discrete transform is more challenging to construct and involves an interesting new discretization of time–frequency–direction space in order to obtain frame bounds for functions inL²(A) whereAis a compact set of Rⁿ. Ideas underlying these representations are related to Littlewood–Paley theory, wavelet analysis, and group representation theory.     

Topological Properties of the Set of Functions Generated by Neural Networks of Fixed Size

We analyze the topological properties of the set of functions that can be implemented by neural networks of a fixed size. Surprisingly, this set has many undesirable properties. It is highly non-convex, except possibly for a few exotic activation functions. Moreover, the set is not closed with respect to \(L^p\)-norms, \(0< p < \infty \), for all practically used activation functions, and also not closed with respect to the \(L^\infty \)-norm for all practically used activation functions except for the ReLU and the parametric ReLU. Finally, the function that maps a family of weights to the function computed by the associated network is not inverse stable for every practically used activation function. In other words, if \(f_1, f_2\) are two functions realized by neural networks and if \(f_1, f_2\) are close in the sense that \(\Vert f_1 - f_2\Vert _{L^\infty } \le \varepsilon \) for \(\varepsilon > 0\), it is, regardless of the size of \(\varepsilon \), usually not possible to find weights \(w_1, w_2\) close together such that each \(f_i\) is realized by a neural network with weights \(w_i\). Overall, our findings identify potential causes for issues in the training procedure of deep learning such as no guaranteed convergence, explosion of parameters, and slow convergence.

     

非对称细胞神经网络的稳定性分析

This paper describes the problem of stability for one-dimensional Cellular Neural Networks(CNNs). A sufficient condition is presented to ensure complete stability for a class of special CNN's with nonsymmetric templates, where the parameter in the output function is greater than or equal to zero. The main method is analysising the property of the equilibrium point of the CNNs system.     

时滞细胞神经网络的稳定性分析

研究具有时滞的细胞神经网络的稳定性问题，通过构造合适的Lyapunov函数及不等式分析技巧，给出了时滞细胞神经网络全局稳定的新的充分判据，这些结论推广了已知文献中的结果。     

一类具有时滞的Cohen-Grossberg神经网络的指数稳定性

考虑一类具有时滞的Cohen-Grossberg神经网络,利用Lyapunov方法和微分不等式理论,得到了其全局指数稳定性的判别准则.该准则引入了更多的参数,更便于系统的设计与分析.     

Cohen-Grossberg神经网络的全局指数稳定性分析

本文研究了CohenGrossberg神经网络模型的指数稳定性.为避免构造Lyapunov函数的困难,我们采用广义相对Dalquist数方法来分析神经网络的稳定性.借助这一方法,我们不但得到了CohenGrossberg神经网络模型平衡解的存在性、唯一性和全局指数稳定性的新的充分条件,而且给出了神经网络的指数衰减估计.所获结论改进了已有文献的相关结果.     

一类一维混沌映射的拓扑条件

20世纪中期以来,人们在物理、天文、气象等领域中发现了大量的混沌现象.这些新发现引发了近几十年来对混沌现象的研究.由于它的困难程度和在解决实际问题中的巨大价值,对混沌现象的研究成为动力系统乃至数学中的一个长期的前沿和热点研究方向.混沌现象最本质的特征是初值敏感性,保证有初值敏感性的一个充分条件是系统具有正Lyapunov指数.因此研究系统是否具有正Lyapunov指数成为研究系统是否出现混沌的重要方法.从拓扑角度给出了一类一维映射出现混沌现象的充分条件.从拓扑的角度来研究,将加深对此类映射出现混沌的机理的认识.研究此类映射,最重要的是研究临界点、临界点轨道及它们的相互关系.我们采用临界点的逆像建立拓扑工具,使用这一拓扑工具分析临界点轨道与临界点的复杂关系,研究临界点逆轨道的运动形态、相应开集的拓扑特征,进而导出系统出现混沌的拓扑特征及它与Lyapunov指数之间的关系.     

Topological Design of Survivable Mesh-Based Transport Networks

The advent of Sonet and DWDM mesh-restorable networks which contain explicit reservations of spare capacity for restoration presents a new problem in topological network design. On the one hand, the routing of working flows wants a sparse tree-like graph for minimization of the classic fixed charge plus routing (FCR) costs. On the other hand, restorability requires a closed (bi-connected) and preferably high-degree topology for efficient sharing of spare capacity allocations (SCA) for restoration over non-simultaneous failure scenarios. These diametrically opposed considerations underlie the determination of an optimum physical facilities graph for a broadband network provider. Standalone instances of each constituent problem are NP-hard. The full problem of simultaneously optimizing mesh-restorable topology, routing, and sparing is therefore very difficult computationally. Following a comprehensive survey of prior work on topological design problems, we provide a {1–0} MIP formulation for the complete mesh-restorable design problem and also propose a novel three-stage heuristic. The heuristic is based on the hypothesis that the union set of edges obtained from separate FCR and SCA sub-problems constitutes an effective topology space within which to solve a restricted instance of the full problem. Where fully optimal reference solutions are obtainable the heuristic shows less than 8% gaps but runs in minutes as opposed to days. In other test cases the reference problem cannot be solved to optimality and we can only report that heuristic results obtained in minutes are not improved upon by CPLEX running the full problem for 6 to 18 hours. The computational behavior we observe gives insight for further work based on an appreciation of the problem as embodying unexpectedly difficult feasibility apects, as well as optimality aspects.     

Applications of Artificial Neural Networks for Periodic Structures Analysis

The paper presents the nondeterministic, based on artificial neural network application approach analysis of periodic structures. We can distinguish several examples where the problem may be observed: conventional and magnetic railways, high building constructions that consist of repeatable blocks, ship and aeroplane bodies, space-shuttle periodic designs, long-beam antenna structures or mistuned blade disks with friction damping elements. The scope of research is to examine possibilities of use the neural networks for mistuning parameters definition and also to denominate its possible causes. The results obtained via neural network simulator training process are compared with the calculations based on mathematical model. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hopfield型时滞神经网络的稳定性分析

研究了具有时滞的Hopfield型神经网络模型平衡点的全局渐近稳定性。去掉了有关文献中关于f_j在R上有界性、可微性的条件,给出了更弱的实用性较强的判定平衡点的存在唯一性及全局渐近稳定性的条件,增强了模型的适用性。     

Stability Analysis of Gradient-Based Neural Networks for Optimization Problems

The paper introduces a new approach to analyze the stability of neural network models without using any Lyapunov function. With the new approach, we investigate the stability properties of the general gradient-based neural network model for optimization problems. Our discussion includes both isolated equilibrium points and connected equilibrium sets which could be unbounded. For a general optimization problem, if the objective function is bounded below and its gradient is Lipschitz continuous, we prove that (a) any trajectory of the gradient-based neural network converges to an equilibrium point, and (b) the Lyapunov stability is equivalent to the asymptotical stability in the gradient-based neural networks. For a convex optimization problem, under the same assumptions, we show that any trajectory of gradient-based neural networks will converge to an asymptotically stable equilibrium point of the neural networks. For a general nonlinear objective function, we propose a refined gradient-based neural network, whose trajectory with any arbitrary initial point will converge to an equilibrium point, which satisfies the second order necessary optimality conditions for optimization problems. Promising simulation results of a refined gradient-based neural network on some problems are also reported.     

A Comparison of Discriminant Analysis versus Artificial Neural Networks

Artificial Neural Network (ANN) techniques have recently been applied to many different fields and have demonstrated their capabilities in solving complex problems. In a business environment, the techniques have been applied to predict bond ratings and stock price performance. In these applications, ANN techniques outperformed widely-used multivariate statistical techniques. The purpose of this paper is to compare the ANN method with the Discriminant Analysis (DA) method in order to understand the merits of ANN that are responsible for the higher level of performance. The paper provides an overview of the basic concepts of ANN techniques in order to enhance the understanding of this emerging technique. The similarities and differences between ANN and DA techniques in representing their models are described. This study also proposes a method to overcome the limitations of the ANN approach, Finally, a case study using a data set in a business environment demonstrates the superiority of ANN over DA as a method of classification of observations.     

拓扑熵为零的混乱映射的充分条件

线段上的连续自映射,当周期点集为闭集时,其轨道十分简单,当然,动力系统不会是混乱的,因此,研究周期点集的聚点的极限性态与混乱的关系,无疑可以进一步揭示混乱现象产生的原因。文[6]证明了当回归点集非闭,f是混乱的。本文则给出了周期点集非闭时f为混乱的充分条件。这说明了只要周期点集非闭动力系统就可能是混乱的。     

不分明化拓扑群的一致结构

本文引入了不分明化拓扑群的左、右、双一致结构，讨论了此类一致结构在一致连续下的一些性质，给出了此类结构在其子群上的相对结构和在其乘积上的来积结构。     

具有无穷时滞的细胞神经网络的全局稳定性分析

对具有无穷时滞的细胞神经网络平衡点的存在性、唯一性和全局渐近稳定性进行了分析．在放弃了激活函数的有界性、单调性和可微性假设的情况下,得到了系统的平衡点的存在性条件．利用向量Liapunov函数法的思想,构造适当的含有变时滞和无穷时滞的微分-积分不等式,通过对微分-积分不等式的稳定性分析,得到了神经网络系统的全局渐近稳定的充分条件．     

复杂网络混沌系统的最优控制

考虑到控制系统能量限制的要求,确定了一个二次目标函数,基于最优控制理论给出了复杂网络混沌系统的最优控制律,利用Lyapunov稳定性理论证明了闭环系统的稳定性,数值结果证明了该方法的有效性.     

数学规划的求解与广义Hopfield神经网络

神经网络技术最为成功的应用领域之一是用于求解优化问题,本文就近年来的求解优化问题的神经网络方法进行了综述     

运筹学在复杂网络社团结构分析中的应用

社团结构研究是复杂网络这一前沿领域中的重要问题,同运筹学有着密切的关联。本文介绍了传统社团结构问题的基本定义,以及最近十年通过应用运筹学理论对该问题的研究进展。这些进展包括启发式模型,到随后的概率优化模型,以及组合优化模型。通过这些介绍,说明了运筹学方法论和基本工具在复杂系统研究中所起到的重要作用。