Boundedness and Convergence of Online Gradient Method with Penalty for Linear Output Feedforward Neural Networks

This paper investigates an online gradient method with penalty for training feedforward neural networks with linear output. A usual penalty is considered, which is a term proportional to the norm of the weights. The main contribution of this paper is to theoretically prove the boundedness of the weights in the network training process. This boundedness is then used to prove an almost sure convergence of the algorithm to the zero set of the gradient of the error function.     

Sufficient conditions for the preservation of the boundedness in a numerical method for a physical model with transport memory and nonlinear damping

Departing from a finite-difference scheme to approximate solutions of a nonlinear, hyperbolic partial differential equation which generalizes the Burgers–Huxley equation from fluid dynamics, we investigate conditions on the model coefficients and the computational parameters under which positive and bounded initial data evolve into positive and bounded new approximations. The model under investigation includes nonlinear coefficients of damping and advection, and the reaction term extends the reaction law of the classical Fisher–Kolmogorov–Petrovsky–Piscounov equation. The method can be expressed in vector form in terms of a multiplicative matrix which, under certain parametric conditions, becomes an M-matrix. Using the fact that every M-matrix is non-singular and that the entries of its inverse are positive, real numbers, we establish sufficient conditions under which the method provides new, positive and bounded approximations from previous, positive and bounded data and boundary conditions. The numerical results confirm the fact that the conditions derived here are sufficient for the positivity and the boundedness of the approximations; moreover, computational experiments evidence the fact that the method still preserves these properties for values of the model and the numerical parameters outside of the analytic regions of positivity and boundedness. We point out that our simulations show a good agreement between the numerical approximations computed through our method and the corresponding, analytical solutions.     

Decidability and complexity of Petri nets with unordered data

We prove several decidability and undecidability results for ν-PN, an extension of P/T nets with pure name creation and name management. We give a simple proof of undecidability of reachability, by reducing reachability in nets with inhibitor arcs to it. Thus, the expressive power of ν-PN strictly surpasses that of P/T nets. We encode ν-PN into Petri Data Nets, so that coverability, termination and boundedness are decidable. Moreover, we obtain Ackermann-hardness results for all our decidable decision problems. Then we consider two properties, width-boundedness and depth-boundedness, that factorize boundedness. Width-boundedness has already been proven to be decidable. Here we prove that its complexity is also non-primitive recursive. Then we prove undecidability of depth-boundedness. Finally, we prove that the corresponding “place version” of all the boundedness problems is undecidable for ν-PN. These results carry over to Petri Data Nets.     

一类非线性动力系统的定性分析

通过定性分析方法讨论了一类被捕食者种群有密度制约的Ho lling I类捕食系统平衡点的性态、解的有界性、正平衡点的全局渐进稳定性,得到了极限环不存在的条件,并利用Po incare-Bend ixson环域定理得到了极限环存在的充分条件.     

一类饱和反应模型的全局稳定性

本文对一类饱和反应模型进行了研究,得到了其在R~+中的一切解的有界性以及平衡点M(x~*,y~*)在R~+中的全局稳定性的充分条件。     

Boundedness and convergence of online gradient method with penalty and momentum

In this paper, the deterministic convergence of an online gradient method with penalty and momentum is investigated for training two-layer feedforward neural networks. The monotonicity of the new error function with the penalty term in the training iteration is firstly proved. Under this conclusion, we show that the weights are uniformly bounded during the training process and the algorithm is deterministically convergent. Sufficient conditions are also provided for both weak and strong convergence results.     

Dynamics of the difference equation

We study the invariant interval, the character of semicycles, the global stability, and the boundedness of the difference equation

where the initial conditions x_−k,…,x₋₁,x₀ are nonnegative, and p,q>0.     

Petri网化简操作及其在系统验证中的应用

针对柔性制造系统的验证问题，提出了用Petri网化简操作解决问题的方案。给出了两种化简操作。研究了这两种化简操作的动态性质保持问题，给出了化简后的Petri网保持活性、有界性、可回复性的一组充分条件或充要条件。对一个柔性制造系统进行了验证。本文的结果可为Petri网系统静态和动态性质的考察提供有效途径，为复杂大系统的分析提供重要手段，并特别适合于柔性制造系统的验证，具有一定的实用价值。     

广义极大算子的加权Φ－有界性

定义了一类广义极大算子，该算子是Ｈａｒｄｙ－Ｌｉｔｔｌｅｗｏｏｄ极大算子的推广，建立了该算子的加权弱型与强型Φ－有界性，这里Φ是Ｙｏｎｇ函数．     

Petri网共享子网合成及其在系统设计中的应用

针对柔性制造系统的设计和分析,提出了经由Petri网共享子网构成合成网的解决方案;研究了Petri网共享子网合成的动态性质保持问题,给出了合成网保持活性、有界性的一组充分条件;对一个柔性制造系统进行了设计和分析。本文的结果可为Petri网系统静态和动态性质的考察提供有效途径,为Petri网复杂大系统的分析提供重要手段,并特别适合于柔性制造系统的设计,具有一定的实用价值。