Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations

In this paper by using MultiLayer Perceptron and Radial Basis Function (RBF) neural networks, a novel method for solving both kinds of differential equation, ordinary and partial differential equation, is presented. From the differential equation and its boundary conditions, the energy function of the network is prepared which is used in the unsupervised training method to update the network parameters. This method was implemented to solve the nonlinear Schrodinger equation in hydrogen atom and triangle-shaped quantum well. Comparison of this method results with analytical solution and two well-known numerical methods, Runge–kutta and finite element, shows the efficiency of Neural Networks with high accuracy, fast convergence and low use of memory for solving the differential equations.     

On the a-posteriori error bounds for the solution of ordinary nonlinear differential equations

This paper advances a procedure to compute the a-posteriori error bounds for the solution of a wide class of ordinary differential equations with given initial conditions. The method is based on an iterative scheme which yields upper and lower bounds which approach each other. The convergence of these iterations is proved analytically. Application of the proposed method is demonstrated by several examples. The method is independent of the integration scheme used. Also, separate bounds for each component of the solution are specifically available as a function of time compared to the bound on the norm yielded by conventional methods.     

Switching behavior of solutions of ordinary differential equations with abs-factorable right-hand sides

We consider nonsmooth dynamic systems that are formulated as the unique solutions of ordinary differential equations (ODEs) with right-hand side functions that are finite compositions of analytic functions and absolute-value functions. Various non-Zenoness results are obtained for such solutions: in particular, any absolute-value function in the ODE right-hand side can only switch between its two linear pieces finitely many times on any finite duration, even when a discontinuous control input is included. These results are extended to obtain numerically verifiable necessary conditions for the emergence of “valley-tracing modes”, in which the argument of an absolute-value function is identically zero for a nonzero duration. Such valley-tracing modes can create theoretical and numerical complications during sensitivity analysis or optimization. We show that any valley-tracing mode must begin either at the initial time, or when another absolute-value function switches between its two linear pieces.     

On the convergence of variational iteration method for nonlinear coupled system of partial differential equations

In this paper, the sufficient conditions that guarantee the convergence of the variational iteration method when applied to solve a coupled system of nonlinear partial differential equations are presented. Especial attention is given to the error bound of the nth term of the resultant sequence. Numerical examples to show the efficiency of the method are presented.     

Optimal control of large space structures governed by a coupled system of ordinary and partial differential equations

In this paper we consider the problem of optimal regulation of large space structures in the presence of flexible appendages. For simplicity of presentation, we consider a spacecraft consisting of a rigid bus and a flexible beam. The complete dynamics of the system is given by a coupled set of ordinary and partial differential equations. We show that the solution of this hybrid system is defined in a product space of appropriate finite- and infinite-dimensional spaces. We develop necessary conditions for determining the control torque and forces for optimal regulation of attitude maneuvers of the satellite along with simultaneous suppression of elastic vibrations of the flexible beam.     

Adaptive consensus protocol for networks of multiple agents with nonlinear dynamics using neural networks

In this paper, an adaptive protocol is proposed to solve the consensus problem of multi‐agent systems with high‐order nonlinear dynamics by using neural networks (NNs) to approximate the unknown nonlinear system functions. It is derived that all agents achieve consensus if the undirected interaction graph is connected, and the transient performance of the multi‐agent system is also investigated. It shows that the adaptive protocol and the consensus analysis can be easily extended to switching networks by the existing LaSalle's Invariance Principle of switched systems. A numerical simulation illustrates the effectiveness of the proposed consensus protocol. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Equality of partial solutions in the decomposition method for linear or nonlinear partial differential equations

We consider the solution of partial differential equations for initial/boundary conditions using the decomposition method. The partial solutions obtained from the seperate equations for the highest-ordered linear operator terms are shown to be identical when the boundary conditions are general, and asymptotically equal when the boundary conditions in one independent variable are independet of other variables.     

Vectorisation of the multiple shooting method for the nonlinear boundary value problem in ordinary differential equations

This paper describes a vectorised variant of the multiple shooting algorithm for solving boundary value problems in ordinary differential equations. The central idea for achieving vectorisation is to construct an integration mesh that is used subsequently for every Newton iteration, thus enabling one to build vectors whose elements correspond to the different sub-intervals.     

连续非线性动态系统建模的模糊神经网络方法

提出一种适合于一般连续非线性动态系统建模的新的Runge-Kutta模糊神经网络(RKFNN)，证明了RKFNN的存在性。采用传统的Runge-Kutta求积公式构造，实现了对系统的状态变化特性进行学习，解决了直接映射方式对系统的动态轨迹进行学习时存在的精度低等问题，并提出了RKFNN的在线递推学习算法。对连续非线性动态系统进行楚模的仿真结果表明，RKFNN方法是一种较好的方法。     

A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations

In this study, the decomposition method for solving the linear heat equation and nonlinear Burgers equation is implemented with appropriate initial conditions. The application of the method demonstrated that the partial solution in the x-direction requires more computational work when compared with the partial solution developed in the t-direction but the numerical solution in the x-direction are performed extremely well in terms of accuracy and efficiency.     

Parallel implementation of finite-element/newton method for solution of steady-state and transient nonlinear partial differential equations

Domain decomposition by nested dissection for concurrent factorization and storage (CFS) of asymmetric matrices is coupled with finite element and spectral element discretizations and with Newton's method to yield an algorithm for parallel solution of nonlinear initial-and boundary-value problem. The efficiency of the CFS algorithm implemented on a MIMD computer is demonstrated by analysis of the solution of the two-dimensional, Poisson equation discretized using both finite and spectral elements. Computation rates and speedups for the LU-decomposition algorithm, which is the most time consuming portion of the solution algorithm, scale with the number of processors. The spectral element discretization with high-order interpolating polynomials yields especially high speedups because the ratio of communication to computation is lower than for low-order finite element discretizations. The robustness of the parallel implementation of the finite-element/Newton algorithm is demonstrated by solution of steady and transient natural convection in a two-dimensional cavity, a standard test problem for low Prandtl number convection. Time integration is performed using a fully implicit algorithm with a modified Newton's method for solution of nonlinear equations at each time step. The efficiency of the CFS version of the finite-element/Newton algorithm compares well with a spectral element algorithm implemented on a MIMD computer using iterative matrix methods.Submitted toJ. Scientific Computing, August 25, 1994.     

Conjugate gradient algorithms in the solution of optimization problems for nonlinear elliptic partial differential equations

Several variants of the conjugate gradient algorithm are discussed with emphasis on determining the parameters without performing line searches and on using splitting techniques to accelerate convergence. The splittings used here are related to the nonlinear SSOR algorithm. The behavior of the methods is illustrated on a discretization of a nonlinear elliptic partial differential boundary value problem, the minimal surface equation. A conjugate gradient algorithm with splittings is also developed for constrained minimization with upper and lower bounds on the variables, and the method is applied to the obstacle problem for the minimal surface equation.     

An A-stable extended trapezoidal rule for the integration of ordinary differential equations

We examine a single-step implicit-integration algorithm which is obtained by a modification of the well-known trapezoidal rule. The obtained new method is a third-order numerical process and preserves the property of A-stability of the trapezoidal rule. Numerical examples involving stiff linear systems of first-order differential equations are also included to demonstrate the practical usefulness of this new integration procedure.     

Finding the differential characteristics of block ciphers with neural networks

We have developed a model to represent the differential operation of block ciphers in order to help finding differential characteristics. Through this model, the whole space of differential characteristics for a block cipher is represented by a multi-level weighted directed graph. In this way, the problem of finding the best differential characteristic for a block cipher reduces to the problem of finding the minimum-weight multi-branch path between two known nodes in the proposed graph. In this paper, we use recurrent neural networks to find such a path in the differential operation graph of a block cipher. The path is found through minimization of the network cost function. We use the Hopfield network and the Boltzmann machine with and without chaos to minimize the cost function. Chaos is introduced to assist the network to escape from the local minima of the cost function. Experimental results indicate the usefulness of the approach and comparison of the performance of the used techniques shows that the Boltzmann machine algorithm incorporating simulated annealing produces the best result.     

The shooting method for the solution of ordinary differential equations: A control-theoretical perspective

A discrete space state representation is used to provide a feedback control-theoretical formulation for the iterative shooting method used to solve two-point boundary value problems in ordinary differential equations. Preconditioning matrices, error analysis and convergence conditions for these iterative methods are put into a feedback control perspective as well.