一类自适应泛函网络循环结构与算法

Banach压缩映射原理不仅在泛函分析中占有举足轻重的地位,同时也是数值分析中求解代数方程、常微分方程解存在唯一性,以及数学分析中积分方程求解的重要理论依据。它是数学和工程计算中最常用的方法之一。基于Banach压缩映射原理,提出一种自适应泛函网络循环结构和算法,通过训练该结构使其逼近于目标函数的不动点。通过算例分析表明,该算法具有计算精度高、收敛速度快等特点。所获结果对于神经计算方法的研究具有参考价值。     

Smooth function approximation using neural networks

An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input-output and/or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.     

Finding all roots of 2?×?2 nonlinear algebraic systems using back-propagation neural networks

The objective of this research is the numerical estimation of the roots of a complete 2?×?2 nonlinear algebraic system of polynomial equations using a feed forward back-propagation neural network. The main advantage of this approach is the simple solution of the system, by building a structure—including product units—that simulates exactly the nonlinear system under consideration and find its roots via the classical back-propagation approach. Examples of systems with four or multiple roots were used, in order to test the speed of convergence and the accuracy of the training algorithm. Experimental results produced by the network were compared with their theoretical values.     

Reduction of stable differential–algebraic equation systems via projections and system identification

Most large-scale process models derived from first principles are represented by nonlinear differential–algebraic equation (DAE) systems. Since such models are often computationally too expensive for real-time control, techniques for model reduction of these systems need to be investigated. However, models of DAE type have received little attention in the literature on nonlinear model reduction. In order to address this, a new technique for reducing nonlinear DAE systems is presented in this work. This method reduces the order of the differential equations as well as the number and complexity of the algebraic equations. Additionally, the algebraic equations of the resulting system can be replaced by an explicit expression for the algebraic variables such as a feedforward neural network. This last property is important insofar as the reduced model does not require a DAE solver for its solution but system trajectories can instead be computed with regular ODE solvers. This technique is illustrated with a case study where responses of several different reduced-order models of a distillation column with 32 differential equations and 32 algebraic equations are compared.     

A neural-network method for the nonlinear servomechanism problem

The solution of the nonlinear servomechanism problem relies on the solvability of a set of mixed nonlinear partial differential and algebraic equations known as the regulator equations. Due to the nonlinear nature, it is difficult to obtain the exact solution of the regulator equations. This paper proposes to solve the regulator equations based on a class of recurrent neural network, which has the features of a cellular neural network. This research not only represents a novel application of the neural networks to numerical mathematics, but also leads to an effective approach to approximately solving the nonlinear servomechanism problem. The resulting design method is illustrated by application to the well-known ball and beam system.     

Algebra manipulating programs for structural analogies

Two computer programs are described for solving sets of simultaneous equations with coefficients in the form of algebraic expressions.Both programs retain the algebraic nature of these coefficients throughout, although the methods of storage and solution are entirely different. The problems which the programs were originally designed to solve relate to regular structures, for which modular stiffness matrix equations can be written. The coefficients of these matrices are expressions of simple algebraic quantities and finite difference operators. These operators are transformed into differential operators by the programs, the output giving continuum approximations to the structures. Other applications of the programs are discussed.     

On predicting abrupt contraction flows with differential and algebraic viscoelastic models

The numerical simulation of flows through a planar contraction at low Reynolds number is considered for Newtonian and for viscoelastic fluids. The recently proposed algebraic extra-stress model (AESM) derived from the differential constitutive equation for an Oldroyd-B fluid is extended to a Phan-Thien–Tanner fluid. The approach is based on the exact polynomial representation using a three-term tensor basis. It is also shown that the algebraic formulation reproduces exactly the extra-stress tensor components for pure shear and for pure elongation flow. A parameter based on the strain rate and the rotation rate tensors is presented to identify the regions of the flow where the AESM model produces exact results. A second-order numerical scheme accurate in time and space, based on the finite volume method using a staggered grid has been applied to solve the conservation and constitutive equations for the Newtonian and viscoelastic flows. The numerical simulations for the viscoelastic fluids have been done using the classical constitutive equations in a differential form and the algebraic extra-stress model. Excellent agreement between the extra-stress values is obtained with the two different approaches, showing the viability of AESM.     

Balanced realizations via gradient flow techniques

The task of finding a class of balanced minimal realizations is shown to be equivalent to finding limiting solutions of certain gradient flow differential equations. By viewing such algebraic tasks in the context of calculus, they are amenable to analog computational solutions, or parallel processing machines, perhaps even neural networks. The convergence rates of the differential equations is exponential, and consequentially convergence is rapid and numerical stability properties are attractive.     

一种基于代数算法的RBF神经网络优化方法

提出了一种新的RBF神经网络的训练方法,采用动态K-均值方法对RBF 神经网络的隐层中心值和宽度进行了优化,用代数算法训练隐层和输出层之间的权值。在对非线性函数进行逼近的仿真中,验证了该算法的有效性。     

Multiple-page mapping artificial neural network algorithm used for constant tension control

Constant tension control is widely required in industrial applications such as paper machines, coating machines, rewinding and unwinding machines. In a metal film coating machine, which is a multi-input multi-output system, speed and tension have cross coupling and thus desired speed and tension responses are difficult to achieve by applying conventional analogue proportional-plus-integral (PI) control. This paper introduces a multiple-page mapping artificial neural network with back-propagation training algorithm. This method can successfully decouple the speed and tension control loops and both loops can operate quasi-independently. It overcomes the disadvantages of traditional PI control systems. To handle the variation of the rewinding roll diameter, multiple pages of neural networks are applied. Some simulation results show the effectiveness of this control algorithm.     

基于矩阵符号函数求解代数Riccati方程的ANN方法

本文基于矩阵符号函数方法，运用神经网络技术的智能特性，给出了一种求解连续及离散代数Ｒｉｃｃａｔｉ方程的ＡＮＮ方法，最后给出这种方法的应用例子，验证了该方法的有效性及可靠性。     

Derivation of the equations of motion for complex structures by symbolic manipulation

This paper outlines a computer program especially tailored to the task of deriving explicit equations of motion for structures with point-connected substructures. The special purpose program is written in FORTRAN and is designed for performing the specific algebraic operations encountered in the derivation of explicit equations of motion. The derivation is by the Lagrangian approach. Using an orderly kinematical procedure and a discretization and/or truncation scheme, it is possible to write the kinetic and potential energy of each substructure in a compact vector-matrix form. Then, if each element of the matrices and vectors encountered in the kinetic and potential energy is a known algebraic expression, the computer program performs the necessary operations to evaluate the kinetic and potential energy of the system explicitly. Lagrange's equations for small motions about equilibrium can be deduced directly from the explicit form of the system kinetic and potential energy.     

Parallel solutions of simple indexed recurrence equations

We define a new type of recurrence equations called “Simple Indexed Recurrences” (SIR). In this type of equations, ordinary recurrences are generalized to X[g(i)]=op_i(X[f(i)], X[g(i)]), where f, g : {1...n}→{1...m}, op_i(x, y) is a binary associative operator and g is distinct, i.e., ∀i≠j g(i)≠g(j). This enables us to model certain sequential loops as a sequence of SIR equations. A parallel algorithm that solves a set of SIR equations will, in fact, parallelize sequential loops of the above type. Such a parallel SIR algorithm must be efficient enough to compete with the O(n) work complexity of the original loop. We show why efficient parallel algorithms for the related problems of list ranking and tree contraction, which require O(n) work, cannot be applied to solving SIR. We instead use repeated iterations of pointer jumping to compute the final values of X[] in n/p·log p steps and n·log p work, with p processors. A sequence of experiments was performed to test the effect of synchronous and asynchronous executions on the actual performance of the algorithm. These experiments show that pointer jumping requires O(n)) work in most practical cases of SIR loops. An efficient solution is given for the special case where we know how to compute the inverse of op_i, and finally, useful applications of SIR to the well-known Livermore loops benchmark are presented     

Reasoning algebraically about loops

We show how to formalise different kinds of loop constructs within the refinement calculus, and how to use this formalisation to derive general transformation rules for loop constructs. The emphasis is on using algebraic methods for reasoning about equivalence and refinement of loop constructs, rather than operational ways of reasoning about loops in terms of their execution sequences. We apply the algebraic reasoning techniques to derive a collection of transformation rules for action systems and for guarded loops. These include transformation rules that have been found important in practical program derivations: data refinement and atomicity refinement of action systems; and merging, reordering, and data refinement of loops with stuttering transitions. Received: 11 February 1998 / 18 March 1999     

Cooperative binary-real coded genetic algorithms for generating and adapting artificial neural networks

This paper describes a genetic system for designing and training feed-forward artificial neural networks to solve any problem presented as a set of training patterns. This system, called GANN, employs two interconnected genetic algorithms that work parallelly to design and train the better neural network that solves the problem. Designing neural architectures is performed by a genetic algorithm that uses a new indirect binary codification of the neural connections based on an algebraic structure defined in the set of all possible architectures that could solve the problem. A crossover operation, known as Hamming crossover, has been designed to obtain better performance when working with this type of codification. Training neural networks is also accomplished by genetic algorithms but, this time, real number codification is employed. To do so, morphological crossover operation has been developed inspired on the mathematical morphology theory. Experimental results are reported from the application of GANN to the breast cancer diagnosis within a complete computer-aided diagnosis system.     

Driving simulator with double-wishbone suspension using efficient block-triangularized kinematic equations

When modeled with ideal joints, many vehicle suspensions contain closed kinematic chains, or kinematic loops, and are most conveniently modeled using a set of generalized coordinates of cardinality exceeding the degrees-of-freedom of the system. Dependent generalized coordinates add nonlinear algebraic constraint equations to the ordinary differential equations of motion, thereby producing a set of differential-algebraic equations that may be difficult to solve in an efficient yet precise manner. Several methods have been proposed for simulating such systems in real time, including index reduction, model simplification, and constraint stabilization techniques. In this work, the equations of motion for a double-wishbone suspension are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. Symbolic computation is then used to triangularize a subset of the kinematic constraint equations, thereby producing a recursively solvable system for calculating a subset of the dependent generalized coordinates. Thus, the kinematic equations are reduced to a block-triangular form, which results in a more computationally efficient solution strategy than that obtained by iterating over the original constraint equations. The efficiency of this block-triangular kinematic solution is exploited in the real-time simulation of a vehicle with double-wishbone suspensions on both axles, which is implemented in a hardware- and operator-in-the-loop driving simulator.     

Application of boundary-domain element method to the free vibration problem of plate structures

The boundary-domain element method is applied to the free vibration problem of thin-walled plate structures. The static fundamental solutions are used for the derivation of the integral equations for both in-plane and out-of-plane motions. All the integral equations to be implemented are regularized up to an integrable order and then discretized by means of the boundary-domain element method. The entire system of equations for the plate structures composed of thin elastic plates is obtained by assembling the equations for each plate component satisfying the equilibrium and compatibility conditions on the connected edge as well as the boundary conditions. The algebraic eigenvalue equation is derived from this system of equations and is able to be solved by using the standard solver to obtain eigenfrequencies and eigenmodes. Numerical analysis is carried out for a few example problems and the computational aspects are discussed.     

Tracing post-limit-point paths with reduced basis technique

A reduced basis technique and a problem-adaptive computational algorithm are presented for predicting the post-limit-point paths of structures. In the proposed approach the structure is discretized by using displacement finite element models. The nodal displacement vector is expressed as a linear combination of a small number of vectors and a Rayleigh-Ritz technique is used to approximate the finite element equations by a small system of nonlinear algebraic equations.To circumvent the difficulties associated with the singularity of the stiffness matrix at limit points, a constraint equation, defining a generalized arc-length in the solution space, is added to the system of nonlinear algebraic equations and the Rayleigh-Ritz approximation functions (or basis vectors) are chosen to consist of a nonlinear solution of the discretized structure and its various order derivatives with respect to the generalized arc-length. The potential of the proposed approach and its advantages over the reduced basis-load control technique are outlined. The effectiveness of the proposed approach is demonstrated by means of numerical examples of structural problems with snap-through and snap-back phenomena.     

Nonlinear strong model matching

The problem of matching a given input-output behavior for systems described by general nonlinear differential equations is considered. It is shown that, by appropriately modifying the zero-dynamics algorithm, it is possible to obtain a simple, necessary, and sufficient condition for the solvability of the model matching problem, which requires that the initial state be on an appropriate submanifold of the state space. Another condition necessary and sufficient for the solvability of the strong model matching problem is proposed. This last condition is then related to an equality of a list of integers which, under some regularity assumptions, coincide with the algebraic structures at infinity of the process and of a composition of the process and the model. The relation between these conditions and the equality of the algebraic structures at infinity of the process and the model is established     

Multidimensional structured matrices and polynomial systems

We apply and extend some well-known and some recent techniques from algebraic residue theory in order to relate to each other two major subjects of algebraic and numerical computing, that is, computations with structured matrices and solving a system of polynomial equations. In the first part of our paper, we extend the Toeplitz and Hankel structures of matrices and some of their known properties to some new classes of structured (quasi-Hankel and quasi-Toeplitz) matrices, naturally associated to systems of multivariate polynomial equations. In the second part of the paper, we prove some relations between these structured matrices, which extend the classical relations of the univariate case. Supported by NSF Grant CCR 9625344 and PSC CUNY Awards Nos. 667340