Nejib Smaoui

The Box–Jenkins analysis and neural networks: prediction and time series modelling

Abstract Two approaches, namely the Box–Jenkins (BJ) approach and the artificial neural networks (ANN) approach were combined to model time series data of water consumption in Kuwait. The BJ approach was used to predict unrecorded water consumption data from May 1990 to December 1991 due to the Iraqi invasion of Kuwait in August 1990. A supervised feedforward back-propagation neural network was then designed, trained and tested to model and predict water consumption from January 1980 to December 1999. It is interesting to note that the lagged or delayed variables obtained from the BJ approach and used in neural networks provide a better ANN model than the one obtained either blindly in blackbox mode as has been suggested or from traditional known methods.

Timely Communication: Symmetry and the Karhunen--Loève Analysis

The Karhunen--Loeve (K--L) analysis is widely used to generate low-dimensional dynamical systems, which have the same low-dimensional attractors as some large-scale simulations of PDEs. If the PDE is symmetric with respect to a symmetry group G, the dynamical system has to be equivariant under G to capture the full phase space. It is shown that symmetrizing the K--L eigenmodes instead of symmetrizing the data leads to considerable computational savings if the K-L analysis is done in the snapshot method. The feasibility of the approach is demonstrated with an analysis of Kolmogorov flow.

Nonlinear Boundary Control of the Generalized Burgers Equation

In this paper, the adaptive and non-adaptive stabilization of the generalized Burgers equation by nonlinear boundary control are analyzed. For the non-adaptive case, we show that the controlled system is exponentially stable in L2. As for the adaptive case, we present a novel and elegant approach to show the L2 regulation of the solution of the generalized Burgers system. Numerical results supporting and reinforcing the analytical ones of both the controlled and uncontrolled system for the non-adaptive and adaptive cases are presented using the Chebychev collocation method with backward Euler method as a temporal scheme.

IRREGULARLY DECIMATED CHAOTIC MAP(S) FOR BINARY DIGITS GENERATIONS

This paper proposes a new technique for generating random-looking binary digits based on an irregularly decimated chaotic map. We present a class of irregularly decimated chaos-based keystream generators, related to the shrinking generator, for the generation of binary sequences. Each generator consists of two subsystems: a control subsystem and a generating subsystem, where each subsystem is based on a single chaotic map. This chaotic map is presented as a 1-D piecewise chaotic map related to the chaotic logistic map. We conduct an analysis of the dynamical behavior of the proposed map to integrate it as a component in the proposed generators subsystems. The output bits of these keystream generators are produced by applying a threshold function to convert the floating-point iterates of the irregularly decimated map into a binary form. The generated keystream bits are demonstrated to exhibit high level of security, long period length, high linear complexity measure and random-like properties at given certain parameter values. Standard statistical tests on the proposed generators, as well as other keystream generators, are performed and compared.

Linear versus Nonlinear Dimensionality Reduction of High-Dimensional Dynamical Systems

Two techniques for dimensionality reduction of high-dimensional dynamical systems are presented. The first is based on Karhunen--Loeve (K-L) analysis and the second on autoassociative neural networks (ANNs). First, we analyze the dynamics of two partial differential equations, namely, the one-dimensional (1-d) Kuramoto--Sivashinsky (K-S) equation and the two-dimensional (2-d) Navier--Stokes (N-S) equations. For the 1-d K-S equation, one particular dynamical behavior, represented by a heteroclinic connection in phase space, is investigated. As for the 2-d N-S equations, a quasi-periodic behavior is examined. Coherent structures of both dynamics were extracted spanning linear subspaces with minimum information loss. Then we obtain systems of ordinary differential equations based on sophisticated (K-L) Galerkin-type approximation capturing the dynamics of the attractors of both regimes residing on linear manifolds. Using the K-L data coefficients as inputs to autoassociative neural networks, we are able to r...

A new approach combining Karhunen-Loéve decomposition and artificial neural network for estimating tight gas sand permeability

Abstract The Karhunen-Loeve (KL) decomposition, known for its wide applications in scientific problems for data compression, noise filtering, and feature identification, is used to determine an intrinsic coordinate system, or eigenfunctions, that best represents a data set. Projections of the data set onto these eigenfunctions reduces the data set to a set of data coefficients. Processing the data coefficients of the most energetic eigenfunctions through an artificial neural network (ANN) is found to enhance capturing the hidden complex relationships among the data variables. This approach is demonstrated using tight gas sand data to estimate permeability from effective porosity, mean pore size, and mineralogical data. For an arbitrary neural network architecture, combination of KL decomposition and ANN is found to be superior over ANN alone. This combination of two powerful multivariate analysis tools not only correctly estimates the permeability but also eliminates iterative procedures needed for optimizing the neural network topology.

Artificial neural network-based low-dimensional model for spatio-temporally varying cellular flames

Abstract In general obtaining a mathematical model from experimental data of a system with spatio-temporal variation is a challenging task. In this article Karhunen-Loeve (KL) decomposition and artificial neural networks (ANN) are used to extract a simple and accurate dynamic model from video data from experiments of two-dimensional flames of a radial extinction mode regime. The KL decomposition is used to identify coherent structures or eigenfunctions of the system. Projections onto these eigenfunctions reduce the data to a small number of time series. The ANN is then used to process these time series. As a result a low-dimensional, nonlinear dynamic model is obtained.

An artificial neural network noise reduction method for chaotic attractors

A method is presented to reduce noise in chaotic attractors without knowing the underlying maps. The method is based on using Artificial Neural Network (ANN) for moderate levels of additive noise. For high levels of additive noise, a combination of a refinement procedure with ANN is used. In this case, only one refinement is needed for the successful use of ANN. The obtained ANN model is used for long-term predictions of the future behavior of a Henon attractor, using information based only on past values.

Analyzing the Dynamics of Cellular Flames Using Karhunen--Loève Decomposition and Autoassociative Neural Networks

Video data from experiments on the dynamics of two dimensional flames are analyzed. The tools used are Karhunen--Loeve (K-L) decomposition and autoassociative neural networks (ANN). The K-L decomposition, known for its wide applications in scientific problems for data compression, noise filtering, and feature identification, is used to determine an intrinsic coordinate system or orthogonal eigenfunctions that best represent the flame data set. Five eigenfunctions are retained and the rest are disregarded so that reconstructions of the flame data based on the retained eigenfunctions capture most of the dynamics from the original data. The time dependent data coefficients in the expansion of the flame data are used to develop and train an ANN with the task of reducing the dimensionality of the dynamics into a space which reflects the intrinsic dimensionality of the problem.

A Model for the Unstable Manifold of the Bursting Behavior in the 2D Navier--Stokes Flow

Quasi-periodic and bursting behaviors of the two-dimensional (2D) Navier--Stokes flow are analyzed. The tools used are the proper orthogonal decomposition (POD) method and the artificial neural network (ANN) method. The POD is used to extract coherent structures and prominent features from PDE simulations of a quasi-periodic regime and a bursting regime. Eigenfunctions of the two regimes were related by the symmetries of the 2D Navier--Stokes equations. Three eigenfunctions that represent the dynamics of the quasi-periodic regime and two eigenfunctions associated with the unstable manifold of the bursting regime were derived. Calculations of the POD eigenfunctions are performed on the Fourier amplitudes in a comoving frame. Inverse Fourier transform is applied to represent the POD eigenfunctions in both streamfunction and vorticity formulations so that the number of relevant eigenfunctions for streamfunction and vorticity data is the same. Projection onto the two eigenfunctions associated with the unstable manifold reduces the data to two time series. Processing these time series through an ANN results in a low-dimensional model describing the unstable manifold of the bursting regime that can be used to predict the onset of a burst.