Routes to Chaos in Neural Networks with Random Weights

1998 ◽  
Vol 08 (07) ◽  
pp. 1463-1478 ◽  
Author(s):  
D. J. Albers ◽  
J. C. Sprott ◽  
W. D. Dechert
Keyword(s):  
Dynamical System ◽  
Neural Networks ◽  
Function Space ◽  
Dynamic Properties ◽  
Monte Carlo Study ◽  
Random Dynamical System ◽  
Network Function ◽  
The Neural Network ◽  
Quasiperiodic Route To Chaos ◽  
Route To Chaos

Neural networks are dense in the space of dynamical system. We present a Monte Carlo study of the dynamic properties along the route to chaos over random dynamical system function space by randomly sampling the neural network function space. Our results show that as the dimension of the system (the number of dynamical variables) is increased, the probability of chaos approaches unity. We present theoretical and numerical results which show that as the dimension is increased, the quasiperiodic route to chaos is the dominant route. We also qualitatively analyze the dynamics along the route.

Dynamics of Neural Networks as Nonlinear Systems with Several Equilibria

2011 ◽  
pp. 331-357 ◽  
Author(s):  
Daniela Danciu
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Time Delay ◽  
Dynamical Systems ◽  
Qualitative Properties ◽  
The Neural Network ◽  
The Neural Networks ◽  
Global Asymptotics ◽  
Dynamical Properties

Neural networks—both natural and artificial, are characterized by two kinds of dynamics. The first one is concerned with what we would call “learning dynamics”. The second one is the intrinsic dynamics of the neural network viewed as a dynamical system after the weights have been established via learning. The chapter deals with the second kind of dynamics. More precisely, since the emergent computational capabilities of a recurrent neural network can be achieved provided it has suitable dynamical properties when viewed as a system with several equilibria, the chapter deals with those qualitative properties connected to the achievement of such dynamical properties as global asymptotics and gradient-like behavior. In the case of the neural networks with delays, these aspects are reformulated in accordance with the state of the art of the theory of time delay dynamical systems.

scholarly journals ESTIMATION OF THE LYAPUNOV SPECTRUM FROM ONE-DIMENSIONAL OBSERVATIONS USING NEURAL NETWORKS

2014 ◽  
pp. 30-34
Author(s):  
Vladimir Golovko
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Time Series ◽  
Lyapunov Spectrum ◽  
Network Approach ◽  
Neural Network Approach ◽  
One Dimensional ◽  
The Neural Network ◽  
Attractor Dynamics

This paper discusses the neural network approach for computing of Lyapunov spectrum using one dimensional time series from unknown dynamical system. Such an approach is based on the reconstruction of attractor dynamics and applying of multilayer perceptron (MLP) for forecasting the next state of dynamical system from the previous one. It allows for evaluating the Lyapunov spectrum of unknown dynamical system accurately and efficiently only by using one observation. The results of experiments are discussed.

Estimation of Dynamic Properties of Sand Using Artificial Neural Networks

1996 ◽  
Vol 1526 (1) ◽  
pp. 1-5
Author(s):  
S. H. Ni ◽  
C. H. Juang ◽  
P. C. Lu
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Shear Modulus ◽  
Dynamic Properties ◽  
Damping Ratio ◽  
Soil Parameters ◽  
Overburden Pressure ◽  
The Neural Network ◽  
Artificial Neural

Dynamic properties of soils are usually determined by time-consuming laboratory tests. This study presents a method for estimating dynamic soil parameters using artificial neural networks. A simple feedforward neural network with back-propagation training algorithm is used. The neural network is trained with actual laboratory data, which consists of six input variables. They are the standard penetration test value, the void ratio, the unit weight, the water content, the effective overburden pressure, and the mean effective confining pressure. The output layer consists of a single neuron, representing shear modulus or damping ratio. Results of the neural network training and testing show that predictions of shear modulus by the neural network approach is reliable although it is less successful in predicting damping ratio.

scholarly journals Fidelity Assessment of Real-Time Hybrid Substructure Testing: a Review and the Application of Artificial Neural Networks

2021 ◽  
Author(s):  
C. Insam ◽  
D. J. Rixen
Keyword(s):  
Dynamical System ◽  
Neural Networks ◽  
Artificial Neural Networks ◽  
Real Time ◽  
Dynamic Properties ◽  
Reference Solution ◽  
Data Set ◽  
Fidelity Assessment ◽  
Artificial Neural ◽  
Error Indicators

AbstractReal-Time Hybrid Substructure (RTHS) testing is a commonly used method to investigate the dynamical influence of a component on a mechanical system. In RTHS, a part of the dynamical system is tested experimentally, while the remaining structure is simulated numerically in a co-simulation. There are several error sources in the RTHS loop that distort the test outcome. To investigate the reliability of the test, the fidelity of the test must be quantified. In many engineering applications, however, there is no reference solution available to which the test outcome can be validated against. This work reviews currently existing accuracy measures used in RTHS. Furthermore, using Artificial Neural Networks (ANN) to predict the fidelity of the RTHS test outcome when no reference solution is available is proposed. Appropriate input features for the network, such as dynamic properties of the system and existing error indicators, are discussed. ANN training was performed on a data set from a virtual RTHS (vRTHS) simulation of a dynamical system with contact. The training process was successful, meaning that the correlation between the ANN prediction and the true fidelity value was > 99 %. Then, the network was applied to data of experimental RTHS tests of the same dynamical system and achieved a correlation of 98 %, which proves that the relation found by the ANN captured the relation between the chosen input features and the error measure. The application of the trained ANN to data from a linear vRTHS test revealed that further improvement of the network and the choice of input features is necessary. This work suggests that ANNs could be a meaningful tool to predict the fidelity of the RTHS test outcome in the absence of a reference solution, especially if more data from different RTHS tests were aggregated to train them.

Neural Networks and Equilibria, Synchronization, and Time Lags

2011 ◽  
pp. 1219-1225 ◽  
Author(s):  
Daniela Danciu ◽  
Vladimir Rasvan
Keyword(s):  
Neural Network ◽  
Dynamical System ◽  
Neural Networks ◽  
Point Of View ◽  
Forced Oscillations ◽  
Time Lags ◽  
A Posteriori ◽  
Time Dynamics ◽  
The Neural Network ◽  
Spatio Temporal

All neural networks, both natural and artificial, are characterized by two kinds of dynamics. The first one is concerned with what we would call “learning dynamics”, in fact the sequential (discrete time) dynamics of the choice of synaptic weights. The second one is the intrinsic dynamics of the neural network viewed as a dynamical system after the weights have been established via learning. Regarding the second dynamics, the emergent computational capabilities of a recurrent neural network can be achieved provided it has many equilibria. The network task is achieved provided it approaches these equilibria. But the dynamical system has a dynamics induced a posteriori by the learning process that had established the synaptic weights. It is not compulsory that this a posteriori dynamics should have the required properties, hence they have to be checked separately. The standard stability properties (Lyapunov, asymptotic and exponential stability) are defined for a single equilibrium. Their counterpart for several equilibria are: mutability, global asymptotics, gradient behavior. For the definitions of these general concepts the reader is sent to Gelig et. al., (1978), Leonov et. al., (1992). In the last decades, the number of recurrent neural networks’ applications increased, they being designed for classification, identification and complex image, visual and spatio-temporal processing in fields as engineering, chemistry, biology and medicine (see, for instance: Fortuna et. al., 2001; Fink, 2004; Atencia et. al., 2004; Iwahori et. al., 2005; Maurer et. al., 2005; Guirguis & Ghoneimy, 2007). All these applications are mainly based on the existence of several equilibria for such networks, requiring them the “good behavior” properties above discussed. Another aspect of the qualitative analysis is the so-called synchronization problem, when an external stimulus, in most cases periodic or almost periodic has to be tracked (Gelig, 1982; Danciu, 2002). This problem is, from the mathematical point of view, nothing more but existence, uniqueness and global stability of forced oscillations.

Study on Neural Networks in Parallel Computing Environments

2012 ◽  
Vol 198-199 ◽  
pp. 707-710
Author(s):  
Yu Hu
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Time Series ◽  
Parallel Computing ◽  
Time Series Forecasting ◽  
Optimal Solutions ◽  
Network Function ◽  
The Neural Network ◽  
Computing Environments ◽  
Function Approximator

Neurons are highly interconnected with each other and are communicating via sending and receiving electrochemical signals, thus composing sophisticated network of interconnected and communicating neurons. This paper discuss the structure of the neural network function approximator and the time series forecasting with neural network, the results could help us to obtain the optimal solutions to higher complexity of the problem.

SCORING MODELING BASED ON NEURAL NETWORKS FOR DETERMINING A BANK BORROWER'S RATING

2020 ◽  
Vol 2020 (10) ◽  
pp. 54-62
Author(s):  
Oleksii VASYLIEV ◽  
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Network Architecture ◽  
Statistical Data ◽  
Activation Function ◽  
Decision Making Process ◽  
Neural Network Architecture ◽  
Acceptable Accuracy ◽  
The Neural Network ◽  
Sigmoid Activation Function

The problem of applying neural networks to calculate ratings used in banking in the decision-making process on granting or not granting loans to borrowers is considered. The task is to determine the rating function of the borrower based on a set of statistical data on the effectiveness of loans provided by the bank. When constructing a regression model to calculate the rating function, it is necessary to know its general form. If so, the task is to calculate the parameters that are included in the expression for the rating function. In contrast to this approach, in the case of using neural networks, there is no need to specify the general form for the rating function. Instead, certain neural network architecture is chosen and parameters are calculated for it on the basis of statistical data. Importantly, the same neural network architecture can be used to process different sets of statistical data. The disadvantages of using neural networks include the need to calculate a large number of parameters. There is also no universal algorithm that would determine the optimal neural network architecture. As an example of the use of neural networks to determine the borrower's rating, a model system is considered, in which the borrower's rating is determined by a known non-analytical rating function. A neural network with two inner layers, which contain, respectively, three and two neurons and have a sigmoid activation function, is used for modeling. It is shown that the use of the neural network allows restoring the borrower's rating function with quite acceptable accuracy.

scholarly journals Discretization and machine learning approximation of BSDEs with a constraint on the Gains-process

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Idris Kharroubi ◽  
Thomas Lim ◽  
Xavier Warin
Keyword(s):  
Neural Network ◽  
Machine Learning ◽  
Neural Networks ◽  
Differential Equations ◽  
Numerical Experiments ◽  
Optimization Problem ◽  
Learning Approach ◽  
The Neural Network ◽  
Machine Learning Approach ◽  
Mesh Grid

AbstractWe study the approximation of backward stochastic differential equations (BSDEs for short) with a constraint on the gains process. We first discretize the constraint by applying a so-called facelift operator at times of a grid. We show that this discretely constrained BSDE converges to the continuously constrained one as the mesh grid converges to zero. We then focus on the approximation of the discretely constrained BSDE. For that we adopt a machine learning approach. We show that the facelift can be approximated by an optimization problem over a class of neural networks under constraints on the neural network and its derivative. We then derive an algorithm converging to the discretely constrained BSDE as the number of neurons goes to infinity. We end by numerical experiments.

Application of neural networks in predictions of brake wear particulate matter emission

2021 ◽  
pp. 095440702110363
Author(s):  
Saša Vasiljević ◽  
Jasna Glišović ◽  
Nadica Stojanović ◽  
Ivan Grujić
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Particulate Matter ◽  
Traffic Control ◽  
World Health ◽  
Vehicle Speed ◽  
System Pressure ◽  
The Neural Network ◽  
Vehicle Systems ◽  
Brake Wear

According to the World Health Organization, air pollution with PM10 and PM2.5 (PM-particulate matter) is a significant problem that can have serious consequences for human health. Vehicles, as one of the main sources of PM10 and PM2.5 emissions, pollute the air and the environment both by creating particles by burning fuel in the engine, and by wearing of various elements in some vehicle systems. In this paper, the authors conducted the prediction of the formation of PM10 and PM2.5 particles generated by the wear of the braking system using a neural network (Artificial Neural Networks (ANN)). In this case, the neural network model was created based on the generated particles that were measured experimentally, while the validity of the created neural network was checked by means of a comparative analysis of the experimentally measured amount of particles and the prediction results. The experimental results were obtained by testing on an inertial braking dynamometer, where braking was performed in several modes, that is under different braking parameters (simulated vehicle speed, brake system pressure, temperature, braking time, braking torque). During braking, the concentration of PM10 and PM2.5 particles was measured simultaneously. The total of 196 measurements were performed and these data were used for training, validation, and verification of the neural network. When it comes to simulation, a comparison of two types of neural networks was performed with one output and with two outputs. For each type, network training was conducted using three different algorithms of backpropagation methods. For each neural network, a comparison of the obtained experimental and simulation results was performed. More accurate prediction results were obtained by the single-output neural network for both particulate sizes, while the smallest error was found in the case of a trained neural network using the Levenberg-Marquardt backward propagation algorithm. The aim of creating such a prediction model is to prove that by using neural networks it is possible to predict the emission of particles generated by brake wear, which can be further used for modern traffic systems such as traffic control. In addition, this wear algorithm could be applied on other vehicle systems, such as a clutch or tires.

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