A Complex-Valued Hopfield Neural Network

2009 ◽  
pp. 79-103 ◽  
Author(s):  
V. Srinivasa Chakravarthy
Keyword(s):  
Neural Network ◽  
Fixed Point ◽  
Continuous Time ◽  
Optimization Problems ◽  
Hopfield Neural Network ◽  
Memory Capacity ◽  
Chaotic Synchronization ◽  
Hopfield Network ◽  
Oscillatory Dynamics ◽  
Complex Valued

This chapter describes Complex Hopfield Neural Network (CHNN), a complex-variable version of the Hopfield neural network, which can exist in both fixed point and oscillatory modes. Memories can be stored by a complex version of Hebbs rule. In the fixed-point mode, CHNN is similar to a continuous-time Hopfield network. In the oscillatory mode, when multiple patterns are stored, the network wanders chaotically among patterns. Presence of chaos in this mode is verified by appropriate time series analysis. It is shown that adaptive connections can be used to control chaos and increase memory capacity. Electronic realization of the network in oscillatory dynamics, with fixed and adaptive connections shows an interesting tradeoff between energy expenditure and retrieval performance. It is shown how the intrinsic chaos in CHNN can be used as a mechanism for annealing when the network is used for solving quadratic optimization problems. The networks applicability to chaotic synchronization is described.

scholarly journals Combinatorial optimization by weight annealing in memristive hopfield networks

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Z. Fahimi ◽  
M. R. Mahmoodi ◽  
H. Nili ◽  
Valentin Polishchuk ◽  
D. B. Strukov
Keyword(s):  
Neural Network ◽  
Combinatorial Optimization ◽  
Optimization Problems ◽  
Main Idea ◽  
Hopfield Neural Network ◽  
Combinatorial Problems ◽  
Hopfield Network ◽  
Global Minima ◽  
Mixed Signal ◽  
Combinatorial Optimization Problems

AbstractThe increasing utility of specialized circuits and growing applications of optimization call for the development of efficient hardware accelerator for solving optimization problems. Hopfield neural network is a promising approach for solving combinatorial optimization problems due to the recent demonstrations of efficient mixed-signal implementation based on emerging non-volatile memory devices. Such mixed-signal accelerators also enable very efficient implementation of various annealing techniques, which are essential for finding optimal solutions. Here we propose a “weight annealing” approach, whose main idea is to ease convergence to the global minima by keeping the network close to its ground state. This is achieved by initially setting all synaptic weights to zero, thus ensuring a quick transition of the Hopfield network to its trivial global minima state and then gradually introducing weights during the annealing process. The extensive numerical simulations show that our approach leads to a better, on average, solutions for several representative combinatorial problems compared to prior Hopfield neural network solvers with chaotic or stochastic annealing. As a proof of concept, a 13-node graph partitioning problem and a 7-node maximum-weight independent set problem are solved experimentally using mixed-signal circuits based on, correspondingly, a 20 × 20 analog-grade TiO2 memristive crossbar and a 12 × 10 eFlash memory array.

Hopfield Lagrange Network for Economic Load Dispatch

2011 ◽  
pp. 57-94 ◽  
Author(s):  
Vo Ngoc Dieu ◽  
Weerakorn Ongsakul
Keyword(s):  
Neural Network ◽  
Power Systems ◽  
Energy Function ◽  
Optimization Problems ◽  
Hopfield Neural Network ◽  
Lagrangian Function ◽  
Hopfield Network ◽  
Convergence Time ◽  
Economic Load Dispatch ◽  
Convergence Process

In this chapter, a Hopfield Lagrange network (HLN) is proposed for solving economic load dispatch (ELD) problems. HLN is a combination of Lagrangian function and continuous Hopfield neural network where the Lagrangian function is directly used as the energy function for the continuous Hopfield neural network. In the HLN method, its energy function augmented by Hopfield terms from the continuous Hopfield network could damp out oscillation of the conventional Hopfield network during the convergence process. Consequently, the proposed HLN can overcome the disadvantages of the conventional Hopfield network in solving optimization problems for its simpler implementation, better global solution, faster convergence time, and larger scale applications. The proposed method has been tested on different ELD problems including all thermal units, thermal units with fuel constraint, and both thermal and hydro units. The obtained results from the test cases have shown that the proposed method is effective and efficient for solving the ELD problems. Therefore, the HLN method is the new contribution to the development of new methods for solving optimization problems in power systems.

CHAOTIC SYNCHRONIZATION USING A NETWORK OF NEURAL OSCILLATORS

2008 ◽  
Vol 18 (02) ◽  
pp. 157-164 ◽  
Author(s):  
V. SRINIVASA CHAKRAVARTHY ◽  
NEELIMA GUPTE ◽  
S. YOGESH ◽  
ATUL SALHOTRA
Keyword(s):  
Neural Network ◽  
Fixed Point ◽  
Chaotic Synchronization ◽  
Oscillatory Behavior ◽  
Hopfield Network ◽  
High Dimensional ◽  
Secure Communications ◽  
Chaotic Neural Network ◽  
Neural Oscillators ◽  
Low Dimensional

Synchronization of chaotic low-dimensional systems has been a topic of much recent research. Such systems have found applications for secure communications. In this work we show how synchronization can be achieved in a high-dimensional chaotic neural network. The network used in our studies is an extension of the Hopfield Network, known as the Complex Hopfield Network (CHN). The CHN, also an associative memory, has both fixed point and limit cycle or oscillatory behavior. In the oscillatory mode, the network wanders chaotically from one stored pattern to another. We show how a pair of identical high-dimensional CHNs can be synchronized by communicating only a subset of state vector components. The synchronizability of such a system is characterized through simulations.

scholarly journals An Optimal Implementation on FPGA of a Hopfield Neural Network

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
W. Mansour ◽  
R. Ayoubi ◽  
H. Ziade ◽  
R. Velazco ◽  
W. EL Falou
Keyword(s):  
Neural Network ◽  
Artificial Neural Network ◽  
High Performance ◽  
Optimization Problems ◽  
Parallel Architecture ◽  
Hopfield Neural Network ◽  
Noise Removal ◽  
Combinatorial Optimization Problems ◽  
Artificial Neural Network Ann ◽  
Optimal Implementation

The associative Hopfield memory is a form of recurrent Artificial Neural Network (ANN) that can be used in applications such as pattern recognition, noise removal, information retrieval, and combinatorial optimization problems. This paper presents the implementation of the Hopfield Neural Network (HNN) parallel architecture on a SRAM-based FPGA. The main advantage of the proposed implementation is its high performance and cost effectiveness: it requires O(1) multiplications and O(log⁡ N) additions, whereas most others require O(N) multiplications and O(N) additions.

scholarly journals DEVELOPMENT OF A METHODOLOGY FOR FORMALIZING THE INVESTMENT DECISION-MAKING PROCESS BASED ON THE HOPFIELD NEURAL NETWORK

2019 ◽  
pp. 65-72
Author(s):  
Olga RUZAKOVA
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Decision Making ◽  
Investment Decision ◽  
Methodological Approach ◽  
Hopfield Neural Network ◽  
Hopfield Network ◽  
Decision Making Process ◽  
Mathematical Apparatus ◽  
Investment Process

The article presents a methodological approach to assessing the investment attractiveness of an enterprise based on the Hopfield neural network mathematical apparatus. An extended set of evaluation parameters of the investment process has been compiled. An algorithm for formalizing the decision-making process regarding the investment attractiveness of the enterprise based on the mathematical apparatus of neural networks has been developed. The proposed approach allows taking into account the constantly changing sets of quantitative and qualitative parameters, identifying the appropriate level of investment attractiveness of the enterprise with minimal money and time expenses – one of the standards of the Hopfield network, which is most similar to the one that characterizes the activity of the enterprise. Developed complex formalization of the investment process allows you to make investment decisions in the context of incompleteness and heterogeneity of information, based on the methodological tools of neural networks.

scholarly journals Odia Characters and Numerals Recognition using Hopfield Neural Network Based on Zoning Feature

2019 ◽  
Vol 8 (2) ◽  
pp. 4928-4937 ◽  
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Character Recognition ◽  
Optical Character Recognition ◽  
Hopfield Neural Network ◽  
Hopfield Network ◽  
Research Areas ◽  
Different Types ◽  
Classification Tasks ◽  
Text Version

Odia character and digits recognition area are vital issues of these days in computer vision. In this paper a Hope field neural network design to solve the printed Odia character recognition has been discussed. Optical Character Recognition (OCR) is the principle of applying conversion of the pictures from handwritten, printed or typewritten to machine encoded text version. Artificial Neural Networks (ANNs) trained as a classifier and it had been trained, supported the rule of Hopfield Network by exploitation code designed within the MATLAB. Preprocessing of data (image acquisition, binarization, skeletonization, skew detection and correction, image cropping, resizing, implementation and digitalization) all these activities have been carried out using MATLAB. The OCR, designed a number of the thought accuses non-standard speech for different types of languages. Segmentation, feature extraction, classification tasks is the well-known techniques for reviewing of Odia characters and outlined with their weaknesses, relative strengths. It is expected that who are interested to figure within the field of recognition of Odia characters are described in this paper. Recognition of Odia printed characters, numerals, machine characters of research areas finds costly applications within the banks, industries, offices. In this proposed work we devolve an efficient and robust mechanism in which Odia characters are recognized by the Hopfield Neural Networks (HNN).

scholarly journals Beyond the Maximum Storage Capacity Limit in Hopfield Recurrent Neural Networks

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 726 ◽  
Author(s):  
Giorgio Gosti ◽  
Viola Folli ◽  
Marco Leonetti ◽  
Giancarlo Ruocco
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Storage Capacity ◽  
Hamming Distance ◽  
Basin Of Attraction ◽  
Stable State ◽  
Hopfield Neural Network ◽  
Hopfield Network ◽  
Stable States ◽  
Limit States

In a neural network, an autapse is a particular kind of synapse that links a neuron onto itself. Autapses are almost always not allowed neither in artificial nor in biological neural networks. Moreover, redundant or similar stored states tend to interact destructively. This paper shows how autapses together with stable state redundancy can improve the storage capacity of a recurrent neural network. Recent research shows how, in an N-node Hopfield neural network with autapses, the number of stored patterns (P) is not limited to the well known bound 0.14 N , as it is for networks without autapses. More precisely, it describes how, as the number of stored patterns increases well over the 0.14 N threshold, for P much greater than N, the retrieval error asymptotically approaches a value below the unit. Consequently, the reduction of retrieval errors allows a number of stored memories, which largely exceeds what was previously considered possible. Unfortunately, soon after, new results showed that, in the thermodynamic limit, given a network with autapses in this high-storage regime, the basin of attraction of the stored memories shrinks to a single state. This means that, for each stable state associated with a stored memory, even a single bit error in the initial pattern would lead the system to a stationary state associated with a different memory state. This thus limits the potential use of this kind of Hopfield network as an associative memory. This paper presents a strategy to overcome this limitation by improving the error correcting characteristics of the Hopfield neural network. The proposed strategy allows us to form what we call an absorbing-neighborhood of state surrounding each stored memory. An absorbing-neighborhood is a set defined by a Hamming distance surrounding a network state, which is an absorbing because, in the long-time limit, states inside it are absorbed by stable states in the set. We show that this strategy allows the network to store an exponential number of memory patterns, each surrounded with an absorbing-neighborhood with an exponentially growing size.

Storage Capacity of Quaternion-Valued Hopfield Neural Networks with Dual Connections

2021 ◽  
pp. 1-15
Author(s):  
Masaki Kobayashi
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Stochastic Analysis ◽  
Storage Capacity ◽  
Hopfield Neural Network ◽  
Activation Function ◽  
Hopfield Neural Networks ◽  
Noise Tolerance ◽  
Complex Valued ◽  
Projection Rule

Abstract A complex-valued Hopfield neural network (CHNN) is a multistate Hopfield model. A quaternion-valued Hopfield neural network (QHNN) with a twin-multistate activation function was proposed to reduce the number of weight parameters of CHNN. Dual connections (DCs) are introduced to the QHNNs to improve the noise tolerance. The DCs take advantage of the noncommutativity of quaternions and consist of two weights between neurons. A QHNN with DCs provides much better noise tolerance than a CHNN. Although a CHNN and a QHNN with DCs have the samenumber of weight parameters, the storage capacity of projection rule for QHNNs with DCs is half of that for CHNNs and equals that of conventional QHNNs. The small storage capacity of QHNNs with DCs is caused by projection rule, not the architecture. In this work, the ebbian rule is introduced and proved by stochastic analysis that the storage capacity of a QHNN with DCs is 0.8 times as many as that of a CHNN.

Study of Reconfiguration Algorithm in Distribution System Based on Hopfield Network

2012 ◽  
Vol 241-244 ◽  
pp. 1900-1903
Author(s):  
Na Wei ◽  
Zhe Cheng ◽  
Xiao Meng Wu
Keyword(s):  
Neural Network ◽  
Energy Function ◽  
Distribution System ◽  
Distribution Network ◽  
Hopfield Neural Network ◽  
Hopfield Network ◽  
Network Reconfiguration ◽  
Power Sources ◽  
Distribution Network Reconfiguration ◽  
The Neural Network

In accordance with the characteristic of radial running an algorithm for distribution network reconfiguration based on Hopfield neural network is put forward. The in-degree of each node is determined by Hopfield neural network, it is determined whether the lines run according to the in-degree of the nodes, and the state of each loop switch is determined according to whether the lines run, and thus the distribution network reconfiguration scheme is determined finally. The energy function of the neural network and its solution method are presented. In the energy function are considered the radial running of distribution network, the lowest distribution network loss and no loop switch in some lines. The IEEE distribution network structure with three power sources obtained by the algorithm is basically consistent to that obtained by genetic algorithm, but the time spent using the former is shorter than that the latter.

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