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1

Accuracy influence of the hardware implementation of the Hopfield network on the solution quality for the travelling salesman problem

Nagórny Z.

Journal of Applied Computer Science

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2012

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Vol. 20, nr 1

51-67

EN

The objective of this work was to study the accuracy influence of the hardware implementation of the Hopfield network on the solution quality for the travelling salesman problem. In this work the 8-bit accuracy influence of the hardware implementation of weights, activation functions, and external input signals on the quality of achieved solutions for 100 randomly generated instances of the 10-city TSP was studied and comparable results in comparison with the simulation in which the network was simulated using double precision floating point numbers were obtained. The results show that the hardware implementation of the Hopfield network with the 8-bit accuracy allows to obtain satisfactory solutions for the TSP. It should be also noted that the network described in this work utilizes the novel method of auto-tuning of Hopfield network parameters and thanks to this method, in contrast to other works, none of the network parameters is tuned for a given solved TSP on the basis of preliminary simulations. The Hopfield network presented in this work is destined for the hardware implementation. The application of the hardware implementation of the network could significantly decrease the time required to obtain the combinatorial problem solution in comparison with methods using von Neumann architecture computers.

2

Discrete-time impulsive Hopfield neural networks with finite distributed delays

Akça H., Alassar R., Covachev V., Yurtsever H. A.

Computer Assisted Mechanics and Engineering Sciences

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2007

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Vol. 14, No. 2

145-158

EN

The discrete counterpart of a class of Hopfield neural networks with periodic impulses and finite distributed delays is introduced. A sufficient condition for the existence and global exponential stability of a unique periodic solution of the discrete system considered is obtained.

3

Sieci Hopfielda dla problemu komiwojażera

Klimek M.

Zagadnienia Techniczno-Ekonomiczne

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2005

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T. 50, z. 1

65-75

PL

W artykule opisano koncepcję sieci neuronowych typu Hopfielda do rozwiązywania problemów optymalizacyjnych na przykładzie problemu komiwojażera TSP (Traveling Salesman Problem). Zaimplementowano trzy algorytmy rozwiązujące TSP przy użyciu sieci Hopfielda. Wydajność tych algorytmów zbadano za pomocą eksperymentów, których wyniki są przedstawione w tym artykule. Na ich podstawie można stwierdzić, że sieci Hopfielda nie są dobrym narzędziem do rozwiązywania problemu komiwojażera.

EN

In article is described an idea Hopfield neural network for solving combinatorial optimization problems, specifically for the Traveling Salesman Problem (TSP). It was implemented three algorithms for solving TSP with Hopfield nets. The performance of these algorithms was examined in simulation study. Results of experiments are described in this work. According to these results, the conclusion is that the Hopfield nets are not applicable for solving TSP.

4

Rozpoznawanie obrazów obrabiarek z zastosowaniem sieci neuronowej Hopfielda

Gałuszka E., Głośny J., Sokołowski A.

Prace Naukowe Katedry Budowy Maszyn / Politechnika Śląska

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2005

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nr 4

167-176

PL

Celem prezentowanych badań jest przybliżenie podstawowych zagadnień związanych z sieciami neuronowymi Hopfielda oraz próby zastosowania do rozpoznawania obrazów obrabiarek. Rozpoznawanie obrazów przez sieć Hopfielda może być wykorzystane np. przy pomiarze pola temperatur i prowadzeniu analiz odkształceń termicznych obrabiarek.

EN

The paper presents some results of the research aiming at determining applicability of Hopfield artificial neural network for pattern recognition. In our research, the Hopfield neural network was tested for digital image correction. In order to perform the tests, four digital images were generated. The tests revealed that the network correctly recognized (corrected) all noisy patterns.

5

Optymalizacja z wykorzystaniem zmodyfikowanej sieci Hopfielda

Nagórny Z., Kos A.

Kwartalnik Elektroniki i Telekomunikacji

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2005

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Vol. 51, z. 2

255--275

PL

W pracy przedstawiono oryginalną metodę rozwiązywania problemów optymalizacyjnych z wykorzystaniem zmodyfikowanej sieci neuronowej typu Hopfielda. W klasycznym rozwiązaniu wartości wag połączeń między neuronami w sieci Hopfielda są obliczane przed symulacją i nie ulegają zmianie. W niniejszej pracy zbadano wpływ modyfikacji wartości sygnałów wejściowych neuronów lub wag podczas symulacji sieci, na otrzymywane rozwiązania. Zauważono, że modyfikacja sygnałów wejściowych lub wag umożliwia uzyskanie lepszych rozwiązań. W pracy przedstawiono rezultaty otrzymane zmodyfikowaną siecią dla problemu komiwojażera. Dla problemów o wymiarze równym 10, sieć dla 100% prób generowała optymalne rozwiązanie. Dla większych problemów rozwiązania są lepsze od otrzymanych klasyczną siecią. Przedstawiono wnioski płynące z zastosowania opisanej metody.

EN

The paper presents a novel method for solving optimization problems by using a modified Hopfield network. In a conventional Hopfield network weight values of the net are calculated before simulation and are constant. Our method is a novel method, because the values of input signals or weights are modified during simulation. The method makes use of the Hopfield net with continuous function of neurons according to Eq. (2). The model Hopfield net in electronic components is shown in Figure 1. An energy function of the neural net is described by Eq. (3). Comparing Eq. (3) and Eq. (4), which is a general form of an optimization problem cost function, we get weight and external input signal values. The Hopfield net is implemented in software in this work. A simulating program makes use of Eq. (11) to calculate the input of each neuron in the net. During the simulation the input signals are modified in accordance with Eq. (12). The duration of input signals modifying is defined by a random value nnar. Finally, a number of iterations to achieve a stable state of the net are done. A number of trials are performed for each optimization problem and the best results are chosen. Figure 2 shows the algorithm of the method. In this work, simulations were done for six examples of the travelling salesman problem. A cost function of the travelling salesman problem is described by Eq. (17). This function consists of four components: the total length of the salesman's tour, two terms, which ensure that the salesman's tour is valid and the term, which forces neurons to have the output signal equal zero or one. The method with input signal values modifying during simulation gives better results than the conventional one. Results are performed in Table 1 and 2. For ten-city problems the modified Hopfield net finds the optimal solution with 100% success rate. Some conclusions coming from using this method are presented.

6

Minimalizacja długości połączeń w układach elektronicznych z wykorzystaniem sieci Hopfielda

Kos A., Nagórny Z.

Kwartalnik Elektroniki i Telekomunikacji

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2005

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Vol. 51, z. 1

55--72

PL

W pracy przedstawiono oryginalną metodę rozmieszczania elementów w układzie elektronicznym VLSI z wykorzystaniem sieci neuronowej typu Hopfielda. Celem rozmieszczenia elementów jest zapewnienie minimalnej sumarycznej długości połączeń w układzie. Określono postać funkcji energii, która jest minimalizowana przez sieć. W klasycznym rozwiązaniu wartości wag połączeń między neuronami w sieci są obliczane przed symulacją i nie ulegają zmianie. W niniejszej pracy zbadano wpływ zmian wartości wag podczas symulacji pracy sieci, na otrzymywane rozwiązania. Zauważono, że zmiana wartości wag umożliwia uzyskanie lepszych rozwiązań. Przedstawiono przykłady i wnioski płynące z zastosowania tej metody.

EN

The paper presents a novel method for solving two-dimensional assignment problems in electronic circuits. The method makes use of the Hopfield neural network. The aim of component placement is the minimization of the total length of interconnections in electronic circuits. The method makes use of the Hopfield net with continuous function of neurons according to Eq. (4). An energy function of the neural net is described by Eq. (9). This function consists of three components: the total length of interconnections between components in an electronic circuit and two terms, which make that all components are placed in separate cells of a substrate. Comparing Eq. (9) and Eq. (5), which is a general form of neural net energy function, we get Eqs. (10) and (11) for weight and external input signal values. We force the weight matrix to have zeros on the diagonal according to Eq. (14). The model Hopfield net in electronic components is shown in Figure 3. The Hopfield net is implemented in software in this work, a simulating program makes use of Eq. (21) to calculate the output of each neuron in the net. In conventional method weight values of the net are calculated before simulation and are constant. Our method is a novel method, because the weights are changed during simulation according to the algorithm shown in Figure 4. During the simulation the values of weights are changed in a linear way in accordance with Eq. (22). The speed of weight values changing is defined by a random value. Finally, a number of iterations to achieve a stable state of the net are done. A number of triaIs are performed for each assignment problem and the best results are chosen. Simulations were done for four examples of electronic circuits. The method with weight values changing during simulation gives better results than the conventional one. Results are performed in Table 3. Some conclusions coming from using this method are presented.

7

Neural networks for the N-Queens Problem : a review

Mańdziuk J.

Control and Cybernetics

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2002

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Vol. 31, no 2

217-248

EN

Neural networks can be successfully applied to solving certain types of combinatorial optimization problems. In this paper several neural approaches to solving constrained optimization problems are presented and their properties discussed. The main goal of the paper is to present various improvements to the wellknown Hopfield models which are intensively used in combinatorial optimization domain. These improvements include deterministic modifications (binary Hopfield model with negative self-feedback connections and Maximum Neural Network model), stochastic modifications (Gaussian Machine), chaotic Hopfield-based models (Chaotic Neural Network and Transiently Chaotic Neural Network), hybrid approaches (Dual-mode Dynamic Neural Network and Harmony Theory approach) and finally modifications motivated by digital implementation feasibility (Strictly Digital Neural Network). All these models are compared based on a commonly used benchmark prohlem - the N-Queens Problem (NQP). Numerical results indicate that each of modified Hopfield models can be effectively used to solving the NQP. Coonvergence to solutions rate of these methods is very high - usually close to 100%. Experimental time requirements are generally low - polynomial in most casos. Some discussion of non-neural, heuristic approaches to solving the NQP is also presented in the paper.

8

Neural network models for combinatorial optimization : a survey of deterministic, stochastic and chaotic approaches

Smith K., Potvin J., Kwok T.

Control and Cybernetics

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2002

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Vol. 31, no 2

183-216

EN

This paper serves as a tutorial on the use of neural networks for solving combinatorial optimization problems. It reviews the two main classes of neural network models : the gradient-based neural networks such as the Hopfield network, and the deformable template approaches such as the elastic net method and self organizing maps. In each class, the original model is presented, its limitations discussed, and subsequent developments and extensions are reviewed. Particular emphasis is placed on stochastic and chaotic variations on the neural network models designed to improve the optimization performance. Finally, the performance of these neural network models is compared and discussed relative to other heuristic approaches.