Due to feedback connections, recurrent neural networks (RNNs) are dynamic models. RNNs can provide more\r\ncompact structure for approximating dynamic systems compared to feedforward neural networks (FNNs). For some\r\nRNN models such as the Hopfield model and the Boltzmann machine, the fixed-point property of the dynamic systems\r\ncan be used for optimization and associative memory. The Hopfield model is the most important RNN model, and\r\nthe Boltzmann machine as well as some other stochastic dynamic models are proposed as its generalization. These\r\nmodels are especially useful for dealing with combinatorial optimization problems (COPs), which are notorious NP-\r\ncomplete problems. In this paper, we provide a state-of-the-art introduction to these RNN models, their learning\r\nalgorithms as well as their analog implementations. Associative memory, COPs, simulated annealing (SA), chaotic\r\nneural networks and multilevel Hopfield models are also important topics treated in this paper.
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