By Zhang Yi
Since the exceptional and pioneering study paintings of Hopfield on recurrent neural networks (RNNs) within the early 80s of the final century, neural networks have rekindled robust pursuits in scientists and researchers. fresh years have recorded a impressive enhance in learn and improvement paintings on RNNs, either in theoretical learn as weIl as real functions. the sector of RNNs is now turning out to be a whole and self reliant topic. From conception to software, from software program to undefined, new and interesting effects are rising every day, reflecting the willing curiosity RNNs have instilled in all people, from researchers to practitioners. RNNs include suggestions connections one of the neurons, a phenomenon which has led quite clearly to RNNs being considered as dynamical platforms. RNNs may be defined by way of non-stop time differential platforms, discrete time structures, or practical differential platforms, and extra quite often, when it comes to non linear platforms. therefore, RNNs need to their disposal, a big set of mathematical instruments in relation to dynamical procedure concept which has tumed out to be very invaluable in allowing a rigorous research of RNNs.
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Additional resources for Convergence Analysis of Recurrent Neural Networks
The details are thus omitted. The proof is completed. 4 Global Exponential Stability and Convergence Rate Estimation Global exponential stability (GES) is a good convergence property of dynamic systems. It can provide a dear understanding of the convergence, which is especially interesting in engineering field. 2). Meanwhile, the GES convergence rate estimation will also be given. 2) is ca lied GES, ifthere exist constants E > 0 and M ~ 1 such that forall t ~ O. 2) is GES. Moreover, for all t ~ 0 and (i = 1"" ,n), where _..
3), the matrix - U R191 zUl) + Tu T2I [ ·· · T 12 - u~ R2g2 U2) Tnl .. Tn2 ... + T 22 TIn T 2n ... - R n9:'(Un) + Tnn should be singular in some domain of Rn. 2), c1early, this is impossible. The proof is completed. 3. Complete Stability In this section, complete stability ofHopfield RNNs will be studied by using Hopfield energy functions. Throughout this section, denote T = (Tij)nxn. 1) is completely stable. 1) is bounded. l), it follows that for t ~ 0 and (i = 1, ... , n). Then, for t ~ 0 and (i = 1,···, n).
T ij can be both positive or negative, representing excitation or inhibition connection between neurons. I i is the external constant input to neuron i. Ci > 0 and ~ > 0 are the neuron amplifier input capacitance and resistance, respectively. 1. 1. Invertlng ampllfler • reslstor Electric circuit of Hopfield RNNs. 2 shows the sigmoid function 9(S) 9(0). 1 -. 2. = 1/(1 + e- S) and 9(0). This chapter is organised as follows. In Section 2, the equilibria analysis is given. Complete stability analysis using Hopfield energy functions is discussed in Section 3.
Convergence Analysis of Recurrent Neural Networks by Zhang Yi