scholarly journals Two novel finite time convergent recurrent neural networks for tackling complex-valued systems of linear equation

Filomat ◽  
2020 ◽  
Vol 34 (15) ◽  
pp. 5009-5018
Author(s):  
Lei Ding ◽  
Lin Xiao ◽  
Kaiqing Zhou ◽  
Yonghong Lan ◽  
Yongsheng Zhang
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Linear Equation ◽  
Recurrent Neural Networks ◽  
Activation Function ◽  
Convergence Speed ◽  
The Other ◽  
Activation Functions ◽  
Complex Valued ◽  
Nonlinear Activation Function

Compared to the linear activation function, a suitable nonlinear activation function can accelerate the convergence speed. Based on this finding, we propose two modified Zhang neural network (ZNN) models using different nonlinear activation functions to tackle the complex-valued systems of linear equation (CVSLE) problems in this paper. To fulfill this goal, we first propose a novel neural network called NRNN-SBP model by introducing the sign-bi-power activation function. Then, we propose another novel neural network called NRNN-IRN model by introducing the tunable activation function. Finally, simulative results demonstrate that the convergence speed of NRNN-SBP and the NRNN-IRN is faster than that of the FTRNN model. On the other hand, these results also reveal that different nonlinear activation function will have a different effect on the convergence rate for different CVSLE problems.

Relating the Slope of the Activation Function and the Learning Rate Within a Recurrent Neural Network

1999 ◽  
Vol 11 (5) ◽  
pp. 1069-1077 ◽  
Author(s):  
Danilo P. Mandic ◽  
Jonathon A. Chambers
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Recurrent Neural Network ◽  
Recurrent Neural Networks ◽  
Degrees Of Freedom ◽  
Learning Algorithm ◽  
Activation Function ◽  
Learning Rate ◽  
Optimization Task ◽  
Nonlinear Activation Function

A relationship between the learning rate η in the learning algorithm, and the slope β in the nonlinear activation function, for a class of recurrent neural networks (RNNs) trained by the real-time recurrent learning algorithm is provided. It is shown that an arbitrary RNN can be obtained via the referent RNN, with some deterministic rules imposed on its weights and the learning rate. Such relationships reduce the number of degrees of freedom when solving the nonlinear optimization task of finding the optimal RNN parameters.

scholarly journals Differential Equation Units: Learning Functional Forms of Activation Functions from Data

2020 ◽  
Vol 34 (04) ◽  
pp. 6030-6037
Author(s):  
MohamadAli Torkamani ◽  
Shiv Shankar ◽  
Amirmohammad Rooshenas ◽  
Phillip Wallis
Keyword(s):  
Differential Equation ◽  
Neural Networks ◽  
Deep Neural Networks ◽  
Functional Form ◽  
Activation Function ◽  
The Other ◽  
Superior Performance ◽  
Activation Functions ◽  
Functional Forms ◽  
Nonlinear Activation Function

Most deep neural networks use simple, fixed activation functions, such as sigmoids or rectified linear units, regardless of domain or network structure. We introduce differential equation units (DEUs), an improvement to modern neural networks, which enables each neuron to learn a particular nonlinear activation function from a family of solutions to an ordinary differential equation. Specifically, each neuron may change its functional form during training based on the behavior of the other parts of the network. We show that using neurons with DEU activation functions results in a more compact network capable of achieving comparable, if not superior, performance when compared to much larger networks.

scholarly journals Stacked Heterogeneous Neural Networks for Time Series Forecasting

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Florin Leon ◽  
Mihai Horia Zaharia
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Time Series ◽  
Case Studies ◽  
Multilayer Perceptron ◽  
Neural Model ◽  
Activation Function ◽  
Time Series Forecasting ◽  
The Other ◽  
Activation Functions

A hybrid model for time series forecasting is proposed. It is a stacked neural network, containing one normal multilayer perceptron with bipolar sigmoid activation functions, and the other with an exponential activation function in the output layer. As shown by the case studies, the proposed stacked hybrid neural model performs well on a variety of benchmark time series. The combination of weights of the two stack components that leads to optimal performance is also studied.

Deep neural network based on generalized neo-fuzzy neurons and its learning based on backpropagation

2021 ◽  
Vol 26 (jai2021.26(1)) ◽  
pp. 32-41
Author(s):  
Bodyanskiy Y ◽  
◽  
Antonenko T ◽  
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Learning Process ◽  
Activation Function ◽  
Point Of View ◽  
Basic Unit ◽  
Theoretical Point ◽  
Activation Functions ◽  
Approximation Properties ◽  
Network Training

Modern approaches in deep neural networks have a number of issues related to the learning process and computational costs. This article considers the architecture grounded on an alternative approach to the basic unit of the neural network. This approach achieves optimization in the calculations and gives rise to an alternative way to solve the problems of the vanishing and exploding gradient. The main issue of the article is the usage of the deep stacked neo-fuzzy system, which uses a generalized neo-fuzzy neuron to optimize the learning process. This approach is non-standard from a theoretical point of view, so the paper presents the necessary mathematical calculations and describes all the intricacies of using this architecture from a practical point of view. From a theoretical point, the network learning process is fully disclosed. Derived all necessary calculations for the use of the backpropagation algorithm for network training. A feature of the network is the rapid calculation of the derivative for the activation functions of neurons. This is achieved through the use of fuzzy membership functions. The paper shows that the derivative of such function is a constant, and this is a reason for the statement of increasing in the optimization rate in comparison with neural networks which use neurons with more common activation functions (ReLU, sigmoid). The paper highlights the main points that can be improved in further theoretical developments on this topic. In general, these issues are related to the calculation of the activation function. The proposed methods cope with these points and allow approximation using the network, but the authors already have theoretical justifications for improving the speed and approximation properties of the network. The results of the comparison of the proposed network with standard neural network architectures are shown

scholarly journals Binary and Multiclass Text Classification by Means of Separable Convolutional Neural Network

Inventions ◽  
2021 ◽  
Vol 6 (4) ◽  
pp. 70
Author(s):  
Elena Solovyeva ◽  
Ali Abdullah
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Convolutional Neural Network ◽  
Recurrent Neural Networks ◽  
Low Cost ◽  
Computational Cost ◽  
High Accuracy ◽  
Activation Functions ◽  
Fully Connected ◽  
Fully Connected Networks

In this paper, the structure of a separable convolutional neural network that consists of an embedding layer, separable convolutional layers, convolutional layer and global average pooling is represented for binary and multiclass text classifications. The advantage of the proposed structure is the absence of multiple fully connected layers, which is used to increase the classification accuracy but raises the computational cost. The combination of low-cost separable convolutional layers and a convolutional layer is proposed to gain high accuracy and, simultaneously, to reduce the complexity of neural classifiers. Advantages are demonstrated at binary and multiclass classifications of written texts by means of the proposed networks under the sigmoid and Softmax activation functions in convolutional layer. At binary and multiclass classifications, the accuracy obtained by separable convolutional neural networks is higher in comparison with some investigated types of recurrent neural networks and fully connected networks.

scholarly journals Finite-Time Synchronization for Complex-Valued Recurrent Neural Networks with Time Delays

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-14 ◽  
Author(s):  
Ziye Zhang ◽  
Xiaoping Liu ◽  
Chong Lin ◽  
Bing Chen
Keyword(s):  
Neural Networks ◽  
Lyapunov Function ◽  
Recurrent Neural Networks ◽  
Finite Time ◽  
Time Delays ◽  
Time Synchronization ◽  
Great Difficulty ◽  
Activation Functions ◽  
Complex Valued ◽  
Independent Parameters

This paper focuses on the finite-time synchronization analysis for complex-valued recurrent neural networks with time delays. First, two kinds of common activation functions appearing in the existing references are combined together and more general assumptions are given. To achieve our aim, a nonlinear delayed controller with two independent parameters different from the existing ones is provided, which leads to great difficulty. To overcome it, a newly developed inequality is used. Then, via Lyapunov function approach, some criteria are derived to guarantee the finite-time synchronization of the considered system, and the settling time for synchronization is also estimated. Finally, two numerical simulations are given to support the effectiveness and advantages of the obtained results.

scholarly journals Interval universal approximation for neural networks

2022 ◽  
Vol 6 (POPL) ◽  
pp. 1-29
Author(s):  
Zi Wang ◽  
Aws Albarghouthi ◽  
Gautam Prakriya ◽  
Somesh Jha
Keyword(s):  
Neural Network ◽  
Neural Networks ◽  
Approximation Problem ◽  
Activation Function ◽  
Constructive Proof ◽  
Universal Approximation ◽  
Activation Functions ◽  
Complete Problems ◽  
Robust Network ◽  
Interval Approximation

To verify safety and robustness of neural networks, researchers have successfully applied abstract interpretation , primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain for neural-network verification. First, we introduce the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function f (universal approximation) as we have known for decades, but we can find a neural network, using any well-behaved activation function, whose interval bounds are an arbitrarily close approximation of the set semantics of f (the result of applying f to a set of inputs). We call this notion of approximation interval approximation . Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we call squashable functions . Additionally, the IUA theorem implies that we can always construct provably robust neural networks under ℓ ∞ -norm using almost any practical activation function. Second, we study the computational complexity of constructing neural networks that are amenable to precise interval analysis. This is a crucial question, as our constructive proof of IUA is exponential in the size of the approximation domain. We boil this question down to the problem of approximating the range of a neural network with squashable activation functions. We show that the range approximation problem (RA) is a Δ 2 -intermediate problem, which is strictly harder than NP -complete problems, assuming coNP ⊄ NP . As a result, IUA is an inherently hard problem : No matter what abstract domain or computational tools we consider to achieve interval approximation, there is no efficient construction of such a universal approximator. This implies that it is hard to construct a provably robust network, even if we have a robust network to start with.

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.

Data-Reusing Recurrent Neural Adaptive Filters

2002 ◽  
Vol 14 (11) ◽  
pp. 2693-2707 ◽  
Author(s):  
Danilo P. Mandic
Keyword(s):  
Real Time ◽  
Recurrent Neural Networks ◽  
Adaptive Filters ◽  
Linear Structure ◽  
Activation Function ◽  
Infinite Impulse Response ◽  
Training Algorithm ◽  
Activation Functions ◽  
Convergence Conditions ◽  
Nonlinear Activation Function

A class of data-reusing learning algorithms for real-time recurrent neural networks (RNNs) is analyzed. The analysis is undertaken for a general sigmoid nonlinear activation function of a neuron for the real time recurrent learning training algorithm. Error bounds and convergence conditions for such data-reusing algorithms are provided for both contractive and expansive activation functions. The analysis is undertaken for various configurations that are generalizations of a linear structure infinite impulse response adaptive filter.

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