Recent Advances in Fractional-Order Neural Networks: Theory and Application

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: 31 July 2024 | Viewed by 26880

Special Issue Editors


E-Mail Website
Guest Editor
School of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan 430074, China
Interests: chaos theory; bifurcation; hidden attractors; non-smooth systems

E-Mail Website
Guest Editor
School of Mathematics and Physics, China University of Geosciences (Wuhan), Wuhan 430074, China
Interests: computational neuroscience; system biology

Special Issue Information

Dear Colleagues,

The field of fractional-order neural networks refers to research that incorporates the concepts of fractional calculus into the related theory and application of neural networks. It is introduced to accurately describe the physical process and system state with heredity and memorability. With the in-depth study of fractional-order neural network models and dynamics analysis (e.g., stability, synchronization, bifurcation), more control methods and control techniques are applied to these systems, which will enrich the theoretical system of fractional-order neural networks. Additionally, fractional-order neural networks have infinite memory properties, which can further improve the design, characterization, and control capabilities of network models for many practical problems. These concepts have great application prospects and research value in biological nervous systems, circuit design and simulation, artificial intelligence, and other fields.

The focus of this Special Issue is to continue to advance research on topics relating to the theory, control, design, and application of fractional-order neural networks. Topics that are invited for submission include (but are not limited to):

  • Fractional-order neural network model;
  • Dynamic analysis and control of fractional-order neural networks;
  • Circuit design and simulation of fractional-order neural networks;
  • Applications of fractional-order neural networks for biology and biomedicine;
  • Applications of fractional-order circuit models for artificial intelligence.

Prof. Dr. Zhouchao Wei
Dr. Lulu Lu
Guest Editors

Manuscript Submission Information

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Keywords

  • fractional calculus
  • dynamics analysis
  • biological nervous system
  • circuit design and simulation
  • artificial intelligence

Published Papers (10 papers)

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Research

16 pages, 6211 KiB  
Article
Remaining Useful Life Prediction of a Planetary Gearbox Based on Meta Representation Learning and Adaptive Fractional Generalized Pareto Motion
by Hongqing Zheng, Wujin Deng, Wanqing Song, Wei Cheng, Piercarlo Cattani and Francesco Villecco
Fractal Fract. 2024, 8(1), 14; https://doi.org/10.3390/fractalfract8010014 - 22 Dec 2023
Viewed by 969
Abstract
The remaining useful life (RUL) prediction of wind turbine planetary gearboxes is crucial for the reliable operation of new energy power systems. However, the interpretability of the current RUL prediction models is not satisfactory. To this end, a multi-stage RUL prediction model is [...] Read more.
The remaining useful life (RUL) prediction of wind turbine planetary gearboxes is crucial for the reliable operation of new energy power systems. However, the interpretability of the current RUL prediction models is not satisfactory. To this end, a multi-stage RUL prediction model is proposed in this work, with an interpretable metric-based feature selection algorithm. In the proposed model, the advantages of neural networks and long-range-dependent stochastic processes are combined. In the offline training stage, a general representation of the degradation trend is learned with the meta-long short-term memory neural network (meta-LSTM) model. The inevitable measurement error in the sensor reading is modelled by white Gaussian noise. During the online RUL prediction stage, fractional generalized Pareto motion (fGPm) with an adaptive diffusion is employed to model the stochasticity of the planetary gearbox degradation. In the case study, real planetary gearbox degradation data are used for the model validation. Full article
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15 pages, 1153 KiB  
Article
Complex Dynamics Analysis and Chaos Control of a Fractional-Order Three-Population Food Chain Model
by Zhuang Cui, Yan Zhou and Ruimei Li
Fractal Fract. 2023, 7(7), 548; https://doi.org/10.3390/fractalfract7070548 - 16 Jul 2023
Cited by 3 | Viewed by 981
Abstract
The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator [...] Read more.
The present study investigates the stability analysis and chaos control of a fractional-order three-population food chain model. Previous research has indicated that the predation relationship within a long-established predator–prey system can be influenced by factors such as the prey’s fear of the predator and its carry-over effects. This study examines the state evolution of fractional-order systems and compares their dynamic behavior with integer-order systems. By utilizing the Routh–Hurwitz condition and the stability theory of fractional differential equations, this paper establishes the local stability conditions of the model through the application of the Jacobi matrix and eigenvalue method. Furthermore, the conditions for the Hopf bifurcation generation are determined. Subsequently, chaos control techniques based on the Lyapunov stability theory are employed to stabilize the unstable trajectory at the equilibrium point. The theoretical findings are validated through numerical simulations. These results enhance our understanding of the stability properties and chaos control mechanisms in fractional-order three-population food chain models. Full article
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18 pages, 1263 KiB  
Article
Forecasting Cryptocurrency Prices Using LSTM, GRU, and Bi-Directional LSTM: A Deep Learning Approach
by Phumudzo Lloyd Seabe, Claude Rodrigue Bambe Moutsinga and Edson Pindza
Fractal Fract. 2023, 7(2), 203; https://doi.org/10.3390/fractalfract7020203 - 18 Feb 2023
Cited by 23 | Viewed by 11943
Abstract
Highly accurate cryptocurrency price predictions are of paramount interest to investors and researchers. However, owing to the nonlinearity of the cryptocurrency market, it is difficult to assess the distinct nature of time-series data, resulting in challenges in generating appropriate price predictions. Numerous studies [...] Read more.
Highly accurate cryptocurrency price predictions are of paramount interest to investors and researchers. However, owing to the nonlinearity of the cryptocurrency market, it is difficult to assess the distinct nature of time-series data, resulting in challenges in generating appropriate price predictions. Numerous studies have been conducted on cryptocurrency price prediction using different Deep Learning (DL) based algorithms. This study proposes three types of Recurrent Neural Networks (RNNs): namely, Long Short-Term Memory (LSTM), Gated Recurrent Unit (GRU), and Bi-Directional LSTM (Bi-LSTM) for exchange rate predictions of three major cryptocurrencies in the world, as measured by their market capitalization—Bitcoin (BTC), Ethereum (ETH), and Litecoin (LTC). The experimental results on the three major cryptocurrencies using both Root Mean Squared Error (RMSE) and the Mean Absolute Percentage Error (MAPE) show that the Bi-LSTM performed better in prediction than LSTM and GRU. Therefore, it can be considered the best algorithm. Bi-LSTM presented the most accurate prediction compared to GRU and LSTM, with MAPE values of 0.036, 0.041, and 0.124 for BTC, LTC, and ETH, respectively. The paper suggests that the prediction models presented in it are accurate in predicting cryptocurrency prices and can be beneficial for investors and traders. Additionally, future research should focus on exploring other factors that may influence cryptocurrency prices, such as social media and trading volumes. Full article
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11 pages, 310 KiB  
Article
On Variable-Order Fractional Discrete Neural Networks: Existence, Uniqueness and Stability
by Othman Abdullah Almatroud, Amel Hioual, Adel Ouannas, Mohammed Mossa Sawalha, Saleh Alshammari and Mohammad Alshammari
Fractal Fract. 2023, 7(2), 118; https://doi.org/10.3390/fractalfract7020118 - 26 Jan 2023
Cited by 6 | Viewed by 1209
Abstract
Given the recent advances regarding the studies of discrete fractional calculus, and the fact that the dynamics of discrete-time neural networks in fractional variable-order cases have not been sufficiently documented, herein, we consider a novel class of discrete-time fractional-order neural networks using discrete [...] Read more.
Given the recent advances regarding the studies of discrete fractional calculus, and the fact that the dynamics of discrete-time neural networks in fractional variable-order cases have not been sufficiently documented, herein, we consider a novel class of discrete-time fractional-order neural networks using discrete nabla operator of variable-order. An adequate criterion for the existence of the solution in addition to its uniqueness for such systems is provided with the use of Banach fixed point technique. Moreover, the uniform stability is investigated. We provide at the end two numerical simulations illustrating the relevance of the aforementioned results. Full article
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18 pages, 9203 KiB  
Article
Multistability and Phase Synchronization of Rulkov Neurons Coupled with a Locally Active Discrete Memristor
by Minglin Ma, Yaping Lu, Zhijun Li, Yichuang Sun and Chunhua Wang
Fractal Fract. 2023, 7(1), 82; https://doi.org/10.3390/fractalfract7010082 - 11 Jan 2023
Cited by 41 | Viewed by 2014
Abstract
In order to enrich the dynamic behaviors of discrete neuron models and more effectively mimic biological neural networks, this paper proposes a bistable locally active discrete memristor (LADM) model to mimic synapses. We explored the dynamic behaviors of neural networks by introducing the [...] Read more.
In order to enrich the dynamic behaviors of discrete neuron models and more effectively mimic biological neural networks, this paper proposes a bistable locally active discrete memristor (LADM) model to mimic synapses. We explored the dynamic behaviors of neural networks by introducing the LADM into two identical Rulkov neurons. Based on numerical simulation, the neural network manifested multistability and new firing behaviors under different system parameters and initial values. In addition, the phase synchronization between the neurons was explored. Additionally, it is worth mentioning that the Rulkov neurons showed synchronization transition behavior; that is, anti-phase synchronization changed to in-phase synchronization with the change in the coupling strength. In particular, the anti-phase synchronization of different firing patterns in the neural network was investigated. This can characterize the different firing behaviors of coupled homogeneous neurons in the different functional areas of the brain, which is helpful to understand the formation of functional areas. This paper has a potential research value and lays the foundation for biological neuron experiments and neuron-based engineering applications. Full article
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24 pages, 1958 KiB  
Article
Adaptive Quantized Synchronization of Fractional-Order Output-Coupling Multiplex Networks
by Yunzhan Bai, Juan Yu and Cheng Hu
Fractal Fract. 2023, 7(1), 22; https://doi.org/10.3390/fractalfract7010022 - 26 Dec 2022
Viewed by 1023
Abstract
This paper is devoted to investigating the synchronization of fractional-order output-coupling multiplex networks (FOOCMNs). Firstly, a type of fractional-order multiplex network is introduced, where the intra-layer coupling and the inter-layer coupling are described separately, and nodes communicate with each other by their outputs, [...] Read more.
This paper is devoted to investigating the synchronization of fractional-order output-coupling multiplex networks (FOOCMNs). Firstly, a type of fractional-order multiplex network is introduced, where the intra-layer coupling and the inter-layer coupling are described separately, and nodes communicate with each other by their outputs, which is more realistic when the node states are unmeasured. By using the Lyapunov method and the fractional differential inequality, sufficient conditions are provided for achieving asymptotic synchronization based on the designed adaptive control, where the synchronized state of each layer is different. Furthermore, a quantized adaptive controller is developed to realize the synchronization of FOOCMNs, which effectively reduces signal transmission frequency and improves the effective utilization rate of network resources. Two numerical examples are given at last to support the theoretical analysis. Full article
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23 pages, 26086 KiB  
Article
Characteristic Analysis and Circuit Implementation of a Novel Fractional-Order Memristor-Based Clamping Voltage Drift
by Huaigu Tian, Jindong Liu, Zhen Wang, Fei Xie and Zelin Cao
Fractal Fract. 2023, 7(1), 2; https://doi.org/10.3390/fractalfract7010002 - 20 Dec 2022
Cited by 27 | Viewed by 1588
Abstract
The ideal magnetic flux-controlled memristor was introduced into a four-dimensional chaotic system and combined with fractional calculus theory, and a novel four-dimensional commensurate fractional-order system was proposed and solved using the Adomian decomposition method. The system orders, parameters, and initial values were studied [...] Read more.
The ideal magnetic flux-controlled memristor was introduced into a four-dimensional chaotic system and combined with fractional calculus theory, and a novel four-dimensional commensurate fractional-order system was proposed and solved using the Adomian decomposition method. The system orders, parameters, and initial values were studied as independent variables in the bifurcation diagram and Lyapunov exponents spectrum, and it was discovered that changing these variables can cause the system to exhibit more complex and rich dynamical behaviors. The system had an offset boosting, which was discovered by adding a constant term after the decoupled linear term. Finally, the results of the numerical simulation were verified through the use of analog circuits and FPGA designs, and a control scheme for the system circuit was also suggested. Full article
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17 pages, 2900 KiB  
Article
Multiscale Approach for Bounded Deformation Image Registration
by Yunfeng Du and Huan Han
Fractal Fract. 2022, 6(11), 681; https://doi.org/10.3390/fractalfract6110681 - 18 Nov 2022
Viewed by 1394
Abstract
Deformable image registration is a very important topic in the field of image processing. It is widely used in image fusion and shape analysis. Generally speaking, image registration models can be divided into two categories: smooth registration and non-smooth registration. During the last [...] Read more.
Deformable image registration is a very important topic in the field of image processing. It is widely used in image fusion and shape analysis. Generally speaking, image registration models can be divided into two categories: smooth registration and non-smooth registration. During the last decades, many smooth registration models (i.e., diffeomorphic registration) were proposed. However, image with strong noise may lead to discontinuous deformation, which cannot be modelled by smooth registration. To simulate this kind of deformation, some non-smooth registration models were also proposed. However, numerical algorithms for these models are easily trapped into a local minimum because of the nonconvexity of the object functional. To overcome the local minimum of the object functional, we propose a multiscale approach for a non-smooth registration model: the bounded deformation (BD) model. The convergence of the approach is shown, and numerical tests are also performed to show the good performance of the proposed multiscale approach. Full article
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16 pages, 25100 KiB  
Article
Dynamical Analysis of a Novel Fractional-Order Chaotic System Based on Memcapacitor and Meminductor
by Xingce Liu, Jun Mou, Jue Wang, Santo Banerjee and Peng Li
Fractal Fract. 2022, 6(11), 671; https://doi.org/10.3390/fractalfract6110671 - 13 Nov 2022
Cited by 23 | Viewed by 1305
Abstract
In this paper, a chaotic circuit based on a memcapacitor and meminductor is constructed, and its dynamic equation is obtained. Then, the mathematical model is obtained by normalization, and the system is decomposed and summed by an Adomian decomposition method (ADM) algorithm. So [...] Read more.
In this paper, a chaotic circuit based on a memcapacitor and meminductor is constructed, and its dynamic equation is obtained. Then, the mathematical model is obtained by normalization, and the system is decomposed and summed by an Adomian decomposition method (ADM) algorithm. So as to study the dynamic behavior in detail, not only the equilibrium stability of the system is analyzed, but also the dynamic characteristics are analyzed by means of a Bifurcation diagram and Lyapunov exponents (Les). By analyzing the dynamic behavior of the system, some special phenomena, such as the coexistence of attractor and state transition, are found in the system. In the end, the circuit implementation of the system is implemented on a Digital Signal Processing (DSP) platform. According to the numerical simulation results of the system, it is found that the system has abundant dynamical characteristics. Full article
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19 pages, 8154 KiB  
Article
Study on the Complex Dynamical Behavior of the Fractional-Order Hopfield Neural Network System and Its Implementation
by Tao Ma, Jun Mou, Bo Li, Santo Banerjee and Huizhen Yan
Fractal Fract. 2022, 6(11), 637; https://doi.org/10.3390/fractalfract6110637 - 1 Nov 2022
Cited by 28 | Viewed by 2046
Abstract
The complex dynamics analysis of fractional-order neural networks is a cutting-edge topic in the field of neural network research. In this paper, a fractional-order Hopfield neural network (FOHNN) system is proposed, which contains four neurons. Using the Adomian decomposition method, the FOHNN system [...] Read more.
The complex dynamics analysis of fractional-order neural networks is a cutting-edge topic in the field of neural network research. In this paper, a fractional-order Hopfield neural network (FOHNN) system is proposed, which contains four neurons. Using the Adomian decomposition method, the FOHNN system is solved. The dissipative characteristics of the system are discussed, as well as the equilibrium point is resolved. The characteristics of the dynamics through the phase diagram, the bifurcation diagram, the Lyapunov exponential spectrum, and the Lyapunov dimension of the system are investigated. The circuit of the system was also designed, based on the Multisim simulation platform, and the simulation of the circuit was realized. The simulation results show that the proposed FOHNN system exhibits many interesting phenomena, which provides more basis for the study of complex brain working patterns, and more references for the design, as well as the hardware implementation of the realized fractional-order neural network circuit. Full article
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