Nonlinear Theory and Its Applications, IEICE
Online ISSN : 2185-4106
ISSN-L : 2185-4106
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Editorial Section
  • Masaharu Adachi
    Type: FOREWORD
    2021 Volume 12 Issue 3 Pages 236
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS
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  • Hiroyuki Asahara, Takuji Kousaka
    Type: Invited Review Paper
    2021 Volume 12 Issue 3 Pages 237-256
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    This study reports a stability analysis method based on the monodromy matrix for switched dynamical systems. First, we focus on nonlinear autonomous interrupted systems. In the electrical field, for example, most circuit equations are described in the form of a linear ordinary differential equation. However, with the growth of clean energy power generation devices, the nonlinear term appears in the circuit equation as represented by the photovoltaic module. From this point of view, we discuss a stability analysis method based on the monodromy matrix for nonlinear interrupted systems. Second, we consider impacting systems with a periodic threshold. The previous stability analysis method based on the monodromy matrix is only applicable to impacting systems with fixed threshold. We improved this method considering the periodic threshold. Many impacting systems in the engineering field have a periodic threshold; hence, we introduce this method herein with a mechanical application example. We aim to contribute to the development of the nonlinear theory using above-mentioned two topics.

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  • Eiji Okamoto
    Type: Invited Review Paper
    2021 Volume 12 Issue 3 Pages 257-274
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    In recent years, there have been several advances in wireless communication systems, including the increase in transmission rate, and as a result, they are playing an increasingly significant role in our lives. To meet the widespread application scope, the commercialization of the fifth-generation mobile communications system (5G) has been initialized. This paper gives an overview of the 5G system and discusses the nonlinear signal processing technologies that support its performance improvement. Here, the term “nonlinear signal processing” is defined as an algorithm for transmitting and receiving data in communications that uses a nonlinear mechanism. Because there is a significant shortage of frequency bands in wireless communications, advanced technologies are being integrated to maximize frequency utilization. In addition, unlike in the 4G era, there are additional use cases that require performance guarantees for quality, delay, and the number of multiple connections in the 5G system, and the performance requirements for wireless systems are becoming more stringent. Nonlinear signal processing is key to improving the performance of these systems, and this paper outlines the nonlinear technology used in 5G wireless access. We will introduce multiple-input multiple-output transmission and non-orthogonal multiple access (NOMA), which will help improve the performance of wireless systems. In addition, we introduce a radio wave-encrypted NOMA with a physical layer security that is proposed by the authors. Furthermore, the technical trend of 6G is briefly introduced.

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  • Kosuke Sanada
    Type: Invited Review Paper
    2021 Volume 12 Issue 3 Pages 275-293
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Full Duplex (FD) communication can potentially double the throughput of a point-to-point link in wireless communication. Additionally, FD communication can mitigate collisions induced by hidden nodes. Though exchanging control frames also mitigate hidden node collisions, FD communication has a smaller overhead for one data transmission than the control-frame scheme. Although FD communication is expected to provide system-performance improvement, the enhancement highly depends on its network topology or system parameters. Besides, it is not easy work to comprehend the relationships between system parameters and network performance. Theoretical analysis is useful for evaluating network performance and the optimal design for the system. This paper introduces theoretical analysis for Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA)-based wireless FD networks, mainly author's researches, and discusses their performance. In this paper, star-topology single-hop networks and string topology multi-hop networks are considered as analysis subjects. This paper evaluates FD communication potential for each network through the analytical model by comparing network performances with exchanging control frames and FD communication.

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Special Section on Recent Progress in Nonlinear Theory and Its Applications
  • Hidehiro Nakano, Yasuhiro Tsubo
    Type: FOREWORD
    2021 Volume 12 Issue 3 Pages 294
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS
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  • Mizuka Komatsu, Takaharu Yaguchi, Kenji Kamada, Gen Izumisawa
    Type: Invited Paper
    2021 Volume 12 Issue 3 Pages 295-308
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Realization problems of impulse responses for linear time-invariant (LTI) systems are well-studied. In particular, the realizations with the least order of such systems are said to be minimal. Apart from these problems, parameter estimations of pre-defined LTI models with a specific parametrization of the system matrices are also important. In this paper, we propose a parameter estimation method for such LTI models by transforming a minimal realization obtained through black-box identifications. Our approach is based on the minimal realization theory, exhaustive modelling, and algebraic elimination. Contrary to the existing methods, the proposed method allows polynomial parametrizations of the system.

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  • Keisuke Fukuda, Yoshihiko Horio, Takemori Orima, Koji Kiyoyama, Mitsum ...
    Type: Invited Paper
    2021 Volume 12 Issue 3 Pages 309-322
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Reservoir computing is a computational model inspired by the information processing of the brain. In particular, it shows high performance in time-series processing using recurrent neural network dynamics despite its simple structure. Furthermore, a simple learning algorithm only in the output layer is sufficient for training the entire network. Therefore, its efficient hardware implementation is highly expected. However, it is important for a reservoir network to have a rich variety of dynamics to deal with complex time-series information. To introduce rich dynamics in the reservoir network without degrading the network stability, a chaotic neural network reservoir was proposed. In this paper, we propose a cyclic reservoir neural network circuit suitable for a stacked three-dimensional (3D) integrated circuit (IC). Through 3D IC fabrication technology, in which several semiconductor substrates are vertically stacked and connected by through-silicon vias (TSVs), we can efficiently integrate the chaotic neural network reservoir circuit. We designed and fabricated a prototype IC chip of the proposed circuit with a TSMC 180 nm CMOS semiconductor process. We verified its operation through SPICE and MATLAB simulations and preliminary experiments with the fabricated prototype chip.

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  • Indrapriyadarsini S, Shahrzad Mahboubi, Hiroshi Ninomiya, Takeshi Kami ...
    Type: Invited Paper
    2021 Volume 12 Issue 3 Pages 323-335
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Recent advances in deep reinforcement learning has led to its application in a number of real-world problems. One of the most popularly used deep reinforcement learning algorithms is the deep Q-learning method which uses neural networks to approximate the estimation of the action-value function. Training of deep Q-networks (DQN) is usually restricted to first order gradient based methods. Though second order methods have shown to have faster convergence in several supervised learning problems, their application in deep reinforcement learning is limited. This paper attempts to accelerate the training of deep Q-networks by introducing a second order Nesterov's accelerated quasi-Newton method and verify the feasibility of second order methods in deep reinforcement learning. We evaluate the performance on deep reinforcement learning using double DQNs for VLSI global routing. The results show that the proposed method can obtain better routing solutions compared to the DQNs trained with conventional first order algorithms.

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  • Kentaro Takeda, Hiroyuki Torikai
    Type: Invited Paper
    2021 Volume 12 Issue 3 Pages 336-355
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    In this paper, a central pattern generator (CPG) model based on asynchronous coupling of cellular automaton (CA) phase oscillators for a hexapod robot is presented. The presented CPG model is composed of the CA phase oscillators whose discrete state transitions are triggered by multiple asynchronous clocks. Then, evaluation functions to quantify synchronization states for target gait patterns in the presented CPG model are introduced. Analyzing the synchronizations using the evaluation functions, this paper clarifies that the presented CPG model is suitable to perform smooth gait transitions for the hexapod robot than a CPG model whose discrete state transitions are triggered by a single clock (i.e., synchronous coupling). The presented CPG model is implemented in a field programmable gate array (FPGA); experiments verify that the hexapod robot mounted with the FPGA, in which the presented CPG model is implemented, can perform smooth gait transitions.

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  • Hideyuki Kato
    Type: Paper
    2021 Volume 12 Issue 3 Pages 356-376
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Synaptic responses in activity-dependent manners are thought to play crucial roles in neuronal information processing and are called short-term synaptic plasticity. Typically, two major types of short-term plasticity are observed in the cortical or the hippocampal synapses: facilitation and depression. Although these synaptic responses are reproducible in a canonical phenomenological model, synaptic responses are diverse and target-cell specific, and then the phenomenological model does not cover synaptic responses such as facilitation growth and facilitation-depression. In contrast, while detailed models of short-term plasticity can trace the two synaptic responses, its computational costs are too heavy to simulate large-scale neuronal networks whose simulations are necessary to clarify functional roles and mechanisms of information processing in the cortical or the hippocampal fine-scale circuits. To understand them, the realization of the synaptic responses by a simple STP model is an important issue. In the current study, it is demonstrated a small extension of the canonical phenomenological model makes it possible to realize the synaptic responses of the facilitation growth and the facilitation-depression. Besides, the current study conducts basic analyses of responses in the synapses to both regular and random action potential trains and provides results of the analyses because the nature of the synapses is not clarified yet.

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  • Tomu Katsumata, Muneki Yasuda
    Type: Paper
    2021 Volume 12 Issue 3 Pages 377-390
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    A deep Boltzmann machine (DBM) is a probabilistic deep learning model; DBM learning consists pretraining and fine-tuning stages. This study focuses on the fine-tuning stage, and it develops a new and effective fine-tuning method based on spatial Monte Carlo integration (SMCI), which is an extension of the standard Monte Carlo integration (MCI). It has been proved that SMCI is statistically more accurate than the standard MCI. Fine-tuning methods based on first-order and semi-second-order SMCI methods are formulated. The numerical experiments demonstrate that the proposed fine-tuning methods are superior to the conventional method in terms of both training and generalization errors.

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  • Yoshitaka Itoh, Masaharu Adachi
    Type: Paper
    2021 Volume 12 Issue 3 Pages 391-398
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    We reconstruct bifurcation diagrams of all components in the Rössler equations only from time-series data sets, thereby estimating the attractors when the parameter values are changed. In this study, we show that the bifurcation diagrams of all components can be reconstructed from time-series data of all components. In addition, we estimate the Lyapunov spectrum of the reconstructed bifurcation diagrams. We expect that the reconstruction requires a shorter length of training data when using time-series data sets of all components compared with one component. Accordingly, in numerical experiments, we reconstruct the bifurcation diagrams using training data whose length is shorter than when a bifurcation diagram is reconstructed using training data of one component.

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  • Alkanan Mohannad, Chihiro Shibata, Kohei Miyata, Toshiro Imamura, Shin ...
    Type: Paper
    2021 Volume 12 Issue 3 Pages 399-411
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Apgar score is a test applied 1 minute after birth to check the infant health and can be performed as much as needed. The goal of this paper is to apply a deep learning (DL) method called convolutional neural network (CNN) to predict infants with potentially low Apgar score. Our CNN is a multi-input model that accepts denoised cardiotocography (CTG) images and gestational age. In the first half of the paper, we use basic machine learning (ML) techniques to explore what features and target labels are most effective. In the latter half, we verify to what extent the prediction accuracies can be improved by using our CNN model. Using 5-folds cross validation (CV), the CNN model performance scored an Area Under Curve (AUC) of 0.958 when classifying infants with Apgar score 5 minutes < 7 and AUC of 0.955 if Apgar score 1 or 5 minutes < 6 without using feature extraction algorithms. We conclude that the built model can be utilized as a prognosis tool to predict fetuses with a low Apgar score. Still, we think that a one model isn't enough as obstetricians could benefit more from multiple models that help predict different risks to fetuses.

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  • Hirofumi Miyajima, Noritaka Shigei, Hiromi Miyajima, Norio Shiratori
    Type: Paper
    2021 Volume 12 Issue 3 Pages 412-423
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Edge (or fog) computing is known as a method for improving the conventional cloud system. The basic idea is to consider a system that places edges (servers) between the cloud and the terminals (things). Then, how should machine learning be realized on the edge system? Fast and secure learning methods are desired for machine learning. Secure systems using distributed processing have attracted attention. SMC (Secure Multiparty Computation) is one of the typical models to realize secure learning. Horizontally and vertically partitioned data are known for SMC. The latter is a method consisting of dividing the dataset into element-separated subsets. It is desired to develop a method for directly executing learning using element-separated subsets. Vertically partitioned data (VPD) is considered to be a data structure that realizes such learning. In the previous papers, we proposed learning methods for BP (Back Propagation) and NG (Neural Gas) using VPD. There, we did not consider about the amount of data transferred between servers. In this paper, simplified learning methods that eliminate wasteful data transfer compared to the method in the previous papers are proposed, and its effectiveness is shown. That is, the data transfer from the central server to each server was reduced to 1/L, where L is the number of training data.

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  • Takahiro Goto, Yasuhiro Sugimoto, Daisuke Nakanishi, Keisuke Naniwa, K ...
    Type: Paper
    2021 Volume 12 Issue 3 Pages 424-441
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Cooperative relationships exist among various physical phenomena. Several cooperative phenomena have been analyzed theoretically by considering them as synchronous phenomena. We focus on the cooperative motion obtained in the antagonistic structures by considering the McKibben pneumatic actuator (MPA) as one of the synchronous phenomena. This study aims to realize the various movements of a robot with MPAs based on the autonomous coordination between MPAs. In a previous study, we proposed a novel tension feedback control inspired by animal motion and confirmed that this control law generates some cooperative relationship between the MPAs. In this paper, we mathematically analyze the realized cooperative relationships as a synchronous phenomenon, and clarify the mechanism by which the delay in the change in the MPA length with the change in the MPA pressure generates the transition of phase difference. This result suggests that multiple targeted motions can be generated by a single control law.

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  • Songyuan Zhao, Yoshiki Sugitani, Teruyuki Miyajima
    Type: Paper
    2021 Volume 12 Issue 3 Pages 442-452
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    This paper proposes a novel distributed blind adaptive equalization algorithm for sensor networks, in which each node estimates the data disturbed by inter-symbol interference using time-domain filtering. We apply the minimum variance distortionless response, which is known to be useful for centralized blind equalization, to distributed equalization with multiple distortionless response constraints. Unlike conventional methods, in the proposed approach, each node requires only one filter and sends one signal to the other nodes. Simulation results show the superior performance of the proposed algorithm compared with the conventional algorithm.

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  • Taishi Iriyama, Masatoshi Sato, Hisashi Aomori, Tsuyoshi Otake
    Type: Paper
    2021 Volume 12 Issue 3 Pages 453-463
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    The effectiveness of utilizing inter-channel correlation and self-similarity for demosaicking has been reported in many literatures. On the other hand, many convolutional neural network (CNN)-based demosaicking techniques have also been proposed to achieve state-of-the-art accuracy. In CNN-based demosaicking, one of the most important issue is how to consider the correlations using neural network. In this paper, we propose a novel CNN-based demosaicking method that considers an effective combination of both inter-channel correlation and self-similarity. Specifically, we apply the CNN to predict the color differences R-G and B-G, then the demosaicked image is obtained from the predicted color differences and the input color filter array (CFA) image. At the same time, our network considers the self-similarity in the color difference domain by applying non-local attention for high-level feature map. Experimental results show that our method provides the better accuracy and visual performance compared with conventional demosaicking methods. In addition, the versatility of the proposed framework is demonstrated by experiments with images sampled by various CFA patterns.

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  • Seung-Il Cho, Minami Tsuchiya, Atsushi Tanaka, Muneki Yasuda, Tomochik ...
    Type: Paper
    2021 Volume 12 Issue 3 Pages 464-474
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    In this paper, we discuss the association between the phase of a 90-minute periodic signal (90-MPS) and the subjective quality of sleep in sleep experiments of multi-subjects using sensors for the wake-up support system. When 90-MPS is extracted from the sensor data using Fast Fourier Transform (FFT) and filtering, the change in period appears as a phase shift of 90-MPS. Therefore, it is necessary to focus on the phase of extracted 90-MPS and analyze the association between that and subjective quality. The sympathetic nervous system index that has data with a clear 90-minute period from the sensor data during sleep was selected using FFT. Furthermore, a 90-MPS is extracted by using the filter, and the association between the phase of 90-MPS at the wake-up time and quality of sleep using the Oguri-Shirakawa-Azumi sleep inventory, middle-aged and aged version (OSA-MA) was analyzed. When quality of sleep was the highest, the distribution was centered on phase III of 90-MPS at four categories excluding refreshing (OSA-MA 4) among the five categories of OSA-MA. It has been confirmed that the highest quality of sleep can be obtained when the waking up phase is III.

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  • Masayuki Kato, Yukifumi Oda, Sanggook Lee, Katsuhiro Hirata
    Type: Paper
    2021 Volume 12 Issue 3 Pages 475-488
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    Intrinsic localized mode (ILM) has been observed in various physical systems. In particular, electromagnetic ILM has been observed on an LC ladder circuit array. This paper focused on the electromagnetic ILM and aims to evaluate the feasibility of a new electric motor driven by the electromagnetic ILM numerically. First, we propose two types of nonlinear inductors with different geometries, and their magnetic saturation characteristics are clarified from the magnetic field analysis. Second, numerical simulations using the obtained parameters show that the moving velocity of a moving ILM varies with an initial voltage disturbance. Finally, we propose an inductor with permanent magnets placed on a surface of a rotor core, aiming to generate the electromagnetic ILM more efficiently. Numerical simulations show that the behavior of the moving ILM is affected by the rotational speed of the rotor.

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  • Shiki Kanamaru, Yutaka Shimada, Kantaro Fujiwara, Tohru Ikeguchi
    Type: Paper
    2021 Volume 12 Issue 3 Pages 489-499
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    One of the engineering applications of chaotic nonlinear dynamical systems is a pseudorandom number generator. Pseudorandom numbers generated from chaotic dynamical systems are called chaotic random numbers. A logistic map which exhibits a chaotic response can be used as such a chaotic random number. However, an important issue exists when we use such dynamical systems as a pseudorandom number generator by a digital computer: when the chaotic response of a logistic map is reproduced numerically, the number of iterations that the chaotic response is sustained depends on the precision of the numerical calculation, because the precision of the numerical calculation affects the size of the numerical error. In this paper, we extended the logistic map to an integer logistic map to reduce such numerical errors. We investigated the performance of chaotic random numbers obtained from the integer logistic map with varying numerical precisions and transforming them into binary random numbers. We then used NIST SP 800-22 to evaluate the performance of the random numbers. The results show that a numerical precision of 20 orders of magnitude or more is desirable to the generation of a well-performing chaotic random numbers from the response of the integer logistic map.

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  • Kotaro Kasahara, Yutaka Shimada, Kantaro Fujiwara, Tohru Ikeguchi
    Type: Paper
    2021 Volume 12 Issue 3 Pages 500-511
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    It has been reported that the activation of the Ca2+-permeable transient receptor potential melastatin2 (TRPM2) channel enhances a pancreatic β-cell's insulin secretion. Further, it increases the resting potential, and prolongs the burst duration of the pancreatic β-cell. However, the mechanism by which the TRPM2 channel activation enhances insulin secretion is unknown. Therefore, in this paper, using a mathematical model that represents the dynamics of the membrane potential, we investigate the reproducibility of the results obtained from physiological experiments, and reveal the mechanism of enhancing insulin secretion. We demonstrate that the TRPM2 channel activation prolongs the burst duration when the TRPM2 reversal potential is close to the value observed in the physiological experiments. This could be a plausible theoretical explanation of the experimental value of the TRPM2 reversal potential. In addition, we reveal that the TRPM2 channel activates the voltage-dependent and Ca2+-sensitive channels with large conductance, thereby inducing insulin secretion.

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  • Yuichi Tanji, Ken'ichi Fujimoto
    Type: Paper
    2021 Volume 12 Issue 3 Pages 512-525
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    As an image reconstruction method for computed tomography, continuous-time nonlinear dynamical systems were proposed. However, as the nonlinear dynamical systems are very large-scale, the numerical analysis is costly and the reconstructed images are not easily obtained. Thus, we apply nonlinear model order reduction algorithm to the nonlinear dynamical systems, based on proper orthogonal decomposition. For the continuous-time image reconstruction systems, stability analysis of the equilibria is extremely important, because the reconstructed images are obtained by the equilibria. Hence, stability analysis of the reduced-order systems is presented, in which the equilibria are proved to be asymptotically stable. In the numerical examples, robustness and efficacy of the proposed reduced-order systems will be demonstrated.

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  • Yoshiharu Yamagishi, Tatsuya Kaneko, Megumi Akai-Kasaya, Tetsuya Asai
    Type: Paper
    2021 Volume 12 Issue 3 Pages 526-544
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    A new deep reinforcement learning enhancement is proposed for edge computing. This work focuses on deep Q-networks (DQNs), which are used in deep reinforcement learning. Although DQNs are typically improved through a software-based approach, hardware-specific knowledge such as that on data paths and pipelines is used for improving a DQN. The DQN performance is improved and the number of resources are reduced through an efficient hardware design that considers the learning flow and parameter search. As the scale of the problem increases, the amount of reduction in the use of resources also increases. For example, when the size of the block catch game is 5 × 10, the memory requirement is reduced by approximately 50% compared to a previous DQN. The proposed hardware-oriented approach can be applied to any software technology. This study facilitates the development of novel technologies that can be realized through edge computing.

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  • Keita Koyama, Hiroyasu Ando, Kantaro Fujiwara
    Type: Paper
    2021 Volume 12 Issue 3 Pages 545-553
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    A bursting oscillator network is a typical model of a biological system, and analyzing its dynamics is important for understanding biological phenomena and their functions. In this study, we investigate the interaction of the dynamics of noise perturbation, periodic force, and coupling strength in bursting oscillator networks. We numerically analyze the effect of the interaction on the complete synchronization of the network. It was found that appropriate interactions among these forces make it possible to synchronize the system with a small amount of perturbations. It was also observed that multiple transitions occur between synchronization and de-synchronization with regard to the noise strength.

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  • Shahrzad Mahboubi, Indrapriyadarsini S, Hiroshi Ninomiya, Hideki Asai
    Type: Paper
    2021 Volume 12 Issue 3 Pages 554-574
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    This paper describes a momentum acceleration technique for quasi-Newton (QN) based neural network training and verifies its performance and computational complexity. Recently, Nesterov's accelerated quasi-Newton method (NAQ) has been introduced and shown that the momentum term is effective in reducing the number of iterations and the total training time by incorporating Nesterov's accelerated gradient into QN. However, the gradients had to be calculated two times in one iteration in the NAQ training. This increased the computation time of a training loop compared with the conventional QN. The proposed technique is an improvement to NAQ done by approximating the Nesterov's accelerated gradient as a linear combination of the current and previous gradients. As a result, the gradient is calculated only once per iteration similar to that of QN. The performance of the proposed algorithm is evaluated in comparison to conventional algorithms in neural networks training on two types of problems - function approximation problems with high nonlinearity and classification problems. The results show a significant acceleration in the computation time without losing the quality of the solution compared with conventional training algorithms.

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Regular Section
  • Shin'ichi Oishi, Kouta Sekine
    Type: Invited Paper
    2021 Volume 12 Issue 3 Pages 575-610
    Published: 2021
    Released: July 01, 2021
    JOURNALS FREE ACCESS

    A computer assisted proof is presented for the existence of various periodic solutions for forced Suarez-Schopf's equation, which are delay differential equations modeling El Niño. Tight inclusions of periodic solutions are calculated through numerical verification method by utilizing a structure of Galerkin's equation for forced Suarez-Schopf's equation effectively. The existence of various periodic solutions has been proved via computer assisted proofs including various subharmonics. Especially, coexistence of several subharmonics are proved and numerical simulations are presented suggesting an appearance of chaos.

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